|
|
- \(b\)-Adic Fractions [1110]
- \(C^{n}\)-Diffeomorphism [6206]
- \(C^n\) Differentiable Function [6254]
- \(n\) times Continuously Differentiable Functions [6205]
- Absolute Value of Complex Numbers [1247]
- Absolute Value of Integers [1080]
- Absolute Value of Rational Numbers [1081]
- Absolute Value of Real Numbers (Modulus) [583]
- Absolutely Convergent Complex Series [1725]
- Absolutely Convergent Series [198]
- Accumulation Point (Real Numbers) [174]
- Accumulation Points (Metric Spaces) [306]
- Addition of Complex Numbers [1657]
- Addition of Ideals [1068]
- Adjacency List Representation [1215]
- Adjacency Matrix [1213]
- Affine Basis, Affine Coordinate System [434]
- Affine Space [6277]
- Affine Subspace [414]
- Affinely Dependent and Affinely Independent Points [6280]
- Algebra Homomorphism [6235]
- Algebra over a Ring [6213]
- Algebraic Element [6255]
- Algorithm [1777]
- Algorithm Solving a Problem [1778]
- Alphabet, Letter, Concatenation, String, Empty String, Formal Language [708]
- Alternating Multilinear Map [6338]
- Altitude of a Triangle [923]
- Atomic Formulae in Predicate Logic [6226]
- Automorphism [432]
- Average Velocity [6309]
- Banach Space [6264]
- Bernoulli Experiment [1812]
- Biconnected Graphs, \(k\)-Connected Graphs [1227]
- Big O Notation [1087]
- Bijective Function [771]
- Bilinear Form [6229]
- Binary Operation [6188]
- Binomial Coefficients [993]
- Bipartite Graph [1236]
- Boolean Algebra of Propositional Logic [187]
- Boolean Terms, Variables and Connectors [1307]
- Boundary Point, Boundary [1202]
- Bounded Affine Set [6293]
- Bounded and Unbounded Functions [302]
- Bounded Complex Sequences [1714]
- Bounded Real Sequences, Upper and Lower Bounds for a Real Sequence [1136]
- Bounded Sequence [6591]
- Bounded Subset of a Metric Space [6574]
- Bounded Subsets of Real Numbers [584]
- C.N. 4: Congruence [2781]
- Cancellation Property [837]
- Cancellative Semigroups [1102]
- Canonical Projection of an Equivalence Relation [6330]
- Canonical Representation of Positive Integers [803]
- Canonical Representation of Positive Rational Numbers [804]
- Carrier Set [6658]
- Cartesian Product [748]
- Cauchy Sequence [1072]
- Certain and Impossible Event [183]
- Characteristic of a Ring [881]
- Classification of Differential Equations
- First-Order Ordinary Differential Equation (ODE) [247]
- Clopen Set [853]
- Closed Curve, Open Curve [1211]
- Closed Walks, Closed Trails, and Cycles [1165]
- Collinear Points, Segments, Rays [649]
- Combinations [209]
- Commutative (Abelian) Group [553]
- Commutative (Unit) Ring [880]
- Commutative Monoid [706]
- Commutative Semigroup [1103]
- Comparing Cardinal Numbers [984]
- Complement Graph [6346]
- Complete Bipartite Graph [6372]
- Complete Graph [6343]
- Complete Metric Space [377]
- Complete Ordered Field [6193]
- Complex Cauchy Sequence [1250]
- Complex Conjugate [1245]
- Complex Infinite Series [1724]
- Complex Polynomials [252]
- Complex Sequence [1249]
- Composition of Relations [1309]
- Computational Problem - a Formal Definition [1776]
- Concentric Circles [2784]
- Conclusion (Conditional), Antecedent, Consequent [1304]
- Conditional Probability [428]
- Conjunction [712]
- Connected and Disconnected Graphs, Bridges and Cutvertices [1166]
- Connected Vertices [1223]
- Constant Function [6342]
- Constant Function Real Case [1371]
- Continuous Complex Functions [251]
- Continuous Functions at Single Complex Numbers [1742]
- Continuous Functions at Single Real Numbers [219]
- Continuous Functions in Metric Spaces [1205]
- Continuous Random Variables [225]
- Continuous Real Functions [1260]
- Contradiction, Invalid Boolean Function [1321]
- Contrapositive [1306]
- Convergent Complex Sequence [1700]
- Convergent Complex Series [147]
- Convergent Rational Sequence [1572]
- Convergent Real Sequence [141]
- Convergent Real Series [175]
- Convergent Sequences and Limits [148]
- Convex Affine Set [6287]
- Convex Hull [6292]
- Cosets [827]
- Cosine of a Real Variable [1745]
- Cotangent Bundle [6332]
- Countable Set [772]
- Curves In the Multidimensional Space \(\mathbb R^n\) [1208]
- Cycle Graph [6344]
- Cyclic Group [807]
- Cyclic, Acyclic Graph [6376]
- Decagon [6571]
- Decimal Representation of Real Numbers [6653]
- Def. 1.01: Point [631]
- Def. 1.02: Line, Curve [636]
- Def. 1.03: Intersections of Lines [644]
- Def. 1.04: Straight Line, Segment and Ray [645]
- Def. 1.05: Surface [646]
- Def. 1.06: Intersections of Surfaces [648]
- Def. 1.07: Plane [647]
- Def. 1.08: Plane Angle [6425]
- Def. 1.09: Angle, Rectilinear, Vertex, Legs [650]
- Def. 1.1: Area of Rectangle, Rectangle Contained by Adjacent Sides [1014]
- Def. 1.10: Right Angle, Perpendicular Straight Lines [653]
- Def. 1.11: Obtuse Angle [689]
- Def. 1.12: Acute Angle [6426]
- Def. 1.13: Boundary [907]
- Def. 1.14: Plane Figure [6427]
- Def. 1.15: Circle, Circumference, Radius [690]
- Def. 1.16: Center of the Circle [6428]
- Def. 1.17: Diameter of the Circle [6429]
- Def. 1.18: Semicircle [6430]
- Def. 1.19: Rectilinear Figure, Sides, n-Sided Figure [687]
- Def. 1.2: Gnomon [2776]
- Def. 1.20: Equilateral Triangle, Isosceles Triangle, Scalene Triangle [688]
- Def. 1.21: Right Triangle, Obtuse Triangle, Acute Triangle [6431]
- Def. 1.22: Square, Rectangle, Rhombus, Rhomboid, Trapezium [909]
- Def. 1.23: Parallel Straight Lines [788]
- Def. 10.01: Magnitudes Commensurable and Incommensurable in Length [1095]
- Def. 10.02: Magnitudes Commensurable and Incommensurable in Square [2082]
- Def. 10.03: Rational and Irrational Magnitudes [2083]
- Def. 10.04: Rational and Irrational Magnitudes in Square [2084]
- Def. 10.05: First Binomial [2085]
- Def. 10.06: Second Binomial [2086]
- Def. 10.07: Third Binomial [2087]
- Def. 10.08: Fourth Binomial [2088]
- Def. 10.09: Fifth Binomial [2089]
- Def. 10.10: Sixth Binomial [2090]
- Def. 10.11: First Apotome [2091]
- Def. 10.12: Second Apotome [2092]
- Def. 10.13: Third Apotome [2093]
- Def. 10.14: Fourth Apotome [2094]
- Def. 10.15: Fifth Apotome [6445]
- Def. 10.16: Sixth Apotome [6446]
- Def. 11.01: Solid Figures, Three-Dimensional Polyhedra [2210]
- Def. 11.02: Surface of a Solid Figure [2211]
- Def. 11.03: Straight Line at Right Angles To a Plane [2212]
- Def. 11.04: Plane at Right Angles to a Plane [2213]
- Def. 11.05: Inclination of a Straight Line to a Plane [2214]
- Def. 11.06: Inclination of a Plane to a Plane [2215]
- Def. 11.07: Similarly Inclined Planes [2216]
- Def. 11.08: Parallel Planes [2217]
- Def. 11.09: Similar Solid Figures [2218]
- Def. 11.10: Equal Solid Figures [2219]
- Def. 11.11: Solid Angle [2220]
- Def. 11.12: Pyramid, Tetrahedron [2221]
- Def. 11.13: Prism, Parallelepiped [2222]
- Def. 11.14: Sphere [2223]
- Def. 11.15: Axis of a Sphere [2224]
- Def. 11.16: Center of a Sphere [2225]
- Def. 11.17: Diameter of a Sphere [2226]
- Def. 11.18: Cone [2227]
- Def. 11.19: Axis of a Cone [2228]
- Def. 11.20: Base of a Cone [2229]
- Def. 11.21: Cylinder [2230]
- Def. 11.22: Axis of a Cylinder [2231]
- Def. 11.23: Bases of a Cylinder [2232]
- Def. 11.24: Similar Cones, Similar Cylinders [2233]
- Def. 11.25: Cube [2234]
- Def. 11.26: Octahedron [2235]
- Def. 11.27: Icosahedron [2236]
- Def. 11.28: Dodecahedron [2237]
- Def. 3.01: Congruent Circles [1850]
- Def. 3.02: Tangent to the Circle, Straight-Line Touching The Circle [1853]
- Def. 3.03: Circles Touching One Another [1851]
- Def. 3.04: Chords Equally Far From the Center of a Circle [1854]
- Def. 3.05: Chords Being Further from the Center of a Circle [6433]
- Def. 3.06: Segment of a Circle [1852]
- Def. 3.07: Angle of a Segment [1855]
- Def. 3.08: Angle in the Segment [6434]
- Def. 3.09: Angle Standing Upon An Arc [6435]
- Def. 3.10: Circular Sector [2360]
- Def. 4.1: Rectilinear Figure Inscribed in Another Rectilinear Figure [1918]
- Def. 4.2: Rectilinear Figure Circumscribed in Another Rectilinear Figure [1919]
- Def. 4.3: Inscribing Rectilinear Figures in Circles [1920]
- Def. 4.4: Circumscribing Rectilinear Figures about Circles [1921]
- Def. 4.5: Inscribing Circles in Rectilinear Figures [6439]
- Def. 4.6: Circumscribing Circles about Rectilinear Figures [6438]
- Def. 4.7: Chord and Secant [1012]
- Def. 5.01: Aliquot Part [2316]
- Def. 5.02: Multiple of a Real Number [6440]
- Def. 5.03: Ratio [1943]
- Def. 5.04: Having a Ratio [6441]
- Def. 5.05: Having the Same Ratio [1945]
- Def. 5.06: Proportional Magnitudes [6442]
- Def. 5.07: Having a Greater Ratio [1946]
- Def. 5.08: Proportion in Three Terms [1947]
- Def. 5.09: Squared Ratio [1948]
- Def. 5.10: Cubed Ratio [1949]
- Def. 5.11: Corresponding Magnitudes [1950]
- Def. 5.12: Alternate Ratio [1951]
- Def. 5.13: Inverse Ratio [1952]
- Def. 5.14: Composition of a Ratio [1953]
- Def. 5.15: Separation of a Ratio [1954]
- Def. 5.16: Conversion of a Ratio [1955]
- Def. 5.17: Ratio ex Aequali [1956]
- Def. 5.18: Perturbed Proportion [1957]
- Def. 6.01: Similar Rectilineal Figures [1983]
- Def. 6.02: Cut in Extreme and Mean Ratio [1985]
- Def. 6.03: Height of a Figure [1986]
- Def. 7.01: Unit [2314]
- Def. 7.02: Number [2315]
- Def. 7.03: Proper Divisor [703]
- Def. 7.04: Aliquant Part, a Number Being Not a Divisor of Another Number [2323]
- Def. 7.05: Multiple, Number Multiplying another Number [1275]
- Def. 7.06: Even Number [2317]
- Def. 7.07: Odd Number [2318]
- Def. 7.11: Prime Number [704]
- Def. 7.12: Co-prime (Relatively Prime) Numbers [1288]
- Def. 7.13: Composite Number [6436]
- Def. 7.14: Not Co-prime Numbers [2322]
- Def. 7.15: Multiplication of Numbers [6437]
- Def. 7.16: Rectangular Number, Plane Number [2324]
- Def. 7.17: Cuboidal Number, Solid Number [2325]
- Def. 7.18: Square Number [2326]
- Def. 7.19: Cubic Number, Cube Number [2327]
- Def. 7.20: Proportional Numbers [2328]
- Def. 7.21: Similar Rectangles and Similar Cuboids, Similar Plane and Solid Numbers [2329]
- Def. 7.22: Perfect Number [2330]
- Defining Property of the Field of Real Numbers [6194]
- Definition of Complex Numbers [216]
- Definition of Irrational Numbers [6663]
- Degree Sequence [6350]
- Derivative of an n-Dimensional Curve [6247]
- Derivative, Differentiable Functions [1370]
- Diagonal [908]
- Diameter In Metric Spaces [6581]
- Difference Quotient [1369]
- Differentiable Manifold, Atlas [6207]
- Differential Form of Degree k [6335]
- Digraph, Initial and Terminal Vertices, Loops, Parallel and Inverse Edges, Simple Digraph [524]
- Dimension of a Vector Space [1041]
- Dimension of an Affine Space [6281]
- Direct Sum of Vector Spaces [6320]
- Directional Derivative [256]
- Discrete Random Variables [182]
- Disjunction [713]
- Divergent Sequences [1332]
- Divergent Series [217]
- Divisibility of Ideals [1065]
- Division of Real Numbers [6635]
- Divisor-Closed Subsets of Natural Numbers [6406]
- Divisor, Complementary Divisor [700]
- Domain of Discourse [6219]
- Dot Product of Complex Numbers [1246]
- Dot Product, Inner Product, Scalar Product (Complex Case) [6266]
- Dot Product, Inner Product, Scalar Product (General Field Case) [1049]
- Dual Planar Graph [6391]
- Eigenvalue [6250]
- Eigenvector [6339]
- Ellipse [6300]
- Embedding, Inclusion Map [6241]
- Equality, Inequality [1539]
- Equipotent Sets [978]
- Equivalence Relation, Equivalent Classes, Partitions, Representative Elements, Quotient Sets [574]
- Equivalence [1305]
- Euclidean Affine Space [6278]
- Euclidean Movement - Isometry [2777]
- Euler's Constant [1344]
- Eulerian Graph [6348]
- Eulerian Tour [391]
- Even and Odd Complex Functions [352]
- Even and Odd Functions [416]
- Exponential Function of General Base [1603]
- Exponentiation [673]
- Extended Real Numbers [6668]
- Exterior Algebra, Alternating Product, Universal Alternating Map [6333]
- Exterior, Interior, Alternate and Corresponding Angles [910]
- Face, Infinite Face [6373]
- Falling And Rising Factorial Powers [1399]
- Field [557]
- Field Extension [6211]
- Field Homomorphism [559]
- Finite and Infinite Graphs [6354]
- Finite and Sigma-Finite Measure [6237]
- Finite and Sigma-Finite Pre-measure [6238]
- Finite Field Extension [6228]
- Finite Set, Infinite Set [985]
- First Order Predicate Logic [186]
- Fixed Point, Fixed Point Property [6702]
- Floors and Ceilings [280]
- Frame of Reference [6294]
- Function, Arity and Constant [6222]
- Functional [6725]
- Functional Equation [6726]
- Generalized Polynomial Function [6337]
- Generating Systems [279]
- Geometric Probability [1801]
- Geometric Progression, Continued Proportion [6552]
- Girth and Circumference [6375]
- GOTO Command, GOTO Program, Index [1182]
- GOTO-Computable Functions [1197]
- Graph Decomposable Into \(k\) Trees [6392]
- Graph of a Real Function [6679]
- Greatest Common Divisor [1280]
- Group [671]
- Group Homomorphism [679]
- Group Operation [6253]
- Groupoid (Magma) [836]
- Hamiltonian Cycle [330]
- Hamiltonian Graph [6349]
- Harmonic Series [237]
- Hausdorff Space [6199]
- Heine-Borel Property defines Compact Subsets [6575]
- Hexagon [6448]
- Higher Order Directional Derivative [6204]
- Hilbert Space [6265]
- Homeomorphism, Homeomorphic Spaces [6197]
- Homogeneous and Inhomogeneous Linear Equations with \(n\) Unknowns [1043]
- Homogeneous and Inhomogeneous Systems of Linear Equations with \(n\) Unknowns [1044]
- Homomorphism [401]
- Hyperbolic Cosine [6687]
- Hyperbolic Sine [6688]
- Ideal [1062]
- Identity Matrix [1051]
- Identity, Neutral Element, Left Identity, Right Identity [661]
- Incidence, Adjacency, Neighbours [525]
- Incidence, Adjacency, Predecessor and Successor Vertices, Neighbours [544]
- Independent Events [395]
- Index Set, Family of Sets [6198]
- Inertial and Noninertial Frames of Reference [6295]
- Infimum [1755]
- Infimum of Extended Real Numbers [6670]
- Injective Function [769]
- Instantaneous Velocity [6310]
- Integral Closure [6322]
- Integral Element [6258]
- Interlacing Pieces with Respect to a Cycle, Interlacement Graph [1235]
- International System of Units (SI)
- Meter [6274]
- Second [6273]
- Invertible and Inverse Matrix [1055]
- Invertible Functions, Inverse Functions [407]
- Irreducible, Prime [822]
- Irreflexive, Asymmetric and Antisymmetric Relation [575]
- Isometry [2778]
- Isomorphic Digraphs [1178]
- Isomorphic Semigroups [838]
- Isomorphic Undirected Graphs [1177]
- Isomorphism [412]
- Jordan Arc (Simple Curve) [1210]
- Knot [6362]
- Knot Diagram, Classical Crossing, Virtual Crossing [6358]
- Laplace Experiments and Elementary Events [973]
- Leaf [6366]
- Limit Inferior [6674]
- Limit of a Function [6203]
- Limit Ordinal [780]
- Limit Superior [6673]
- Limits of Real Functions [6683]
- Linear Combination [1035]
- Linear Function [1377]
- Linear Map [403]
- Linear Order [6191]
- Linear Span [1037]
- Linearly Dependent and Linearly Independent Vectors, Zero Vector [1036]
- Linked List, List Nodes [1214]
- LOOP Command, LOOP Program [1180]
- LOOP-Computable Functions [1183]
- Manifold [6200]
- Matrix Multiplication [1050]
- Matrix, Set of Matrices over a Field [1048]
- Maximal Ideal [6243]
- Maximum [6602]
- Measurable Set [6230]
- Measurable Space [6239]
- Measure [6232]
- Measureable Function [6231]
- Metric (Distance) [614]
- Metric Space [617]
- Minimal Polynomial [6321]
- Minimal Tree Decomposability [6393]
- Minimum [6603]
- Module [6233]
- Modulo Operation, Modulus [1283]
- Modulus of Continuity of a Continuous Function [6704]
- Monic Polynomial [6257]
- Monoid [659]
- Monotonic Functions [282]
- Monotonic Sequences [1155]
- Multilinear Map [6319]
- Multiplication of Complex Numbers [1668]
- Multiplication of Natural Numbers [876]
- Multiplicative System [6234]
- Mutually Disjoint Sets [6227]
- Mutually Exclusive and Collectively Exhaustive Events [859]
- Mutually Independent Events [1808]
- Negation [714]
- Norm, Normed Vector Space [846]
- Normal Subgroups [273]
- Null Graph [6345]
- Number of Distinct Divisors [702]
- Open Ball, Neighborhood [849]
- Open Cover [150]
- Open Function, Closed Function [6242]
- Open Set, Closed Set [852]
- Order of a Graph [6353]
- Order Relation [6190]
- Order Relation for Integers - Positive and Negative Integers [1075]
- Order Relation for Natural Numbers [697]
- Order Relation for Rational Numbers - Positive and Negative Rational Numbers [1076]
- Order Relation for Real Numbers [1107]
- Order Relation for Step Functions [1758]
- Ordered Field [6192]
- Ordered Pair, n-Tuple [747]
- Ordinal Number [723]
- Pairwise Independent Events [1809]
- Parallelogram - Defining Property IV [940]
- Partial Maps (Functions) [123]
- Pentagon [6447]
- Permutations [188]
- Pieces of a Graph With Respect to A Cycle [1231]
- Planar Drawing (Embedding) [1212]
- Planar Graph [1226]
- Point of Division, Point of External Division [1013]
- Points, Lines, Planes, Hyperplanes [6282]
- Pointwise and Uniform Convergence [173]
- Polynomial over a Ring, Degree, Variable [487]
- Polynomial Ring [6323]
- Polynomials [199]
- Positive and Negative Real Numbers [585]
- Pre-measure [6236]
- Predicate [6223]
- Preorder (Quasiorder), Partial and Total Order, Poset and Chain [576]
- Prime Ideal [6240]
- Principal Ideal [1063]
- Principal Ideal Domain [6340]
- Principal Ideal Ring [1064]
- Probability and its Axioms [858]
- Probability Distribution [1815]
- Probability Mass Function [1824]
- Properties of Relations Between Two Sets [1308]
- Quantifier, Bound Variables, Free Variables [6221]
- Random Experiments and Random Events [857]
- Random Variable, Realization, Population and Sample [1813]
- Ratio of Two Real Numbers [6634]
- Rational Cauchy Sequence [1485]
- Rational Functions [218]
- Rational Sequence [1484]
- Real Absolute Value Function [6681]
- Real Cauchy Sequence [1704]
- Real Identity Function [6680]
- Real Infinite Series, Partial Sums [1109]
- Real Intervals [1153]
- Real Sequence [875]
- Real Subsequence [6610]
- Rearrangement of Infinite Series [1363]
- Recursive Definition of the Determinant [6252]
- Reflexive, Symmetric and Transitive Relation [572]
- Regular Graph [6351]
- Reidemeister Moves, Planar Isotopy Moves, Diagrammatic Moves [6359]
- Relation [571]
- Relative and Absolute Frequency [1837]
- Restriction [1170]
- Riemann Integrable Functions [1763]
- Riemann Sum With Respect to a Partition [1781]
- Right Inverse [6325]
- Ring [683]
- Ring Homomorphism [885]
- Ring of Integers [6324]
- Ring of Sets (measure-theoretic definition) [6216]
- Section over a Base Space [6334]
- Sematics, Proposition [710]
- Semi-Eulerian Graph [6387]
- Semi-Eulerian Tour, Open Trail [6386]
- Semi-Hamiltonian Graph [6390]
- Semi-Hamiltonian Path [6389]
- Semigroup [660]
- Separating and Non-Separating Cycles [1232]
- Sequence [874]
- Sequences Tending To Infinity [1345]
- Set of Binary Logical Values (True and False) [707]
- Set of Natural Numbers (Peano) [664]
- Set-theoretic Definition of Order Relation for Natural Numbers [719]
- Set-theoretic Definitions of Natural Numbers [718]
- Set, Set Element, Empty Set [550]
- Sieve of Eratosthenes [6402]
- Sigma-Algebra [6212]
- Signature [6224]
- Similarity [2782]
- Simplex [6286]
- Sine of a Real Variable [1746]
- Size of a Graph [6352]
- Solution of Ordinary DE [6341]
- Spacetime Diagram [6307]
- Spanning Subgraph [6347]
- Spanning Tree [6365]
- Spectrum of a Commutative Ring [6245]
- Square Matrix [1056]
- Step Functions [1751]
- Stirling Numbers of the First Kind [1004]
- Subadditive Function [6705]
- Subdigraphs and Superdigraphs; Induced Subdigraph [1171]
- Subdivision of a Graph [6377]
- Subfield [887]
- Subgraphs and Supergraphs; Induced Subgraph [390]
- Subgroup [554]
- Submonoid [6210]
- Subring [884]
- Subsequence [1151]
- Subset, Superset, Union, Intersection, Set Difference, Set Complement, Power Set [552]
- Subsets of Natural Numbers Relatively Prime To a Natural Number [6405]
- Subsets of Prime Numbers Not Dividing a Natural Number [6404]
- Subspace [562]
- Subtraction of Complex Numbers [1701]
- Subtraction of Integers [1585]
- Subtraction of Rational Numbers [1586]
- Subtraction of Real Numbers [1588]
- Sum of Angles [651]
- Sums [261]
- Supplemental Angles [652]
- Suppressing Vertices, Suppressed Multigraph [1169]
- Supremum [1754]
- Supremum of Extended Real Numbers [6669]
- Surjective Function [770]
- Symmetric Bilinear Form [6336]
- Symmetric Matrix [1779]
- Syntax [709]
- Tangent Bundle [6326]
- Tautology, Valid Boolean Functions [1318]
- Terms in Predicate Logic [6225]
- Topological Chart [6201]
- Topological Space, Topology [6189]
- Total Maps (Functions) [592]
- Totally Differentiable Functions, Total Derivative [6215]
- Transcendental Element [6256]
- Transition Map [6202]
- Transitive Set [720]
- Transposed Matrix [1054]
- Trees and Forests [96]
- Triangle [6432]
- Twin Prime Numbers [233]
- Uncountable Set [6660]
- Undirected Graph, Vertices, Edges, Simple Graph [523]
- Uniformly Continuous Functions (General Metric Spaces Case) [6612]
- Uniformly Continuous Functions (Real Case) [6611]
- Unit and Unit Group, Zero Divisor and Integral Domain [821]
- Unit-Cost Random Access Machine [1179]
- Unitary Affine Space [6279]
- Unknot [6364]
- Upper and Lower Triangular Matrix [1053]
- Variable [6220]
- Vector Field [222]
- Vector Space, Vector, Vector Addition, Skalar Multiplication [560]
- Vector Spaces
- Basis, Coordinate System [299]
- Vertex Degrees for Digraphs [1172]
- Vertex Degrees for Undirected Graphs [362]
- Walks, Trails, and Paths [1164]
- Weakly and Strongly Connected Digraphs [1219]
- WHILE Command, WHILE Program [1181]
- WHILE-Computable Functions [1184]
- Zariski Topology of a Commutative Ring [6246]
- Zero (Absorbing, Annihilating) Element, Left Zero, Right Zero [662]
- Zero Matrix [1052]
- Zero Ring [879]
- Zero Vector [6734]
- (Real) Exponential Function Is Always Positive [1419]
- \(-(-x)=x\) [522]
- \(-(x+y)=-x-y\) [535]
- \(-0=0\) [499]
- \((-x)(-y)=xy\) [531]
- \((-x)y=-(xy)\) [530]
- \((x^{-1})^{-1}=x\) [534]
- \((xy)^{-1}=x^{-1}y^{-1}\) [536]
- \(\epsilon\)-\(\delta\) Definition of Continuity [1254]
- \(\exp(0)=1\) [1423]
- \(\exp(0)=1\) (Complex Case) [1739]
- \(0x=0\) [521]
- \(1^{-1}=1\). [500]
- \(b\)-Adic Fractions Are Real Cauchy Sequences [1111]
- A Criterion for Isosceles Triangles [749]
- A Criterion for Subsets of Real Numbers to be Bounded [6667]
- A General Criterion for the Convergence of Infinite Series [1148]
- A Necessary and a Sufficient Condition for Riemann Integrable Functions [1764]
- A Necessary But Not Sufficient Condition For Convergence Of Infinite Series [1264]
- A product of two real numbers is zero if and only if at least one of these numbers is zero. [528]
- A proposition cannot be both, true and false [1322]
- A proposition cannot be equivalent to its negation [1323]
- Absolute Value of Complex Conjugate [6728]
- Absolute Value of the Product of Complex Numbers [6729]
- Addition and Scalar Multiplication of Riemann Upper and Lower Integrals [1770]
- Addition of Complex Numbers Is Associative [1658]
- Addition of Complex Numbers Is Commutative [1660]
- Addition of Integers [890]
- Addition of Integers Is Associative [1443]
- Addition of Integers Is Cancellative [1462]
- Addition of Integers Is Commutative [1460]
- Addition Of Natural Numbers [842]
- Addition Of Natural Numbers Is Associative [1428]
- Addition of Natural Numbers Is Cancellative [1432]
- Addition of Natural Numbers Is Cancellative With Respect To Inequalities [1551]
- Addition of Natural Numbers Is Commutative [1430]
- Addition of Rational Cauchy Sequences [1486]
- Addition of Rational Cauchy Sequences Is Associative [1494]
- Addition of Rational Cauchy Sequences Is Cancellative [1569]
- Addition of Rational Cauchy Sequences Is Commutative [1496]
- Addition Of Rational Numbers [1446]
- Addition of Rational Numbers Is Associative [1447]
- Addition of Rational Numbers Is Cancellative [1471]
- Addition of Rational Numbers Is Commutative [1469]
- Addition of Real Numbers [1514]
- Addition Of Real Numbers Is Associative [31]
- Addition of Real Numbers Is Cancellative [1574]
- Addition Of Real Numbers Is Commutative [33]
- Additivity Theorems For Cosine and Sine [6730]
- Algebraic Structure of Complex Numbers Together with Addition [1666]
- Algebraic Structure of Complex Numbers Together with Addition and Multiplication [1690]
- Algebraic Structure of Integers Together with Addition [1654]
- Algebraic Structure of Integers Together with Addition and Multiplication [892]
- Algebraic Structure Of Natural Numbers Together With Addition [841]
- Algebraic Structure Of Natural Numbers Together With Multiplication [877]
- Algebraic Structure of Non-Zero Complex Numbers Together with Multiplication [1688]
- Algebraic Structure of Non-Zero Rational Numbers Together with Multiplication [1646]
- Algebraic Structure of Non-Zero Real Numbers Together with Multiplication [1640]
- Algebraic Structure of Rational Numbers Together with Addition [1645]
- Algebraic Structure of Rational Numbers Together with Addition and Multiplication [1647]
- Algebraic Structure of Real Numbers Together with Addition [1639]
- Algebraic Structure of Real Numbers Together with Addition and Multiplication [1638]
- All Cauchy Sequences Converge in the Set of Real Numbers (Completeness Principle) [1108]
- All Convergent Real Sequences Are Cauchy Sequences [1394]
- All Uniformly Continuous Functions are Continuous [6700]
- Alternating Sum of Binomial Coefficients [1407]
- Angles and Sides in a Triangle V [903]
- Angles of a Right And Isosceles Triangle [926]
- Angles of Right Triangle [930]
- Any Positive Characteristic Is a Prime Number [882]
- Approximability of Continuous Real Functions On Closed Intervals By Step Functions [6619]
- Barycentric Coordinates, Barycenter [6283]
- Basic Rules of Manipulating Finite Sums [1114]
- Basis Arithmetic Operations Involving Differentiable Functions, Product Rule, Quotient Rule [1375]
- Bayes' Theorem [464]
- Bernoulli's Inequality [590]
- Biconnectivity is a Necessary Condition for a Hamiltonian Graph [6396]
- Binomial Distribution [450]
- Binomial Theorem [1397]
- Bisectors of Two Supplemental Angles Are Right Angle To Each Other [766]
- Boolean Function [1316]
- Bounds for Partial Sums of Exponential Series [6641]
- Bounds for the Minimal Tree Decomposability [6394]
- C.N. 1: Equality is an Equivalence Relation [6420]
- C.N. 2: Adding Equations Preserves Equality [6421]
- C.N. 3: Subtracting Equations Preserves Equality [6422]
- C.N. 5: Comparing the Size of Sets and Their Subsets [6423]
- Calculating the Number of Distinct Positive Divisors [1302]
- Calculating with Complex Conjugates [1251]
- Calculation Rules for General Powers [1628]
- Calculation Rules for the Big O Notation [1167]
- Cancellation Law [823]
- Cardinal Number [980]
- Cauchy Product of Absolutely Convergent Complex Series [1736]
- Cauchy Product of Absolutely Convergent Series [1390]
- Cauchy Product of Convergent Series Is Not Necessarily Convergent [1392]
- Characteristic String [1001]
- Characterization of Biconnected Planar Graphs [1237]
- Characterization of Bipartite Graphs [6370]
- Characterization of Closed Sets by Limits of Sequences [6585]
- Characterization of Cutvertices [1238]
- Characterization of Eulerian Graphs [6381]
- Characterization of Independent Events [1804]
- Characterization of Independent Events II [1806]
- Characterization of Planar Graphs [6380]
- Characterization of Planar Hamiltonian Graphs [6400]
- Characterization of Semi-Eulerian Graphs [6385]
- Closed Formula For Binomial Coefficients [1400]
- Closed Formula for the Maximum and Minimum of Two Numbers [6642]
- Closed n-Dimensional Cuboids Are Compact [6582]
- Closed Real Intervals Are Compact [6583]
- Closed Subsets of Compact Sets are Compact [6594]
- Commutative Group of Multiplicative Functions [506]
- Commutativity of the Greatest Common Divisor [1287]
- Compact Subset of Real Numbers Contains its Maximum and its Minimum [6598]
- Compact Subsets of Metric Spaces Are Bounded and Closed [6589]
- Comparing Natural Numbers Using the Concept of Addition [1547]
- Comparison of Functional Equations For Linear, Logarithmic and Exponential Growth [6727]
- Completeness Principle For Complex Numbers [1709]
- Complex Cauchy Sequences Vs. Real Cauchy Sequences [1705]
- Complex Conjugate of Complex Exponential Function [1747]
- Complex Exponential Function [312]
- Complex Numbers are a Field Extension of Real Numbers [1243]
- Complex Numbers are Two-Dimensional and the Complex Numbers \(1\) and Imaginary Unit \(i\) Form Their Basis [1698]
- Complex Numbers as a Vector Space Over the Field of Real Numbers [1694]
- Composition of Continuous Functions at a Single Point [1606]
- Composition of Relations (Sometimes) Preserves Their Left-Total Property [1312]
- Composition of Relations Preserves Their Right-Uniqueness Property [1310]
- Composition of Total Functions [1314]
- Compositions of Continuous Functions on a Whole Domain [1608]
- Conjunction
- Commutativity of Conjunction [1834]
- Connection between Quotient, Remainder, Modulo and Floor Function [1284]
- Connectivity Is an Equivalence Relation - Components Are a Partition of a Graph [1221]
- Construction of a Light Clock [6275]
- Construction of Fields from Integral Domains [888]
- Construction of Groups from Commutative and Cancellative Semigroups [839]
- Continuity of Complex Exponential Function [1743]
- Continuity of Cosine and Sine [1782]
- Continuity of Exponential Function [1422]
- Continuity of Exponential Function of General Base [1610]
- Continuous Functions Mapping Compact Domains to Real Numbers are Bounded [6606]
- Continuous Functions Mapping Compact Domains to Real Numbers Take Maximum and Minimum Values on these Domains [6604]
- Continuous Functions on Compact Domains are Uniformly Continuous [6614]
- Continuous Real Functions on Closed Intervals are Bounded [6697]
- Continuous Real Functions on Closed Intervals are Riemann-Integrable [1766]
- Continuous Real Functions on Closed Intervals are Uniformly Continuous [6616]
- Continuous Real Functions on Closed Intervals Take Maximum and Minimum Values within these Intervals [6696]
- Contraposition of Cancellative Law for Adding Integers [1561]
- Contraposition of Cancellative Law for Adding Natural Numbers [1545]
- Contraposition of Cancellative Law for Adding Rational Numbers [1565]
- Contraposition of Cancellative Law for Adding Real Numbers [1578]
- Contraposition of Cancellative Law for Multiplying Integers [1563]
- Contraposition of Cancellative Law for Multiplying Natural Numbers [1559]
- Contraposition of Cancellative Law for Multiplying Rational Numbers [1567]
- Contraposition of Cancellative Law of for Multiplying Real Numbers [1580]
- Convergence Behavior of the Inverse of Sequence Members Tending to Infinity [6649]
- Convergence Behavior of the Inverse of Sequence Members Tending to Zero [6650]
- Convergence Behavior of the Sequence \((b^n)\) [1347]
- Convergence Behaviour of Absolutely Convergent Series [1268]
- Convergence of Alternating Harmonic Series [1367]
- Convergence of Complex Conjugate Sequence [1707]
- Convergence of Infinite Series with Non-Negative Terms [1158]
- Convergent Complex Sequences Are Bounded [1716]
- Convergent Complex Sequences Vs. Convergent Real Sequences [1702]
- Convergent Rational Sequences With Limit \(0\) Are a Subgroup of Rational Cauchy Sequences With Respect To Addition [1522]
- Convergent Rational Sequences With Limit \(0\) Are an Ideal Of the Ring of Rational Cauchy Sequences [1524]
- Convergent Rational Sequences With Limit \(0\) Are Rational Cauchy Sequences [1516]
- Convergent Sequence together with Limit is a Compact Subset of Metric Space [6577]
- Convergent Sequence without Limit Is Not a Compact Subset of Metric Space [6579]
- Convergent Sequences are Bounded [6592]
- Convergent Sequences are Bounded [1137]
- Convergent Sequences are Cauchy Sequences [1073]
- Convergent Sequences
- Criteria [307]
- Convergent Series
- A General Criterion for the Convergence of Infinite Complex Series [308]
- Cor. 10.003: Greatest Common Measure of Commensurable Magnitudes [6458]
- Cor. 10.004: Greatest Common Measure of Three Commensurable Magnitudes [6459]
- Cor. 10.006: Magnitudes with Rational Ratio are Commensurable [6460]
- Cor. 10.009: Commensurability of Squares [6461]
- Cor. 10.023: Segment Commensurable with Medial Area is Medial [6462]
- Cor. 10.111: Thirteen Irrational Straight-Lines of Different Order [6463]
- Cor. 10.114: Rectangles With Irrational Sides Can Have Rational Areas [6464]
- Cor. 11.33: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides [6465]
- Cor. 11.35: Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles [6466]
- Cor. 12.07: Prism on Triangular Base divided into Three Equal Tetrahedra [6467]
- Cor. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides [6468]
- Cor. 12.17: Construction of Polyhedron in Outer of Concentric Spheres [6469]
- Cor. 13.16: Construction of Regular Icosahedron within Given Sphere [6470]
- Cor. 13.17: Construction of Regular Dodecahedron within Given Sphere [6471]
- Cor. 3.01: Bisected Chord of a Circle Passes the Center [1060]
- Cor. 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle [6449]
- Cor. 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle [6450]
- Cor. 5.07: Ratios of Equal Magnitudes [6451]
- Cor. 5.19: Proportional Magnitudes have Proportional Remainders [6452]
- Cor. 6.08: Perpendicular in Right-Angled Triangle makes two Similar Triangles, Geometric Mean Theorem, Mean Proportion [6453]
- Cor. 6.19: Ratio of Areas of Similar Triangles [6454]
- Cor. 6.20: Similar Polygons are Composed of Similar Triangles [6455]
- Cor. 7.02: Any Divisor Dividing Two Numbers Divides Their Greatest Common Divisor [6414]
- Cor. 8.02: Construction of Geometric Progression in Lowest Terms [6456]
- Cor. 9.11: Elements of Geometric Progression from One which Divide Later Elements [6457]
- Corollaries From the Group Axioms [555]
- Counting the Set's Elements Using Its Partition [982]
- Criteria for Subgroups [811]
- Criterion for Alternating Infinite Series [1266]
- Cyclic Groups are Abelian [813]
- Decreasing Sequence of Supremum of Extended Real Numbers [6671]
- Def. 3.11: Similar Circular Segments [2783]
- Def. 7.08: Even-Times-Even Number [2319]
- Def. 7.09: Even-Times-Odd Number [2320]
- Def. 7.10: Odd-Times-Odd Number [2321]
- Definition of Continuity Using Open Sets [6195]
- Definition of Integers [844]
- Definition of Rational Numbers [1033]
- Definition of Real Numbers [1105]
- Definition of the Metric Space \(\mathbb R^n\), Euclidean Norm [1206]
- Derivative of a Constant Function [1372]
- Derivative of a Linear Function \(ax+b\) [1378]
- Diagonals of a Rectangle [941]
- Diagonals of a Rhombus [942]
- Difference of Convergent Complex Sequences [1713]
- Difference of Convergent Real Sequences [1133]
- Difference of Convergent Real Series [6645]
- Difference of Squares of Hyperbolic Cosine and Hyperbolic Sine [6691]
- Differentiable Functions are Continuous [1374]
- Direct Comparison Test For Absolutely Convergent Complex Series (Majorant Criterion) [1727]
- Direct Comparison Test For Absolutely Convergent Series (Majorant Criterion) [1270]
- Direct Comparison Test For Divergence Series [1335]
- Discovery of Irrational Numbers [1096]
- Disjunction
- Commutativity of Disjunction [1835]
- Distance in Normed Vector Spaces [847]
- Distributivity Law for Complex Numbers [1678]
- Distributivity Law For Integers [1466]
- Distributivity Law For Natural Numbers [1030]
- Distributivity Law For Rational Cauchy Sequences [1506]
- Distributivity Law For Rational Numbers [1491]
- Distributivity Law For Real Numbers [520]
- Divergence of Harmonic Series [1333]
- Divisibility Laws [508]
- Divisibility of Principal Ideals [1066]
- Division with Quotient and Remainder [818]
- Divisors of a Product Of Many Factors, Co-Prime to All But One Factor, Divide This Factor [1295]
- Divisors of a Product Of Two Factors, Co-Prime to One Factor Divide the Other Factor [1293]
- Divisors of Integers [1273]
- Double Summation [549]
- Dual Graph of a All Faces Contained in a Planar Hamiltonian Cycle is a Tree [6398]
- Equality of Two Ratios [6631]
- Equivalence of Set Inclusion and Element Inclusion of Ordinals [730]
- Equivalence
- Commutativity of Equivalence [1836]
- Equivalency of Vectors in Vector Space If their Difference Forms a Subspace [6328]
- Equivalent Definitions of Trees [1242]
- Equivalent Knot Diagrams [6360]
- Equivalent Statements Regarding Parallel Lines [917]
- Estimate for the Remainder Term of Complex Exponential Function [1732]
- Estimate for the Remainder Term of Exponential Function [1361]
- Estimates for the Remainder Terms of the Infinite Series of Cosine and Sine [6732]
- Euler Characteristic for Planar Graphs [6374]
- Euler's Formula [1783]
- Even Number of Vertices with an Odd Degree in Finite Digraphs [568]
- Even Number of Vertices with an Odd Degree in Finite Graphs [1175]
- Eveness of the Cosine of a Real Variable [1790]
- Every Bounded Real Sequence has a Convergent Subsequence [1152]
- Every Contraposition is a Tautology [1328]
- Every Distance Is Positive Definite [615]
- Every Equilateral Triangle Is Equiangular. [742]
- Every Natural Number Is Greater or Equal Zero [1556]
- Every Proposition Implies Itself [1319]
- Every uniformly convergent sequence of functions is pointwise convergent. [1256]
- Exchanging the Limit of Function Values with the Function Value of the Limit of Arguments [6710]
- Existence of Arbitrarily Small Positive Rational Numbers [1846]
- Existence of Arbitrarily Small Powers [1350]
- Existence of Complex One (Neutral Element of Multiplication of Complex Numbers) [1673]
- Existence of Complex Zero (Neutral Element of Addition of Complex Numbers) [1662]
- Existence of Integer One (Neutral Element of Multiplication of Integers) [1454]
- Existence of Integer Zero (Neutral Element of Addition of Integers) [1452]
- Existence of Integers Exceeding Real Numbers [1342]
- Existence of Inverse Complex Numbers With Respect to Addition [1664]
- Existence of Inverse Complex Numbers With Respect to Multiplication [1675]
- Existence of Inverse Integers With Respect to Addition [1511]
- Existence of Inverse Rational Cauchy Sequences With Respect to Addition [1508]
- Existence of Inverse Rational Numbers With Respect to Addition [1509]
- Existence of Inverse Rational Numbers With Respect to Multiplication [1649]
- Existence of Inverse Real Numbers With Respect to Addition [35]
- Existence of Inverse Real Numbers With Respect to Multiplication [42]
- Existence of Natural Numbers Exceeding Positive Real Numbers [1340]
- Existence of Natural One (Neutral Element of Multiplication of Natural Numbers) [1457]
- Existence of Natural Zero (Neutral Element of Addition of Natural Numbers) [1455]
- Existence of Parallel Straight Lines [786]
- Existence of Powers Exceeding Any Positive Constant [1348]
- Existence of Rational Cauchy Sequence of Ones (Neutral Element of Multiplication of Rational Cauchy Sequences) [1504]
- Existence of Rational Cauchy Sequence of Zeros (Neutral Element of Addition of Rational Cauchy Sequences) [1498]
- Existence of Rational One (Neutral Element of Multiplication of Rational Numbers) [1482]
- Existence of Rational Zero (Neutral Element of Addition of Rational Numbers) [1473]
- Existence of Real One (Neutral Element of Multiplication of Real Numbers) [40]
- Existence of Real Zero (Neutral Element of Addition of Real Numbers) [34]
- Exponential Function [281]
- Exponential Function and the Euler Constant [6657]
- Exponential Function Is Non-Negative (Real Case) [6656]
- Exponential Function Is Strictly Monotonically Increasing [1594]
- Exponential Function of General Base With Integer Exponents [1620]
- Exponential Function of General Base With Natural Exponents [1616]
- Extracting the Real and the Imaginary Part of a Complex Number [1248]
- Factor Groups [191]
- Factor Rings [274]
- Factorial [1005]
- Factorials and Stirling Numbers of the First Kind [1007]
- Fiber of Maximal Ideals [6318]
- Fiber of Prime Ideals [6317]
- Fiber of Prime Ideals Under a Spectrum Function [6261]
- Finite Basis Theorem [1042]
- Finite Cardinal Numbers and Set Operations [988]
- First Law of Planetary Motion [6304]
- Fixed-Point Property of Continuous Functions on Closed Intervals [6703]
- Functional Equation of the Complex Exponential Function [1735]
- Functional Equation of the Exponential Function [1415]
- Functional Equation of the Exponential Function of General Base [1612]
- Functional Equation of the Exponential Function of General Base (Revised) [1630]
- Functional Equation of the Natural Logarithm [1601]
- Functions Continuous at a Point and Identical to Other Functions in a Neighborhood of This Point [6686]
- Functions Continuous at a Point and Non-Zero at this Point are Non-Zero in a Neighborhood of This Point [6698]
- Fundamental Lemma of Homogeneous Systems of Linear Equations [1045]
- General Associative Law of Multiplication [541]
- General Associative Law [540]
- General Commutative Law of Multiplication [543]
- General Commutative Law [542]
- General Powers of Positive Numbers [1626]
- Generalized Euclidean Lemma [1298]
- Generating Co-Prime Numbers Knowing the Greatest Common Divisor [1289]
- Generating the Greatest Common Divisor Knowing Co-Prime Numbers [1291]
- Geometric Distribution [429]
- Greatest Common Divisor and Least Common Multiple of Ideals [1069]
- Greatest Common Divisors Of Integers and Prime Numbers [1296]
- Group Homomorphisms and Normal Subgroups [832]
- Group Homomorphisms with Cyclic Groups [815]
- Handshaking Lemma for Finite Digraphs [565]
- Handshaking Lemma for Finite Graphs [1173]
- Heine-Borel Theorem [6596]
- How Convergence Preserves the Order Relation of Sequence Members [1144]
- How Convergence Preserves Upper and Lower Bounds For Sequence Members [1145]
- How the Boundary Changes the Property of a Set of Being Open [1203]
- Identity Function is Continuous [6685]
- Image of a Compact Set Under a Continuous Function [143]
- Imaginary Unit [1160]
- Increasing Sequence of Infimum of Extended Real Numbers [6672]
- Indefinite Integral, Antiderivative [1768]
- Inequality between Binomial Coefficients and Reciprocals of Factorials [6640]
- Inequality between Powers of $2$ and Factorials [6639]
- Inequality between Square Numbers and Powers of $2$ [6638]
- Inequality of Natural Numbers and Their Successors [1540]
- Inequality of the Artithmetic Mean [589]
- Infinite Geometric Series [1353]
- Infinite Series for Cosine and Sine [6731]
- Infinitesimal Exponential Growth is the Growth of the Identity Function [6720]
- Infinitesimal Growth of Sine is the Growth of the Identity Function [6733]
- Intermediate Root Value Theorem [6692]
- Intermediate Value Theorem [1259]
- Intersection of Convex Affine Sets [6289]
- Inverse Hyperbolic Cosine [6723]
- Inverse Hyperbolic Sine [6722]
- Invertible Functions on Real Intervals [1381]
- Irrational Numbers are Uncountable [6664]
- Isometry is Injective [2779]
- It is true that something can be (either) true or false [1320]
- Kernel and Image of a Group Homomorphism are Subgroups [833]
- Kernel and Image of Group Homomorphism [809]
- Law of Total Probability [449]
- Least Common Multiple [1276]
- Lem. 10.016: Incommensurability of Sum of Incommensurable Magnitudes [6472]
- Lem. 10.021: Medial is Irrational [6473]
- Lem. 10.028.1: Finding Two Squares With Sum Also Square [6474]
- Lem. 10.028.2: Finding Two Squares With Sum Not Square [6475]
- Lem. 10.032: Constructing Medial Commensurable in Square II [6476]
- Lem. 10.041: Side of Sum of Medial Areas is Irrational [6477]
- Lem. 10.053: Construction of Rectangle with Area in Mean Proportion to two Square Areas [6478]
- Lem. 10.059: Sum of Squares on Unequal Pieces of Segment Is Greater than Twice the Rectangle Contained by Them [6479]
- Lem. 10.13: Finding Pythagorean Magnitudes [2371]
- Lem. 11.23: Making a Square Area Equal to the Difference Of Areas of Two Other Incongruent Squares [2372]
- Lem. 12.02: Areas of Circles are as Squares on Diameters [6480]
- Lem. 12.04: Proportion of Sizes of Tetrahedra divided into Two Similar Tetrahedra and Two Equal Prisms [6481]
- Lem. 13.02: Converse of Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio [6482]
- Lem. 13.13: Construction of Regular Tetrahedron within Given Sphere [6483]
- Lem. 13.18: Angle of the Pentagon [2313]
- Limit Inferior is the Infimum of Accumulation Points of a Bounded Real Sequence [6676]
- Limit of 1/n [6713]
- Limit of Exponential Growth as Compared to Polynomial Growth [6715]
- Limit of Logarithmic Growth as Compared to Positive Power Growth [6719]
- Limit of N-th Roots [1624]
- Limit of Nth Root [6709]
- Limit of Nth Root of N [6724]
- Limit Superior is the Supremum of Accumulation Points of a Bounded Real Sequence [6675]
- Limits of General Powers [6717]
- Limits of Logarithm in $[0,+\infty]$ [6716]
- Limits of Polynomials at Infinity [6693]
- Linear Independence of the Imaginary Unit \(i\) and the Complex Number \(1\) [1696]
- Linearity and Monotony of the Riemann Integral [1769]
- Linearity and Monotony of the Riemann Integral for Step Functions [1759]
- Logarithm to a General Base [6721]
- LOOP-Computable Functions are Total [1185]
- Lower Bound of Leaves in a Tree [6367]
- Magnitude of Divisors [1278]
- Mean Value Theorem For Riemann Integrals [1772]
- Metric Spaces and Empty Sets are Clopen [854]
- Metric Spaces are Hausdorff Spaces [850]
- Modulus of Continuity is Continuous [6708]
- Modulus of Continuity is Monotonically Increasing [6707]
- Modulus of Continuity is Subadditive [6706]
- Monotone Convergence [197]
- Monotonic Real Functions on Closed Intervals are Riemann-Integrable [1767]
- Monotonically Increasing Property of Probability Distributions [1816]
- Multinomial Coefficient [1819]
- Multinomial Distribution [481]
- Multinomial Theorem [1822]
- Multiplication of Complex Numbers Is Associative [1669]
- Multiplication of Complex Numbers Is Commutative [1671]
- Multiplication of Integers [891]
- Multiplication of Integers Is Associative [1450]
- Multiplication of Integers Is Cancellative [1464]
- Multiplication of Integers Is Commutative [1448]
- Multiplication of Natural Numbers Is Associative [1434]
- Multiplication of Natural Numbers Is Cancellative [1440]
- Multiplication of Natural Numbers Is Cancellative With Respect to the Order Relation [1583]
- Multiplication of Natural Numbers is Commutative [1435]
- Multiplication Of Rational Cauchy Sequences [1488]
- Multiplication of Rational Cauchy Sequences Is Associative [1500]
- Multiplication of Rational Cauchy Sequences Is Cancellative [1571]
- Multiplication of Rational Cauchy Sequences Is Commutative [1502]
- Multiplication Of Rational Numbers [1475]
- Multiplication of Rational Numbers Is Associative [1476]
- Multiplication Of Rational Numbers Is Cancellative [1480]
- Multiplication Of Rational Numbers Is Commutative [1478]
- Multiplication of Real Numbers [1532]
- Multiplication of Real Numbers Is Associative [37]
- Multiplication of Real Numbers Is Cancellative [1575]
- Multiplication of Real Numbers Is Commutative [38]
- Multiplying Negative and Positive Integers [1589]
- Multiplying Negative and Positive Rational Numbers [1596]
- Multiplying Negative and Positive Real Numbers [1598]
- Natural Logarithm [1421]
- Nested Closed Subset Theorem [127]
- Not all Cauchy sequences converge in the set of rational numbers. [1092]
- Not all Continuous Functions are also Uniformly Continuous [6701]
- Nth Powers [1618]
- Nth Roots of Positive Numbers [46]
- Number of Labeled Spanning Trees [6369]
- Number of Relations on a Finite Set [580]
- Number of Strings With a Fixed Length Over an Alphabet with k Letters [996]
- Number of Subsets of a Finite Set [998]
- Oddness of the Sine of a Real Variable [1792]
- One-to-one Correspondence of Ideals in the Factor Ring and a Commutative Ring [6248]
- Open and Closed Subsets of a Zariski Topology [6262]
- Open Intervals Contain Uncountably Many Irrational Numbers [6665]
- Open Real Intervals are Uncountable [6662]
- Order of Cyclic Group [808]
- Order of Subgroup Divides Order of Finite Group [831]
- Order Relation for Natural Numbers, Revised [1555]
- Ordinals Are Downward Closed [727]
- Parallelogram - Defining Property II [938]
- Parallelogram - Defining Property I [937]
- Planarity of Subdivisions [6378]
- Position of Minus Sign in Rational Numbers Representations [1592]
- Preservation of Continuity with Arithmetic Operations on Continuous Functions [1261]
- Preservation of Continuity with Arithmetic Operations on Continuous Functions on a Whole Domain [1604]
- Primality of the Smallest Non-Trivial Divisor [801]
- Prime Ideals of Multiplicative Systems in Integral Domains [6244]
- Probability of Event Difference [867]
- Probability of Event Union [868]
- Probability of Included Event [865]
- Probability of Joint Events [1802]
- Probability of Laplace Experiments [975]
- Probability of the Complement Event [861]
- Probability of the Impossible Event [862]
- Product of a Complex Number and a Convergent Complex Sequence [1719]
- Product of a Convergent Real Sequence and a Real Sequence Tending to Infinity [6652]
- Product of a Real Number and a Convergent Real Sequence [1140]
- Product of a Real Number and a Convergent Real Series [6647]
- Product of Convegent Complex Sequences [1715]
- Product of Convegent Real Sequences [1135]
- Product of Two Ratios [6633]
- Product of Two Sums (Generalized Distributivity Rule) [6629]
- Prop. 1.01: Constructing an Equilateral Triangle [693]
- Prop. 1.02: Constructing a Segment Equal to an Arbitrary Segment [732]
- Prop. 1.03: Cutting a Segment at a Given Size [736]
- Prop. 1.04: "Side-Angle-Side" Theorem for the Congruence of Triangle [738]
- Prop. 1.05: Isosceles Triagles I [740]
- Prop. 1.06: Isosceles Triagles II [743]
- Prop. 1.07: Uniqueness of Triangles [751]
- Prop. 1.08: "Side-Side-Side" Theorem for the Congruence of Triangles [753]
- Prop. 1.09: Bisecting an Angle [755]
- Prop. 1.10: Bisecting a Segment [757]
- Prop. 1.11: Constructing a Perpendicular Segment to a Straight Line From a Given Point On the Stright Line [759]
- Prop. 1.12: Constructing a Perpendicular Segment to a Straight Line From a Given Point Not On the Straight Line [760]
- Prop. 1.13: Angles at Intersections of Straight Lines [763]
- Prop. 1.14: Combining Rays to Straight Lines [767]
- Prop. 1.15: Opposite Angles on Intersecting Straight Lines [782]
- Prop. 1.16: The Exterior Angle Is Greater Than Either of the Non-Adjacent Interior Angles [784]
- Prop. 1.17: The Sum of Two Angles of a Triangle [789]
- Prop. 1.18: Angles and Sides in a Triangle I [791]
- Prop. 1.19: Angles and Sides in a Triangle II [793]
- Prop. 1.20: The Sum of the Lengths of Any Pair of Sides of a Triangle (Triangle Inequality) [795]
- Prop. 1.21: Triangles within Triangles [893]
- Prop. 1.22: Construction of Triangles From Arbitrary Segments [895]
- Prop. 1.23: Constructing an Angle Equal to an Arbitrary Rectilinear Angle [897]
- Prop. 1.24: Angles and Sides in a Triangle III [899]
- Prop. 1.25: Angles and Sides in a Triangle IV [901]
- Prop. 1.26: "Angle-Side-Angle" and "Angle-Angle-Side" Theorems for the Congruence of Triangles [905]
- Prop. 1.27: Parallel Lines I [911]
- Prop. 1.28: Parallel Lines II [913]
- Prop. 1.29: Parallel Lines III [915]
- Prop. 1.30: Transitivity of Parallel Lines [919]
- Prop. 1.31: Constructing a Parallel Line from a Line and a Point [921]
- Prop. 1.32: Sum Of Angles in a Triangle and Exterior Angle [924]
- Prop. 1.33: Parallel Equal Segments Determine a Parallelogram [931]
- Prop. 1.34: Opposite Sides and Opposite Angles of Parallelograms [933]
- Prop. 1.35: Parallelograms On the Same Base and On the Same Parallels [943]
- Prop. 1.36: Parallelograms on Equal Bases and on the Same Parallels [945]
- Prop. 1.37: Triangles of Equal Area I [947]
- Prop. 1.38: Triangles of Equal Area II [949]
- Prop. 1.39: Triangles of Equal Area III [951]
- Prop. 1.40: Triangles of Equal Area IV [953]
- Prop. 1.41: Parallelograms and Triagles [955]
- Prop. 1.42: Construction of Parallelograms I [957]
- Prop. 1.43: Complementary Segments of Parallelograms [959]
- Prop. 1.44: Construction of Parallelograms II [961]
- Prop. 1.45: Construction of Parallelograms III [963]
- Prop. 1.46: Construction of a Square I [965]
- Prop. 1.47: Pythagorean Theorem [968]
- Prop. 1.48: The Converse of the Pythagorean Theorem [971]
- Prop. 10.001: Existence of Fraction of Number Smaller than Given Number [2095]
- Prop. 10.002: Incommensurable Magnitudes do not Terminate in Euclidean Algorithm [2096]
- Prop. 10.003: Greatest Common Measure of Commensurable Magnitudes [2097]
- Prop. 10.004: Greatest Common Measure of Three Commensurable Magnitudes [2098]
- Prop. 10.005: Ratio of Commensurable Magnitudes [2099]
- Prop. 10.006: Magnitudes with Rational Ratio are Commensurable [2100]
- Prop. 10.007: Incommensurable Magnitudes Have Irrational Ratio [2101]
- Prop. 10.008: Magnitudes with Irrational Ratio are Incommensurable [2102]
- Prop. 10.009: Commensurability of Squares [2103]
- Prop. 10.010: Construction of Incommensurable Lines [2104]
- Prop. 10.011: Commensurability of Elements of Proportional Magnitudes [2105]
- Prop. 10.012: Commensurability is Transitive Relation [2106]
- Prop. 10.013: Commensurable Magnitudes are Incommensurable with Same Magnitude [2107]
- Prop. 10.014: Commensurability of Squares on Proportional Straight Lines [2108]
- Prop. 10.015: Commensurability of Sum of Commensurable Magnitudes [2109]
- Prop. 10.016: Incommensurability of Sum of Incommensurable Magnitudes [2110]
- Prop. 10.017: Condition for Commensurability of Roots of Quadratic Equation [2111]
- Prop. 10.018: Condition for Incommensurability of Roots of Quadratic Equation [2112]
- Prop. 10.019: Product of Rational Numbers is Rational [2113]
- Prop. 10.020: Quotient of Rational Numbers is Rational [2114]
- Prop. 10.021: Medial is Irrational [2115]
- Prop. 10.022: Square on Medial Straight Line [2116]
- Prop. 10.023: Segment Commensurable with Medial Segment is Medial [2117]
- Prop. 10.024: Rectangle Contained by Medial Straight Lines Commensurable in Length is Medial [2118]
- Prop. 10.025: Rationality of Rectangle Contained by Medial Straight Lines Commensurable in Square [2119]
- Prop. 10.026: Medial Area not greater than Medial Area by Rational Area [2120]
- Prop. 10.027: Construction of Components of First Bimedial [2121]
- Prop. 10.028: Construction of Components of Second Bimedial [2122]
- Prop. 10.029: Construction of Rational Straight Lines Commensurable in Square Only whose Square Differences Commensurable with Gre [2123]
- Prop. 10.030: Construction of Rational Straight Lines Commensurable in Square Only whose Square Differences Incommensurable with G [2124]
- Prop. 10.031: Constructing Medial Commensurable in Square I [2125]
- Prop. 10.032: Constructing Medial Commensurable in Square II [2126]
- Prop. 10.033: Construction of Components of Major [2127]
- Prop. 10.034: Construction of Components of Side of Rational plus Medial Area [2128]
- Prop. 10.035: Construction of Components of Side of Sum of Medial Areas [2129]
- Prop. 10.036: Binomial is Irrational [2130]
- Prop. 10.037: First Bimedial is Irrational [2131]
- Prop. 10.038: Second Bimedial is Irrational [2132]
- Prop. 10.039: Major is Irrational [2133]
- Prop. 10.040: Side of Rational plus Medial Area is Irrational [2134]
- Prop. 10.041: Side of Sum of Medial Areas is Irrational [2135]
- Prop. 10.042: Binomial Straight Line is Divisible into Terms Uniquely [2136]
- Prop. 10.043: First Bimedial Straight Line is Divisible Uniquely [2137]
- Prop. 10.044: Second Bimedial Straight Line is Divisible Uniquely [2138]
- Prop. 10.045: Major Straight Line is Divisible Uniquely [2139]
- Prop. 10.046: Side of Rational Plus Medial Area is Divisible Uniquely [2140]
- Prop. 10.047: Side of Sum of Two Medial Areas is Divisible Uniquely [2141]
- Prop. 10.048: Construction of First Binomial Straight Line [2142]
- Prop. 10.049: Construction of Second Binomial Straight Line [2143]
- Prop. 10.050: Construction of Third Binomial Straight Line [2144]
- Prop. 10.051: Construction of Fourth Binomial Straight Line [2145]
- Prop. 10.052: Construction of Fifth Binomial Straight Line [2146]
- Prop. 10.053: Construction of Sixth Binomial Straight Line [2147]
- Prop. 10.054: Root of Area contained by Rational Straight Line and First Binomial [2148]
- Prop. 10.055: Root of Area contained by Rational Straight Line and Second Binomial [2149]
- Prop. 10.056: Root of Area contained by Rational Straight Line and Third Binomial [2150]
- Prop. 10.057: Root of Area contained by Rational Straight Line and Fourth Binomial [2151]
- Prop. 10.058: Root of Area contained by Rational Straight Line and Fifth Binomial [2152]
- Prop. 10.059: Root of Area contained by Rational Straight Line and Sixth Binomial [2153]
- Prop. 10.060: Square on Binomial Straight Line applied to Rational Straight Line [2154]
- Prop. 10.061: Square on First Bimedial Straight Line applied to Rational Straight Line [2155]
- Prop. 10.062: Square on Second Bimedial Straight Line applied to Rational Straight Line [2156]
- Prop. 10.063: Square on Major Straight Line applied to Rational Straight Line [2157]
- Prop. 10.064: Square on Side of Rational plus Medial Area applied to Rational Straight Line [2158]
- Prop. 10.065: Square on Side of Sum of two Medial Area applied to Rational Straight Line [2159]
- Prop. 10.066: Straight Line Commensurable with Binomial Straight Line is Binomial and of Same Order [2160]
- Prop. 10.067: Straight Line Commensurable with Bimedial Straight Line is Bimedial and of Same Order [2161]
- Prop. 10.068: Straight Line Commensurable with Major Straight Line is Major [2162]
- Prop. 10.069: Straight Line Commensurable with Side of Rational plus Medial Area [2163]
- Prop. 10.070: Straight Line Commensurable with Side of Sum of two Medial Areas [2164]
- Prop. 10.071: Sum of Rational Area and Medial Area gives rise to four Irrational Straight Lines [2165]
- Prop. 10.072: Sum of two Incommensurable Medial Areas give rise to two Irrational Straight Lines [2166]
- Prop. 10.073: Apotome is Irrational [2167]
- Prop. 10.074: First Apotome of Medial is Irrational [2168]
- Prop. 10.075: Second Apotome of Medial is Irrational [2169]
- Prop. 10.076: Minor is Irrational [2170]
- Prop. 10.077: That which produces Medial Whole with Rational Area is Irrational [2171]
- Prop. 10.078: That which produces Medial Whole with Medial Area is Irrational [2172]
- Prop. 10.079: Construction of Apotome is Unique [2173]
- Prop. 10.080: Construction of First Apotome of Medial is Unique [2174]
- Prop. 10.081: Construction of Second Apotome of Medial is Unique [2175]
- Prop. 10.082: Construction of Minor is Unique [2176]
- Prop. 10.083: Construction of that which produces Medial Whole with Rational Area is Unique [2177]
- Prop. 10.084: Construction of that which produces Medial Whole with Medial Area is Unique [2178]
- Prop. 10.085: Construction of First Apotome [2179]
- Prop. 10.086: Construction of Second Apotome [2180]
- Prop. 10.087: Construction of Third Apotome [2181]
- Prop. 10.088: Construction of Fourth Apotome [2182]
- Prop. 10.089: Construction of Fifth Apotome [2183]
- Prop. 10.090: Construction of Sixth Apotome [2184]
- Prop. 10.091: Side of Area Contained by Rational Straight Line and First Apotome [2185]
- Prop. 10.092: Side of Area Contained by Rational Straight Line and Second Apotome [2186]
- Prop. 10.093: Side of Area Contained by Rational Straight Line and Third Apotome [2187]
- Prop. 10.094: Side of Area Contained by Rational Straight Line and Fourth Apotome [2188]
- Prop. 10.095: Side of Area Contained by Rational Straight Line and Fifth Apotome [2189]
- Prop. 10.096: Side of Area Contained by Rational Straight Line and Sixth Apotome [2190]
- Prop. 10.097: Square on Apotome applied to Rational Straight Line [2191]
- Prop. 10.098: Square on First Apotome of Medial Straight Line applied to Rational Straight Line [2192]
- Prop. 10.099: Square on Second Apotome of Medial Straight Line applied to Rational Straight Line [2193]
- Prop. 10.100: Square on Minor Straight Line applied to Rational Straight Line [2194]
- Prop. 10.101: Square on Straight Line which produces Medial Whole with Rational Area applied to Rational Straight Line [2195]
- Prop. 10.102: Square on Straight Line which produces Medial Whole with Medial Area applied to Rational Straight Line [2196]
- Prop. 10.103: Straight Line Commensurable with Apotome [2197]
- Prop. 10.104: Straight Line Commensurable with Apotome of Medial Straight Line [2198]
- Prop. 10.105: Straight Line Commensurable with Minor Straight Line [2199]
- Prop. 10.106: Straight Line Commensurable with that which produces Medial Whole with Rational Area [2200]
- Prop. 10.107: Straight Line Commensurable With That Which Produces Medial Whole With Medial Area [2201]
- Prop. 10.108: Side of Remaining Area from Rational Area from which Medial Area Subtracted [2202]
- Prop. 10.109: Two Irrational Straight Lines arising from Medial Area from which Rational Area Subtracted [2203]
- Prop. 10.110: Two Irrational Straight Lines arising from Medial Area from which Medial Area Subtracted [2204]
- Prop. 10.111: Apotome not same with Binomial Straight Line [2205]
- Prop. 10.112: Square on Rational Straight Line applied to Binomial Straight Line [2206]
- Prop. 10.113: Square on Rational Straight Line applied to Apotome [2207]
- Prop. 10.114: Area contained by Apotome and Binomial Straight Line Commensurable with Terms of Apotome and in same Ratio [2208]
- Prop. 10.115: From Medial Straight Line arises Infinite Number of Irrational Straight Lines [2209]
- Prop. 11.01: Straight Line cannot be in Two Planes [2238]
- Prop. 11.02: Two Intersecting Straight Lines are in One Plane [2239]
- Prop. 11.03: Common Section of Two Planes is Straight Line [2240]
- Prop. 11.04: Line Perpendicular to Two Intersecting Lines is Perpendicular to their Plane [2241]
- Prop. 11.05: Three Intersecting Lines Perpendicular to Another Line are in One Plane [2242]
- Prop. 11.06: Two Lines Perpendicular to Same Plane are Parallel [2243]
- Prop. 11.07: Line joining Points on Parallel Lines is in Same Plane [2244]
- Prop. 11.08: Line Parallel to Perpendicular Line to Plane is Perpendicular to Same Plane [2245]
- Prop. 11.09: Lines Parallel to Same Line not in Same Plane are Parallel to each other [2246]
- Prop. 11.10: Two Lines Meeting which are Parallel to Two Other Lines Meeting contain Equal Angles [2247]
- Prop. 11.11: Construction of Straight Line Perpendicular to Plane from point not on Plane [2248]
- Prop. 11.12: Construction of Straight Line Perpendicular to Plane from point on Plane [2249]
- Prop. 11.13: Straight Line Perpendicular to Plane from Point is Unique [2250]
- Prop. 11.14: Planes Perpendicular to same Straight Line are Parallel [2251]
- Prop. 11.15: Planes through Parallel Pairs of Meeting Lines are Parallel [2252]
- Prop. 11.16: Common Sections of Parallel Planes with other Plane are Parallel [2253]
- Prop. 11.17: Straight Lines cut in Same Ratio by Parallel Planes [2254]
- Prop. 11.18: Plane through Straight Line Perpendicular to other Plane is Perpendicular to that Plane [2255]
- Prop. 11.19: Common Section of Planes Perpendicular to other Plane is Perpendicular to that Plane [2256]
- Prop. 11.20: Sum of Two Angles of Three containing Solid Angle is Greater than Other Angle [2257]
- Prop. 11.21: Solid Angle contained by Plane Angles is Less than Four Right Angles [2258]
- Prop. 11.22: Extremities of Line Segments containing three Plane Angles any Two of which are Greater than Other form Triangle [2259]
- Prop. 11.23: Sum of Plane Angles Used to Construct a Solid Angle is Less Than Four Right Angles [2260]
- Prop. 11.24: Opposite Planes of Solid contained by Parallel Planes are Equal Parallelograms [2261]
- Prop. 11.25: Parallelepiped cut by Plane Parallel to Opposite Planes [2262]
- Prop. 11.26: Construction of Solid Angle equal to Given Solid Angle [2263]
- Prop. 11.27: Construction of Parallelepiped Similar to Given Parallelepiped [2264]
- Prop. 11.28: Parallelepiped cut by Plane through Diagonals of Opposite Planes is Bisected [2265]
- Prop. 11.29: Parallelepipeds on Same Base and Same Height whose Extremities are on Same Lines are Equal in Volume [2266]
- Prop. 11.30: Parallelepipeds on Same Base and Same Height whose Extremities are not on Same Lines are Equal in Volume [2267]
- Prop. 11.31: Parallelepipeds on Equal Bases and Same Height are Equal in Volume [2268]
- Prop. 11.32: Parallelepipeds of Same Height have Volume Proportional to Bases [2269]
- Prop. 11.33: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides [2270]
- Prop. 11.34: Parallelepipeds are of Equal Volume iff Bases are in Reciprocal Proportion to Heights [2271]
- Prop. 11.35: Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles [2272]
- Prop. 11.36: Parallelepiped formed from Three Proportional Lines equal to Equilateral Parallelepiped with Equal Angles to it forme [2273]
- Prop. 11.37: Four Straight Lines are Proportional iff Similar Parallelepipeds formed on them are Proportional [2274]
- Prop. 11.38: Common Section of Bisecting Planes of Cube Bisect and are Bisected by Diagonal of Cube [2275]
- Prop. 11.39: Prisms of Equal Height with Parallelogram and Triangle as Base [2276]
- Prop. 12.01: Areas of Similar Polygons Inscribed in Circles are as Squares on Diameters [2277]
- Prop. 12.02: Areas of Circles are as Squares on Diameters [2278]
- Prop. 12.03: Tetrahedron divided into Two Similar Tetrahedra and Two Equal Prisms [2279]
- Prop. 12.04: Proportion of Sizes of Tetrahedra divided into Two Similar Tetrahedra and Two Equal Prisms [2280]
- Prop. 12.05: Sizes of Tetrahedra of Same Height are as Bases [2281]
- Prop. 12.06: Sizes of Pyramids of Same Height with Polygonal Bases are as Bases [2282]
- Prop. 12.07: Prism on Triangular Base divided into Three Equal Tetrahedra [2283]
- Prop. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides [2284]
- Prop. 12.09: Tetrahedra are Equal iff Bases are Reciprocally Proportional to Heights [2285]
- Prop. 12.10: Volume of Cone is Third of Cylinder on Same Base and of Same Height [2286]
- Prop. 12.11: Volume of Cones or Cylinders of Same Height are in Same Ratio as Bases [2287]
- Prop. 12.12: Volumes of Similar Cones and Cylinders are in Triplicate Ratio of Diameters of Bases [2288]
- Prop. 12.13: Volumes of Parts of Cylinder cut by Plane Parallel to Opposite Planes are as Parts of Axis [2289]
- Prop. 12.14: Volumes of Cones or Cylinders on Equal Bases are in Same Ratio as Heights [2290]
- Prop. 12.15: Cones or Cylinders are Equal iff Bases are Reciprocally Proportional to Heights [2291]
- Prop. 12.16: Construction of Equilateral Polygon with Even Number of Sides in Outer of Concentric Circles [2292]
- Prop. 12.17: Construction of Polyhedron in Outer of Concentric Spheres [2293]
- Prop. 12.18: Volumes of Spheres are in Triplicate Ratio of Diameters [2294]
- Prop. 13.01: Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio [2295]
- Prop. 13.02: Converse of Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio [2296]
- Prop. 13.03: Area of Square on Lesser Segment of Straight Line cut in Extreme and Mean Ratio [2297]
- Prop. 13.04: Area of Squares on Whole and Lesser Segment of Straight Line cut in Extreme and Mean Ratio [2298]
- Prop. 13.05: Straight Line cut in Extreme and Mean Ratio plus its Greater Segment [2299]
- Prop. 13.06: Segments of Rational Straight Line cut in Extreme and Mean Ratio are Apotome [2300]
- Prop. 13.07: Equilateral Pentagon is Equiangular if Three Angles are Equal [2301]
- Prop. 13.08: Straight Lines Subtending Two Consecutive Angles in Regular Pentagon cut in Extreme and Mean Ratio [2302]
- Prop. 13.09: Sides Appended of Hexagon and Decagon inscribed in same Circle are cut in Extreme and Mean Ratio [2303]
- Prop. 13.10: Square on Side of Regular Pentagon inscribed in Circle equals Squares on Sides of Hexagon and Decagon inscribed in sa [2304]
- Prop. 13.11: Side of Regular Pentagon inscribed in Circle with Rational Diameter is Minor [2305]
- Prop. 13.12: Square on Side of Equilateral Triangle inscribed in Circle is Triple Square on Radius of Circle [2306]
- Prop. 13.13: Construction of Regular Tetrahedron within Given Sphere [2307]
- Prop. 13.14: Construction of Regular Octahedron within Given Sphere [2308]
- Prop. 13.15: Construction of Cube within Given Sphere [2309]
- Prop. 13.16: Construction of Regular Icosahedron within Given Sphere [2310]
- Prop. 13.17: Construction of Regular Dodecahedron within Given Sphere [2311]
- Prop. 13.18: Comparison of Sides of Platonic Figures - There are only Five Platonic Solids [2312]
- Prop. 2.01: Summing Areas or Rectangles [1015]
- Prop. 2.02: Square is Sum of Two Rectangles [2361]
- Prop. 2.03: Rectangle is Sum of Square and Rectangle [2060]
- Prop. 2.04: Square of Sum [1017]
- Prop. 2.05: Rectangle is Difference of Two Squares [1019]
- Prop. 2.06: Square of Sum with One Halved Summand [1020]
- Prop. 2.07: Sum of Squares [1021]
- Prop. 2.08: Square of Sum with One Doubled Summand [1022]
- Prop. 2.09: Sum of Squares of Sum and Difference [1023]
- Prop. 2.10: Sum of Squares (II) [1024]
- Prop. 2.11: Constructing the Golden Ratio of a Segment [1025]
- Prop. 2.12: Law of Cosines (for Obtuse Angles) [1026]
- Prop. 2.13: Law of Cosines (for Acute Angles) [1027]
- Prop. 2.14: Constructing a Square from a Rectilinear Figure [1028]
- Prop. 3.01: Finding the Centre of a given Circle [1058]
- Prop. 3.02: Chord Lies Inside its Circle [1061]
- Prop. 3.03: Conditions for Diameter to be Perpendicular Bisector [1865]
- Prop. 3.04: Chords do not Bisect Each Other [1866]
- Prop. 3.05: Intersecting Circles have Different Centers [1867]
- Prop. 3.06: Touching Circles have Different Centers [1886]
- Prop. 3.07: Relative Lengths of Lines Inside Circle [1887]
- Prop. 3.08: Relative Lengths of Lines Outside Circle [1888]
- Prop. 3.09: Condition for Point to be Center of Circle [1889]
- Prop. 3.10: Two Circles have at most Two Points of Intersection [1890]
- Prop. 3.11: Line Joining Centers of Two Circles Touching Internally [1891]
- Prop. 3.12: Line Joining Centers of Two Circles Touching Externally [1892]
- Prop. 3.13: Circles Touch at One Point at Most [1893]
- Prop. 3.14: Equal Chords in Circle [1894]
- Prop. 3.15: Relative Lengths of Chords of Circles [1895]
- Prop. 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle [1896]
- Prop. 3.17: Construction of Tangent from Point to Circle [1897]
- Prop. 3.18: Radius at Right Angle to Tangent [1898]
- Prop. 3.19: Right Angle to Tangent of Circle goes through Center [1899]
- Prop. 3.20: Inscribed Angle Theorem [1900]
- Prop. 3.21: Angles in Same Segment of Circle are Equal [1901]
- Prop. 3.22: Opposite Angles of Cyclic Quadrilateral [1902]
- Prop. 3.23: Segment on Given Base Unique [1903]
- Prop. 3.24: Similar Segments on Equal Bases are Equal [1904]
- Prop. 3.25: Construction of Circle from Segment [1905]
- Prop. 3.26: Equal Angles in Equal Circles [1906]
- Prop. 3.27: Angles on Equal Arcs are Equal [1907]
- Prop. 3.28: Straight Lines Cut Off Equal Arcs in Equal Circles [1908]
- Prop. 3.29: Equal Arcs of Circles Subtended by Equal Straight Lines [1909]
- Prop. 3.30: Bisection of Arc [1910]
- Prop. 3.31: Relative Sizes of Angles in Segments [1911]
- Prop. 3.32: Angles made by Chord with Tangent [1912]
- Prop. 3.33: Construction of Segment on Given Line Admitting Given Angle [1913]
- Prop. 3.34: Construction of Segment on Given Circle Admitting Given Angle [1914]
- Prop. 3.35: Intersecting Chord Theorem [1915]
- Prop. 3.36: Tangent Secant Theorem [1916]
- Prop. 3.37: Converse of Tangent Secant Theorem [1917]
- Prop. 4.01: Fitting Chord Into Circle [1925]
- Prop. 4.02: Inscribing in Circle Triangle Equiangular with Given [1926]
- Prop. 4.03: Circumscribing about Circle Triangle Equiangular with Given [1927]
- Prop. 4.04: Inscribing Circle in Triangle [1928]
- Prop. 4.05: Circumscribing Circle about Triangle [1929]
- Prop. 4.06: Inscribing Square in Circle [1930]
- Prop. 4.07: Circumscribing Square about Circle [1931]
- Prop. 4.08: Inscribing Circle in Square [1932]
- Prop. 4.09: Circumscribing Circle about Square [1933]
- Prop. 4.10: Construction of Isosceles Triangle whose Base Angle is Twice Apex [1934]
- Prop. 4.11: Inscribing Regular Pentagon in Circle [1935]
- Prop. 4.12: Circumscribing Regular Pentagon about Circle [1936]
- Prop. 4.13: Inscribing Circle in Regular Pentagon [1937]
- Prop. 4.14: Circumscribing Circle about Regular Pentagon [1938]
- Prop. 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle [1939]
- Prop. 4.16: Inscribing Regular 15-gon in Circle [1940]
- Prop. 5.01: Multiplication of Numbers is Left Distributive over Addition [1958]
- Prop. 5.02: Multiplication of Numbers is Right Distributive over Addition [1959]
- Prop. 5.03: Multiplication of Numbers is Associative [1960]
- Prop. 5.04: Multiples of Terms in Equal Ratios [1961]
- Prop. 5.05: Multiplication of Real Numbers is Left Distributive over Subtraction [1962]
- Prop. 5.06: Multiplication of Real Numbers is Right Distributive over Subtraction [1963]
- Prop. 5.07: Ratios of Equal Magnitudes [1964]
- Prop. 5.08: Relative Sizes of Ratios on Unequal Magnitudes [1965]
- Prop. 5.09: Magnitudes with Same Ratios are Equal [1966]
- Prop. 5.10: Relative Sizes of Magnitudes on Unequal Ratios [1967]
- Prop. 5.11: Equality of Ratios is Transitive [1968]
- Prop. 5.12: Sum of Components of Equal Ratios [1969]
- Prop. 5.13: Relative Sizes of Proportional Magnitudes [1970]
- Prop. 5.14: Relative Sizes of Components of Ratios [1971]
- Prop. 5.15: Ratio Equals its Multiples [1972]
- Prop. 5.16: Proportional Magnitudes are Proportional Alternately [1973]
- Prop. 5.17: Magnitudes Proportional Compounded are Proportional Separated [1974]
- Prop. 5.18: Magnitudes Proportional Separated are Proportional Compounded [1975]
- Prop. 5.19: Proportional Magnitudes have Proportional Remainders [1976]
- Prop. 5.20: Relative Sizes of Successive Ratios [1977]
- Prop. 5.21: Relative Sizes of Elements in Perturbed Proportion [1978]
- Prop. 5.22: Equality of Ratios Ex Aequali [1979]
- Prop. 5.23: Equality of Ratios in Perturbed Proportion [1980]
- Prop. 5.24: Sum of Antecedents of Proportion [1981]
- Prop. 5.25: Sum of Antecedent and Consequent of Proportion [1982]
- Prop. 6.01: Areas of Triangles and Parallelograms Proportional to Base [1987]
- Prop. 6.02: Parallel Line in Triangle Cuts Sides Proportionally [1988]
- Prop. 6.03: Angle Bisector Theorem [1989]
- Prop. 6.04: Equiangular Triangles are Similar [1990]
- Prop. 6.05: Triangles with Proportional Sides are Similar [1991]
- Prop. 6.06: Triangles with One Equal Angle and Two Sides Proportional are Similar [1992]
- Prop. 6.07: Triangles with One Equal Angle and Two Other Sides Proportional are Similar [1993]
- Prop. 6.08: Perpendicular in Right-Angled Triangle makes two Similar Triangles [1994]
- Prop. 6.09: Construction of Part of Line [1995]
- Prop. 6.10: Construction of Similarly Cut Straight Line [1996]
- Prop. 6.11: Construction of Third Proportional Straight Line [1997]
- Prop. 6.12: Construction of Fourth Proportional Straight Line [1998]
- Prop. 6.13: Construction of Mean Proportional [1999]
- Prop. 6.14: Sides of Equal and Equiangular Parallelograms are Reciprocally Proportional [2000]
- Prop. 6.15: Sides of Equiangular Triangles are Reciprocally Proportional [2001]
- Prop. 6.16: Rectangles Contained by Proportional Straight Lines [2002]
- Prop. 6.17: Rectangles Contained by Three Proportional Straight Lines [2003]
- Prop. 6.18: Construction of Similar Polygon [2004]
- Prop. 6.19: Ratio of Areas of Similar Triangles [2005]
- Prop. 6.20: Similar Polygons are Composed of Similar Triangles [2006]
- Prop. 6.21: Similarity of Polygons is Equivalence Relation [2007]
- Prop. 6.22: Similar Figures on Proportional Straight Lines [2008]
- Prop. 6.23: Ratio of Areas of Equiangular Parallelograms [2009]
- Prop. 6.24: Parallelograms About Diameter are Similar [2010]
- Prop. 6.25: Construction of Figure Similar to One and Equal to Another [2011]
- Prop. 6.26: Parallelogram Similar and in Same Angle has Same Diameter [2012]
- Prop. 6.27: Similar Parallelogram on Half a Straight Line [2013]
- Prop. 6.28: Construction of Parallelogram Equal to Given Figure Less a Parallelogram [2014]
- Prop. 6.29: Construction of Parallelogram Equal to Given Figure Exceeding a Parallelogram [2015]
- Prop. 6.30: Construction of Golden Section [2016]
- Prop. 6.31: Similar Figures on Sides of Right-Angled Triangle [2017]
- Prop. 6.32: Triangles with Two Sides Parallel and Equal [2018]
- Prop. 6.33: Angles in Circles have Same Ratio as Arcs [2019]
- Prop. 7.01: Sufficient Condition for Coprimality [2331]
- Prop. 7.02: Greatest Common Divisor of Two Numbers - Euclidean Algorithm [2370]
- Prop. 7.03: Greatest Common Divisor of Three Numbers [2333]
- Prop. 7.04: Natural Number Divisor or Multiple of Divisor of Another [2334]
- Prop. 7.05: Divisors obey Distributive Law [2335]
- Prop. 7.06: Multiples of Divisors Obey Distributive Law [2336]
- Prop. 7.07: Subtraction of Divisors Obeys Distributive Law [2337]
- Prop. 7.08: Subtraction of Multiples of Divisors Obeys Distributive Law [2338]
- Prop. 7.09: Alternate Ratios of Equal Fractions [2339]
- Prop. 7.10: Multiples of Alternate Ratios of Equal Fractions [2340]
- Prop. 7.11: Proportional Numbers have Proportional Differences [2341]
- Prop. 7.12: Ratios of Numbers is Distributive over Addition [2342]
- Prop. 7.13: Proportional Numbers are Proportional Alternately [2343]
- Prop. 7.14: Proportion of Numbers is Transitive [2344]
- Prop. 7.15: Alternate Ratios of Multiples [2345]
- Prop. 7.16: Natural Number Multiplication is Commutative [2346]
- Prop. 7.17: Multiples of Ratios of Numbers [2347]
- Prop. 7.18: Ratios of Multiples of Numbers [2348]
- Prop. 7.19: Relation of Ratios to Products [2349]
- Prop. 7.20: Ratios of Fractions in Lowest Terms [2350]
- Prop. 7.21: Coprime Numbers form Fraction in Lowest Terms [2351]
- Prop. 7.22: Numbers forming Fraction in Lowest Terms are Coprime [2352]
- Prop. 7.23: Divisor of One of Coprime Numbers is Coprime to Other [2353]
- Prop. 7.24: Integer Coprime to all Factors is Coprime to Whole [2354]
- Prop. 7.25: Square of Coprime Number is Coprime [2355]
- Prop. 7.26: Product of Coprime Pairs is Coprime [2356]
- Prop. 7.27: Powers of Coprime Numbers are Coprime [2357]
- Prop. 7.28: Numbers are Coprime iff Sum is Coprime to Both [2358]
- Prop. 7.29: Prime not Divisor implies Coprime [2359]
- Prop. 7.30: Euclidean Lemma [805]
- Prop. 7.31: Existence of Prime Divisors [798]
- Prop. 7.32: Natural Number is Prime or has Prime Factor [2362]
- Prop. 7.33: Least Ratio of Numbers [2363]
- Prop. 7.34: Existence of Lowest Common Multiple [2364]
- Prop. 7.35: Least Common Multiple Divides Common Multiple [2365]
- Prop. 7.36: Least Common Multiple of Three Numbers [2366]
- Prop. 7.37: Integer Divided by Divisor is Integer [2367]
- Prop. 7.38: Divisor is Reciprocal of Divisor of Integer [2368]
- Prop. 7.39: Least Number with Three Given Fractions [2369]
- Prop. 8.01: Geometric Progression with Coprime Extremes is in Lowest Terms [2020]
- Prop. 8.02: Construction of Geometric Progression in Lowest Terms [2021]
- Prop. 8.03: Geometric Progression in Lowest Terms has Coprime Extremes [2022]
- Prop. 8.04: Construction of Sequence of Numbers with Given Ratios [2023]
- Prop. 8.05: Ratio of Products of Sides of Plane Numbers [2024]
- Prop. 8.06: First Element of Geometric Progression not dividing Second [2025]
- Prop. 8.07: First Element of Geometric Progression that divides Last also divides Second [2026]
- Prop. 8.08: Geometric Progressions in Proportion have Same Number of Elements [2027]
- Prop. 8.09: Elements of Geometric Progression between Coprime Numbers [2028]
- Prop. 8.10: Product of Geometric Progressions from One [2029]
- Prop. 8.11: Between two Squares exists one Mean Proportional [2030]
- Prop. 8.12: Between two Cubes exist two Mean Proportionals [2031]
- Prop. 8.13: Powers of Elements of Geometric Progression are in Geometric Progression [2032]
- Prop. 8.14: Number divides Number iff Square divides Square [2033]
- Prop. 8.15: Number divides Number iff Cube divides Cube [2034]
- Prop. 8.16: Number does not divide Number iff Square does not divide Square [2035]
- Prop. 8.17: Number does not divide Number iff Cube does not divide Cube [2036]
- Prop. 8.18: Between two Similar Plane Numbers exists one Mean Proportional [2037]
- Prop. 8.19: Between two Similar Solid Numbers exist two Mean Proportionals [2038]
- Prop. 8.20: Numbers between which exists one Mean Proportional are Similar Plane [2039]
- Prop. 8.21: Numbers between which exist two Mean Proportionals are Similar Solid [2040]
- Prop. 8.22: If First of Three Numbers in Geometric Progression is Square then Third is Square [2041]
- Prop. 8.23: If First of Four Numbers in Geometric Progression is Cube then Fourth is Cube [2042]
- Prop. 8.24: If Ratio of Square to Number is as between Two Squares then Number is Square [2043]
- Prop. 8.25: If Ratio of Cube to Number is as between Two Cubes then Number is Cube [2044]
- Prop. 8.26: Similar Plane Numbers have Same Ratio as between Two Squares [2045]
- Prop. 8.27: Similar Solid Numbers have Same Ratio as between Two Cubes [2046]
- Prop. 9.01: Product of Similar Plane Numbers is Square [2047]
- Prop. 9.02: Numbers whose Product is Square are Similar Plane Numbers [2048]
- Prop. 9.03: Square of Cube Number is Cube [2049]
- Prop. 9.04: Cube Number multiplied by Cube Number is Cube [2050]
- Prop. 9.05: Number multiplied by Cube Number making Cube is itself Cube [2051]
- Prop. 9.06: Number Squared making Cube is itself Cube [2052]
- Prop. 9.07: Product of Composite Number with Number is Solid Number [2053]
- Prop. 9.08: Elements of Geometric Progression from One which are Powers of Number [2054]
- Prop. 9.09: Elements of Geometric Progression from One where First Element is Power of Number [2055]
- Prop. 9.10: Elements of Geometric Progression from One where First Element is not Power of Number [2056]
- Prop. 9.11: Elements of Geometric Progression from One which Divide Later Elements [2057]
- Prop. 9.12: Elements of Geometric Progression from One Divisible by Prime [2058]
- Prop. 9.13: Divisibility of Elements of Geometric Progression from One where First Element is Prime [2059]
- Prop. 9.14: Fundamental Theorem of Arithmetic [800]
- Prop. 9.15: Sum of Pair of Elements of Geometric Progression with Three Elements in Lowest Terms is Coprime to other Element [2061]
- Prop. 9.16: Two Coprime Integers have no Third Integer Proportional [2062]
- Prop. 9.17: Last Element of Geometric Progression with Coprime Extremes has no Integer Proportional as First to Second [2063]
- Prop. 9.18: Condition for Existence of Third Number Proportional to Two Numbers [2064]
- Prop. 9.19: Condition for Existence of Fourth Number Proportional to Three Numbers [2065]
- Prop. 9.20: Infinite Number of Primes [507]
- Prop. 9.21: Sum of Even Numbers is Even [2066]
- Prop. 9.22: Sum of Even Number of Odd Numbers is Even [2067]
- Prop. 9.23: Sum of Odd Number of Odd Numbers is Odd [2068]
- Prop. 9.24: Even Number minus Even Number is Even [2069]
- Prop. 9.25: Even Number minus Odd Number is Odd [2070]
- Prop. 9.26: Odd Number minus Odd Number is Even [2071]
- Prop. 9.27: Odd Number minus Even Number is Odd [2072]
- Prop. 9.28: Odd Number multiplied by Even Number is Even [2073]
- Prop. 9.29: Odd Number multiplied by Odd Number is Odd [2074]
- Prop. 9.30: Odd Divisor of Even Number Also Divides Its Half [2075]
- Prop. 9.31: Odd Number Coprime to Number is also Coprime to its Double [2076]
- Prop. 9.32: Power of Two is Even-Times Even Only [2077]
- Prop. 9.33: Number whose Half is Odd is Even-Times Odd [2078]
- Prop. 9.34: Number neither whose Half is Odd nor Power of Two is both Even-Times Even and Even-Times Odd [2079]
- Prop. 9.35: Sum of Geometric Progression [1123]
- Prop. 9.36: Theorem of Even Perfect Numbers (first part) [2080]
- Properties of a Complex Scalar Product [6251]
- Properties of a Group Homomorphism [680]
- Properties of a Real Scalar Product [6214]
- Properties of Cosets [829]
- Properties of Ordinal Numbers [724]
- Properties of the Absolute Value [619]
- Properties of Transitive Sets [721]
- Pythagorean Identity [1794]
- Quotient of Convergent Complex Sequences [1722]
- Quotient of Convergent Real Sequences [1142]
- Quotient Space [71]
- Ratio of Two Ratios [6636]
- Ratio Test For Absolutely Convergent Complex Series [1729]
- Ratio Test For Absolutely Convergent Series [1337]
- Rational Cauchy Sequence Members Are Bounded [1489]
- Rational Cauchy Sequences Build a Commutative Group With Respect To Addition [1518]
- Rational Cauchy Sequences Build a Commutative Monoid With Respect To Multiplication [1520]
- Rational Functions are Continuous [6684]
- Rational Numbers are Countable [6659]
- Rational Numbers are Dense in the Real Numbers [6666]
- Rational Powers of Positive Numbers [1622]
- Real Numbers are Uncountable [6661]
- Real Numbers Can Be Approximated by Rational Numbers [1127]
- Real Polynomials of Odd Degree Have at Least One Real Root [6694]
- Real Sequences Contain Monotonic Subsequences [6654]
- Rearrangement of Absolutely Convergent Series [1364]
- Rearrangement of Convergent Series [1366]
- Reciprocity Law of Falling And Rising Factorial Powers [1412]
- Reciprocity of Complex Exponential Function, Non-Zero Property [1738]
- Reciprocity of Exponential Function of General Base, Non-Zero Property [1614]
- Reciprocity of Exponential Function, Non-Zero Property [1417]
- Rectangle as a Special Case of a Parallelogram [936]
- Recursive Formula for Binomial Coefficients [994]
- Relationship between Limit, Limit Superior, and Limit Inferior of a Real Sequence [6677]
- Relationship Between Planarity and Biconnectivity of Graphs [1229]
- Relationship Between Planarity and Connectivity of Graphs [1230]
- Relationship Between the Greatest Common Divisor and the Least Common Multiple [1281]
- Replacing Mutually Independent Events by Their Complements [1810]
- Representing Real Cosine by Complex Exponential Function [1786]
- Representing Real Sine by Complex Exponential Function [1788]
- Reverse Triangle Inequalities [6637]
- Rhombus as a Special Case of a Parallelogram [935]
- Riemann Integral for Step Functions [1752]
- Riemann Upper and Riemann Lower Integrals for Bounded Real Functions [1761]
- Right-Distributivity Law For Natural Numbers [1436]
- Rule of Combining Different Sets of Indices [1119]
- Rules for Exponentiation [676]
- Rules of Calculation with Inequalities [594]
- Second Law of Planetary Motion [6305]
- Sieve for Twin Primes [6403]
- Similar Triangles I [927]
- Simple Binomial Identities [1839]
- Simulating LOOP Programs Using WHILE Programs [1199]
- Simulating WHILE Programs Using GOTO Programs (and vice versa) [1201]
- Size of an \(r\)-Regular Graph with \(n\) Vertices [6355]
- Spectrum Function of Commutative Rings [6249]
- Splitting a Graph with Even Degree Vertices into Cycles [6382]
- Square as a Special Case of a Rhombus [966]
- Square Roots [1161]
- Subgroups of Cyclic Groups [817]
- Subgroups of Finite Cyclic Groups [825]
- Subsets of Finite Sets [986]
- Subsets of Natural Numbers Relatively Prime to a Natural Number are Divisor-Closed [6407]
- Successor of Oridinal [774]
- Sum and Difference of Two Ratios [6632]
- Sum of a Convergent Real Sequence and a Real Sequence Tending to Infininty [6651]
- Sum of Arguments of Hyperbolic Cosine [6689]
- Sum of Arguments of Hyperbolic Sine [6690]
- Sum of Arithmetic Progression [1117]
- Sum of Binomial Coefficients [1405]
- Sum of Binomial Coefficients I [1841]
- Sum of Binomial Coefficients II [1843]
- Sum of Binomial Coefficients III [1845]
- Sum of Binomial Coefficients IV [6628]
- Sum of Consecutive Natural Numbers From Zero to a Given Number [6623]
- Sum of Convergent Complex Sequences [1711]
- Sum of Convergent Real Sequences [1131]
- Sum of Convergent Real Series [6643]
- Sum of Cube Numbers [6626]
- Sum of Odd Numbers from One to a Given Number [6624]
- Sum of Squares [6625]
- Sum of Two Supplemental Angles Equals Two Right Angles [765]
- Supremum Property, Infimum Property [1756]
- The absolute value makes the set of rational numbers a metric space. [1090]
- The distance of complex numbers makes complex numbers a metric space. [1253]
- The distance of real numbers makes real numbers a metric space. [618]
- The Fundamental Counting Principle [111]
- The General Perturbation Method [1121]
- The Proving Principle by Contradiction [744]
- The Proving Principle By Contraposition [1330]
- The Proving Principle of Complete Induction (Variant 1) [657]
- The set of WHILE-computable functions is included in the set of partially WHILE-computable functions [1196]
- The supplemental angle of a right angle is another right angle. [654]
- Theorem of Bolzano-Weierstrass [6608]
- Theorem of Large Numbers for Relative Frequencies [1838]
- Third Law of Planetary Motion [6306]
- Time Dilation, Lorentz Factor [6297]
- Transitivity of the Order Relation of Natural Numbers [1549]
- Triangle Inequality [588]
- Triangulation of an N-gon and Sum of Interior Angles [929]
- Triangulation of Quadrilateral and Sum of Angles [928]
- Trichotomy of Ordinals [729]
- Trichotomy of the Order Relation for Natural Numbers [1552]
- Uncountable and Countable Subsets of Natural Numbers [6678]
- Union of Countable Many Countable Sets [796]
- Unique Representation of Real Numbers as \(b\)-adic Fractions [1126]
- Unique Solvability of \(a+x=b\) [516]
- Unique Solvability of \(ax=b\) [517]
- Uniqueness Lemma of a Finite Basis [1039]
- Uniqueness of 1 [48]
- Uniqueness of Complex Zero [1686]
- Uniqueness of Integer Zero [1682]
- Uniqueness of Natural Zero [1680]
- Uniqueness of Negative Numbers [50]
- Uniqueness Of Predecessors Of Natural Numbers [1542]
- Uniqueness of Rational Zero [1684]
- Uniqueness of Real Zero [43]
- Uniqueness of Reciprocal Numbers [51]
- Uniqueness of the Limit of a Sequence [1129]
- Unit Circle [1749]
- Unit Ring of All Rational Cauchy Sequences [1101]
- Urn Model With Replacement [1799]
- Urn Model Without Replacement [1797]
- Value of Zero to the Power of X [6718]
- Well-Ordering Principle [698]
- When is it possible to find a separating cycle in a biconnected graph, given a non-separating cycle? [1233]
- (Real) Exponential Function Is Always Positive
- Proof (related to "(Real) Exponential Function Is Always Positive") [1420]
- \(-(-x)=x\)
- Elementary Proof (related to "\(-(-x)=x\)") [526]
- \(-(x+y)=-x-y\)
- Elementary Proof (related to "\(-(x+y)=-x-y\)") [537]
- \(-0=0\)
- Direct Proof (related to "\(-0=0\)") [502]
- \((-x)(-y)=xy\)
- Elementary Proof (related to "\((-x)(-y)=xy\)") [533]
- \((-x)y=-(xy)\)
- Elementary Proof (related to "\((-x)y=-(xy)\)") [532]
- \((x^{-1})^{-1}=x\)
- Elementary Proof (related to "\((x^{-1})^{-1}=x\)") [539]
- \((xy)^{-1}=x^{-1}y^{-1}\)
- Elementary Proof (related to "\((xy)^{-1}=x^{-1}y^{-1}\)") [538]
- \(\epsilon\)-\(\delta\) Definition of Continuity
- Proof (related to "\(\epsilon\)-\(\delta\) Definition of Continuity") [1255]
- \(\exp(0)=1\)
- Proof (related to "\(\exp(0)=1\)") [1424]
- \(\exp(0)=1\) (Complex Case)
- Proof (related to "\(\exp(0)=1\) (Complex Case)") [1740]
- \(0x=0\)
- Elementary Proof (related to "\(0x=0\)") [527]
- \(1^{-1}=1\).
- Direct Proof (related to "\(1^{-1}=1\).") [501]
- \(b\)-Adic Fractions Are Real Cauchy Sequences
- Proof (related to "\(b\)-Adic Fractions Are Real Cauchy Sequences") [1125]
- A Criterion for Isosceles Triangles
- Proof (related to "A Criterion for Isosceles Triangles") [750]
- A General Criterion for the Convergence of Infinite Complex Series
- Proof (related to "A General Criterion for the Convergence of Infinite Complex Series") [1726]
- A General Criterion for the Convergence of Infinite Series
- Proof (related to "A General Criterion for the Convergence of Infinite Series") [1149]
- A Necessary and a Sufficient Condition for Riemann Integrable Functions
- Proof (related to "A Necessary and a Sufficient Condition for Riemann Integrable Functions") [1765]
- A Necessary But Not Sufficient Condition For Convergence Of Infinite Series
- Proof (related to "A Necessary But Not Sufficient Condition For Convergence Of Infinite Series") [1265]
- A product of two real numbers is zero if and only if at least one of these numbers is zero.
- Proof (related to "A product of two real numbers is zero if and only if at least one of these numbers is zero.") [529]
- A proposition cannot be both, true and false
- Proof (related to "A proposition cannot be both, true and false") [1326]
- A proposition cannot be equivalent to its negation
- Proof (related to "A proposition cannot be equivalent to its negation") [1327]
- Addition of Complex Numbers Is Associative
- Proof (related to "Addition of Complex Numbers Is Associative") [1659]
- Addition of Complex Numbers Is Commutative
- Proof (related to "Addition of Complex Numbers Is Commutative") [1661]
- Addition of Integers
- Proof (related to "Addition of Integers") [1530]
- Addition of Integers Is Associative
- Proof (related to "Addition of Integers Is Associative") [1444]
- Algebraic Proof (related to "Addition of Integers Is Associative") [1445]
- Addition of Integers Is Cancellative
- Proof (related to "Addition of Integers Is Cancellative") [1463]
- Addition of Integers Is Commutative
- Proof (related to "Addition of Integers Is Commutative") [1461]
- Addition Of Natural Numbers
- Proof (related to "Addition Of Natural Numbers") [1544]
- Addition Of Natural Numbers Is Associative
- Proof (related to "Addition Of Natural Numbers Is Associative") [1429]
- Addition of Natural Numbers Is Cancellative
- Proof (related to "Addition of Natural Numbers Is Cancellative") [1433]
- Addition of Natural Numbers Is Cancellative With Respect To Inequalities
- Proof (related to "Addition of Natural Numbers Is Cancellative With Respect To Inequalities") [1554]
- Addition of Natural Numbers Is Commutative
- Proof (related to "Addition of Natural Numbers Is Commutative") [1431]
- Addition of Rational Cauchy Sequences
- Proof (related to "Addition of Rational Cauchy Sequences") [1487]
- Addition of Rational Cauchy Sequences Is Associative
- Proof (related to "Addition of Rational Cauchy Sequences Is Associative") [1495]
- Addition of Rational Cauchy Sequences Is Cancellative
- Proof (related to "Addition of Rational Cauchy Sequences Is Cancellative") [1570]
- Addition of Rational Cauchy Sequences Is Commutative
- Proof (related to "Addition of Rational Cauchy Sequences Is Commutative") [1497]
- Addition Of Rational Numbers
- Proof (related to "Addition Of Rational Numbers") [1515]
- Addition of Rational Numbers Is Associative
- Proof (related to "Addition of Rational Numbers Is Associative") [1468]
- Addition of Rational Numbers Is Cancellative
- Proof (related to "Addition of Rational Numbers Is Cancellative") [1472]
- Addition of Rational Numbers Is Commutative
- Proof (related to "Addition of Rational Numbers Is Commutative") [1470]
- Addition of Real Numbers
- Proof (related to "Addition of Real Numbers") [1526]
- Addition Of Real Numbers Is Associative
- Proof (related to "Addition Of Real Numbers Is Associative") [1527]
- Addition of Real Numbers Is Cancellative
- Proof (related to "Addition of Real Numbers Is Cancellative") [1576]
- Addition Of Real Numbers Is Commutative
- Proof (related to "Addition Of Real Numbers Is Commutative") [1528]
- Algebraic Structure of Complex Numbers Together with Addition
- Proof (related to "Algebraic Structure of Complex Numbers Together with Addition") [1667]
- Algebraic Structure of Complex Numbers Together with Addition and Multiplication
- Proof (related to "Algebraic Structure of Complex Numbers Together with Addition and Multiplication") [1691]
- Algebraic Structure of Integers Together with Addition
- Proof (related to "Algebraic Structure of Integers Together with Addition") [1655]
- Proof (related to "Algebraic Structure of Integers Together with Addition") [1656]
- Algebraic Structure of Integers Together with Addition and Multiplication
- Proof (related to "Algebraic Structure of Integers Together with Addition and Multiplication") [1032]
- Algebraic Structure Of Natural Numbers Together With Addition
- Proof (related to "Algebraic Structure Of Natural Numbers Together With Addition") [843]
- Algebraic Structure Of Natural Numbers Together With Multiplication
- Proof (related to "Algebraic Structure Of Natural Numbers Together With Multiplication") [1442]
- Algebraic Structure of Non-Zero Complex Numbers Together with Multiplication
- Proof (related to "Algebraic Structure of Non-Zero Complex Numbers Together with Multiplication") [1689]
- Algebraic Structure of Non-Zero Rational Numbers Together with Multiplication
- Proof (related to "Algebraic Structure of Non-Zero Rational Numbers Together with Multiplication") [1650]
- Algebraic Structure of Non-Zero Real Numbers Together with Multiplication
- Proof (related to "Algebraic Structure of Non-Zero Real Numbers Together with Multiplication") [1642]
- Algebraic Structure of Rational Numbers Together with Addition
- Proof (related to "Algebraic Structure of Rational Numbers Together with Addition") [1648]
- Algebraic Structure of Rational Numbers Together with Addition and Multiplication
- Proof (related to "Algebraic Structure of Rational Numbers Together with Addition and Multiplication") [1653]
- Proof (related to "Algebraic Structure of Rational Numbers Together with Addition and Multiplication") [1652]
- Algebraic Structure of Real Numbers Together with Addition
- Proof (related to "Algebraic Structure of Real Numbers Together with Addition") [1641]
- Algebraic Structure of Real Numbers Together with Addition and Multiplication
- Proof (related to "Algebraic Structure of Real Numbers Together with Addition and Multiplication") [1643]
- Proof (related to "Algebraic Structure of Real Numbers Together with Addition and Multiplication") [1644]
- All Cauchy Sequences Converge in the Set of Real Numbers (Completeness Principle)
- Proof (related to "All Cauchy Sequences Converge in the Set of Real Numbers (Completeness Principle)") [1128]
- All Convergent Real Sequences Are Cauchy Sequences
- Proof (related to "All Convergent Real Sequences Are Cauchy Sequences") [1395]
- Alternating Sum of Binomial Coefficients
- Proof (related to "Alternating Sum of Binomial Coefficients") [1408]
- Angles and Sides in a Triangle V
- Proof (related to "Angles and Sides in a Triangle V") [904]
- Any Positive Characteristic Is a Prime Number
- Proof by Contradiction (related to "Any Positive Characteristic Is a Prime Number") [883]
- Approximability of Continuous Real Functions On Closed Intervals By Step Functions
- Proof by Construction (related to "Approximability of Continuous Real Functions On Closed Intervals By Step Functions") [6620]
- Barycentric Coordinates, Barycenter
- Proof (related to "Barycentric Coordinates, Barycenter") [6284]
- Basic Rules of Manipulating Finite Sums
- Proof (related to "Basic Rules of Manipulating Finite Sums") [1115]
- Basis Arithmetic Operations Involving Differentiable Functions, Product Rule, Quotient Rule
- Proof (related to "Basis Arithmetic Operations Involving Differentiable Functions, Product Rule, Quotient Rule") [1376]
- Bayes' Theorem
- Proof (related to "Bayes' Theorem") [1833]
- Bernoulli's Inequality
- Proof by Induction (related to "Bernoulli's Inequality") [1338]
- Biconnectivity is a Necessary Condition for a Hamiltonian Graph
- Proof (related to "Biconnectivity is a Necessary Condition for a Hamiltonian Graph") [6397]
- Binomial Distribution
- Proof (related to "Binomial Distribution") [1818]
- Binomial Theorem
- Proof by Induction (related to "Binomial Theorem") [1398]
- Boolean Function
- Proof (related to "Boolean Function") [1317]
- Bounds for the Minimal Tree Decomposability
- Proof (related to "Bounds for the Minimal Tree Decomposability") [6395]
- Calculating the Number of Distinct Positive Divisors
- Proof (related to "Calculating the Number of Distinct Positive Divisors") [1303]
- Calculating with Complex Conjugates
- Proof (related to "Calculating with Complex Conjugates") [1252]
- Calculation Rules for General Powers
- Proof (related to "Calculation Rules for General Powers") [1629]
- Calculation Rules for the Big O Notation
- Proof (related to "Calculation Rules for the Big O Notation") [1168]
- Cancellation Law
- Proof (related to "Cancellation Law") [824]
- Cardinal Number
- Proof (related to "Cardinal Number") [981]
- Cauchy Product of Absolutely Convergent Series
- Proof (related to "Cauchy Product of Absolutely Convergent Series") [1396]
- Cauchy Product of Convergent Series Is Not Necessarily Convergent
- Proof (related to "Cauchy Product of Convergent Series Is Not Necessarily Convergent") [1393]
- Characteristic String
- Proof (related to "Characteristic String") [1002]
- Characterization of Bipartite Graphs
- Proof (related to "Characterization of Bipartite Graphs") [6371]
- Characterization of Closed Sets by Limits of Sequences
- Proof (related to "Characterization of Closed Sets by Limits of Sequences") [6586]
- Characterization of Cutvertices
- Proof (related to "Characterization of Cutvertices") [1239]
- Characterization of Eulerian Graphs
- Proof (related to "Characterization of Eulerian Graphs") [6384]
- Characterization of Independent Events
- Proof (related to "Characterization of Independent Events") [1805]
- Characterization of Independent Events II
- Proof (related to "Characterization of Independent Events II") [1807]
- Characterization of Planar Hamiltonian Graphs
- Proof (related to "Characterization of Planar Hamiltonian Graphs") [6401]
- Characterization of Semi-Eulerian Graphs
- Proof (related to "Characterization of Semi-Eulerian Graphs") [6388]
- Closed Formula For Binomial Coefficients
- Combinatorial Proof (related to "Closed Formula For Binomial Coefficients") [1401]
- Closed n-Dimensional Cuboids Are Compact
- Proof (related to "Closed n-Dimensional Cuboids Are Compact") [6588]
- Closed Real Intervals Are Compact
- Proof (related to "Closed Real Intervals Are Compact") [6584]
- Closed Subsets of Compact Sets are Compact
- Proof (related to "Closed Subsets of Compact Sets are Compact") [6595]
- Compact Subset of Real Numbers Contains its Maximum and its Minimum
- Proof (related to "Compact Subset of Real Numbers Contains its Maximum and its Minimum") [6599]
- Compact Subsets of Metric Spaces Are Bounded and Closed
- Proof (related to "Compact Subsets of Metric Spaces Are Bounded and Closed") [6590]
- Comparing Natural Numbers Using the Concept of Addition
- Proof (related to "Comparing Natural Numbers Using the Concept of Addition") [1548]
- Completeness Principle For Complex Numbers
- Proof (related to "Completeness Principle For Complex Numbers") [1710]
- Complex Cauchy Sequences Vs. Real Cauchy Sequences
- Proof (related to "Complex Cauchy Sequences Vs. Real Cauchy Sequences") [1706]
- Complex Conjugate of Complex Exponential Function
- Proof (related to "Complex Conjugate of Complex Exponential Function") [1748]
- Complex Exponential Function
- Proof (related to "Complex Exponential Function") [1731]
- Complex Numbers are a Field Extension of Real Numbers
- Proof (related to "Complex Numbers are a Field Extension of Real Numbers") [1244]
- Complex Numbers are Two-Dimensional and the Complex Numbers \(1\) and Imaginary Unit \(i\) Form Their Basis
- Proof (related to "Complex Numbers are Two-Dimensional and the Complex Numbers \(1\) and Imaginary Unit \(i\) Form Their Basis") [1699]
- Complex Numbers as a Vector Space Over the Field of Real Numbers
- Proof (related to "Complex Numbers as a Vector Space Over the Field of Real Numbers") [1695]
- Composition of Continuous Functions at a Single Point
- Proof (related to "Composition of Continuous Functions at a Single Point") [1607]
- Composition of Relations (Sometimes) Preserves Their Left-Total Property
- Proof (related to "Composition of Relations (Sometimes) Preserves Their Left-Total Property") [1313]
- Composition of Relations Preserves Their Right-Uniqueness Property
- Proof (related to "Composition of Relations Preserves Their Right-Uniqueness Property") [1311]
- Composition of Total Functions
- Proof (related to "Composition of Total Functions") [1315]
- Compositions of Continuous Functions on a Whole Domain
- Proof (related to "Compositions of Continuous Functions on a Whole Domain") [1609]
- Connection between Quotient, Remainder, Modulo and Floor Function
- Proof (related to "Connection between Quotient, Remainder, Modulo and Floor Function") [1285]
- Connectivity Is an Equivalence Relation - Components Are a Partition of a Graph
- Proof (related to "Connectivity Is an Equivalence Relation - Components Are a Partition of a Graph") [1222]
- Construction of a Light Clock
- Proof (related to "Construction of a Light Clock") [6276]
- Construction of Fields from Integral Domains
- Proof (related to "Construction of Fields from Integral Domains") [889]
- Construction of Groups from Commutative and Cancellative Semigroups
- Proof (related to "Construction of Groups from Commutative and Cancellative Semigroups") [840]
- Continuity of Complex Exponential Function
- Proof (related to "Continuity of Complex Exponential Function") [1744]
- Continuity of Cosine and Sine
- Proof (related to "Continuity of Cosine and Sine") [1785]
- Continuity of Exponential Function
- Proof (related to "Continuity of Exponential Function") [1425]
- Continuity of Exponential Function of General Base
- Proof (related to "Continuity of Exponential Function of General Base") [1611]
- Continuous Functions Mapping Compact Domains to Real Numbers are Bounded
- Proof (related to "Continuous Functions Mapping Compact Domains to Real Numbers are Bounded") [6607]
- Continuous Functions Mapping Compact Domains to Real Numbers Take Maximum and Minimum Values on these Domains
- Proof (related to "Continuous Functions Mapping Compact Domains to Real Numbers Take Maximum and Minimum Values on these Domains") [6605]
- Continuous Functions on Compact Domains are Uniformly Continuous
- Proof (related to "Continuous Functions on Compact Domains are Uniformly Continuous") [6615]
- Continuous Real Functions on Closed Intervals are Riemann-Integrable
- Proof (related to "Continuous Real Functions on Closed Intervals are Riemann-Integrable") [6621]
- Continuous Real Functions on Closed Intervals are Uniformly Continuous
- Proof (related to "Continuous Real Functions on Closed Intervals are Uniformly Continuous") [6617]
- Proof by Contradiction (related to "Continuous Real Functions on Closed Intervals are Uniformly Continuous") [6618]
- Contraposition of Cancellative Law for Adding Integers
- Proof by Contraposition (related to "Contraposition of Cancellative Law for Adding Integers") [1562]
- Contraposition of Cancellative Law for Adding Natural Numbers
- Proof by Contraposition (related to "Contraposition of Cancellative Law for Adding Natural Numbers") [1546]
- Contraposition of Cancellative Law for Adding Rational Numbers
- Proof by Contraposition (related to "Contraposition of Cancellative Law for Adding Rational Numbers") [1566]
- Contraposition of Cancellative Law for Adding Real Numbers
- Proof (related to "Contraposition of Cancellative Law for Adding Real Numbers") [1579]
- Contraposition of Cancellative Law for Multiplying Integers
- Proof by Contraposition (related to "Contraposition of Cancellative Law for Multiplying Integers") [1564]
- Contraposition of Cancellative Law for Multiplying Natural Numbers
- Proof by Contraposition (related to "Contraposition of Cancellative Law for Multiplying Natural Numbers") [1560]
- Contraposition of Cancellative Law for Multiplying Rational Numbers
- Proof by Contraposition (related to "Contraposition of Cancellative Law for Multiplying Rational Numbers") [1568]
- Contraposition of Cancellative Law of for Multiplying Real Numbers
- Proof (related to "Contraposition of Cancellative Law of for Multiplying Real Numbers") [1581]
- Convergence Behavior of the Sequence \((b^n)\)
- Proof (related to "Convergence Behavior of the Sequence \((b^n)\)") [1352]
- Convergence Behaviour of Absolutely Convergent Series
- Proof (related to "Convergence Behaviour of Absolutely Convergent Series") [1269]
- Convergence of Alternating Harmonic Series
- Proof (related to "Convergence of Alternating Harmonic Series") [1368]
- Convergence of Complex Conjugate Sequence
- Proof (related to "Convergence of Complex Conjugate Sequence") [1708]
- Convergence of Infinite Series with Non-Negative Terms
- Proof (related to "Convergence of Infinite Series with Non-Negative Terms") [1159]
- Convergent Complex Sequences Are Bounded
- Proof (related to "Convergent Complex Sequences Are Bounded") [1717]
- Convergent Complex Sequences Vs. Convergent Real Sequences
- Proof (related to "Convergent Complex Sequences Vs. Convergent Real Sequences") [1703]
- Convergent Rational Sequences With Limit \(0\) Are a Subgroup of Rational Cauchy Sequences With Respect To Addition
- Proof (related to "Convergent Rational Sequences With Limit \(0\) Are a Subgroup of Rational Cauchy Sequences With Respect To Addition") [1523]
- Convergent Rational Sequences With Limit \(0\) Are an Ideal Of the Ring of Rational Cauchy Sequences
- Proof (related to "Convergent Rational Sequences With Limit \(0\) Are an Ideal Of the Ring of Rational Cauchy Sequences") [1525]
- Convergent Rational Sequences With Limit \(0\) Are Rational Cauchy Sequences
- Proof (related to "Convergent Rational Sequences With Limit \(0\) Are Rational Cauchy Sequences") [1517]
- Convergent Sequence together with Limit is a Compact Subset of Metric Space
- Proof (related to "Convergent Sequence together with Limit is a Compact Subset of Metric Space") [6578]
- Convergent Sequence without Limit Is Not a Compact Subset of Metric Space
- Proof by Explicit Counterexample (related to "Convergent Sequence without Limit Is Not a Compact Subset of Metric Space") [6580]
- Convergent Sequences are Bounded
- Proof (related to "Convergent Sequences are Bounded") [6593]
- Proof (related to "Convergent Sequences are Bounded") [1138]
- Convergent Sequences are Cauchy Sequences
- Proof (related to "Convergent Sequences are Cauchy Sequences") [1074]
- Cor. 10.111: Thirteen Irrational Straight-Lines of Different Order
- Proof (related to "Cor. 10.111: Thirteen Irrational Straight-Lines of Different Order") [6565]
- Cor. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides
- Proof (related to "Cor. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides") [6568]
- Cor. 12.17: Construction of Polyhedron in Outer of Concentric Spheres
- Proof (related to "Cor. 12.17: Construction of Polyhedron in Outer of Concentric Spheres") [6569]
- Cor. 6.08: Perpendicular in Right-Angled Triangle makes two Similar Triangles, Geometric Mean Theorem, Mean Proportion
- Proof (related to "Cor. 6.08: Perpendicular in Right-Angled Triangle makes two Similar Triangles, Geometric Mean Theorem, Mean Proportion") [6553]
- Cor. 7.02: Any Divisor Dividing Two Numbers Divides Their Greatest Common Divisor
- Proof (related to "Cor. 7.02: Any Divisor Dividing Two Numbers Divides Their Greatest Common Divisor") [6415]
- Corollaries From the Group Axioms
- Proof (related to "Corollaries From the Group Axioms") [556]
- Counting the Set's Elements Using Its Partition
- Proof (related to "Counting the Set's Elements Using Its Partition") [991]
- Criteria for Subgroups
- Proof (related to "Criteria for Subgroups") [812]
- Criterion for Alternating Infinite Series
- Proof (related to "Criterion for Alternating Infinite Series") [1267]
- Cyclic Groups are Abelian
- Proof (related to "Cyclic Groups are Abelian") [814]
- Def. 7.08: Even-Times-Even Number
- Proof (related to "Def. 7.08: Even-Times-Even Number") [6409]
- Def. 7.09: Even-Times-Odd Number
- Proof (related to "Def. 7.09: Even-Times-Odd Number") [6410]
- Def. 7.10: Odd-Times-Odd Number
- Proof (related to "Def. 7.10: Odd-Times-Odd Number") [6411]
- Definition of Continuity Using Open Sets
- Proof (related to "Definition of Continuity Using Open Sets") [6600]
- Definition of Integers
- Proof (related to "Definition of Integers") [845]
- Definition of Rational Numbers
- Proof (related to "Definition of Rational Numbers") [1034]
- Definition of Real Numbers
- Proof (related to "Definition of Real Numbers") [1106]
- Definition of the Metric Space \(\mathbb R^n\), Euclidean Norm
- Proof (related to "Definition of the Metric Space \(\mathbb R^n\), Euclidean Norm") [1207]
- Derivative of a Constant Function
- Proof (related to "Derivative of a Constant Function") [1373]
- Derivative of a Linear Function \(ax+b\)
- Proof (related to "Derivative of a Linear Function \(ax+b\)") [1380]
- Direct Proof (related to "Derivative of a Linear Function \(ax+b\)") [1379]
- Difference of Convergent Complex Sequences
- Proof (related to "Difference of Convergent Complex Sequences") [1721]
- Difference of Convergent Real Sequences
- Proof (related to "Difference of Convergent Real Sequences") [1134]
- Difference of Convergent Real Series
- Proof (related to "Difference of Convergent Real Series") [6646]
- Direct Comparison Test For Absolutely Convergent Complex Series (Majorant Criterion)
- Proof (related to "Direct Comparison Test For Absolutely Convergent Complex Series (Majorant Criterion)") [1728]
- Direct Comparison Test For Absolutely Convergent Series (Majorant Criterion)
- Proof (related to "Direct Comparison Test For Absolutely Convergent Series (Majorant Criterion)") [1271]
- Direct Comparison Test For Divergence Series
- Proof by Contradiction (related to "Direct Comparison Test For Divergence Series") [1336]
- Discovery of Irrational Numbers
- Geometric Proof (related to "Discovery of Irrational Numbers") [1097]
- Distance in Normed Vector Spaces
- Proof (related to "Distance in Normed Vector Spaces") [848]
- Distributivity Law for Complex Numbers
- Proof (related to "Distributivity Law for Complex Numbers") [1679]
- Distributivity Law For Integers
- Proof (related to "Distributivity Law For Integers") [1467]
- Distributivity Law For Natural Numbers
- Proof by Induction (related to "Distributivity Law For Natural Numbers") [1031]
- Distributivity Law For Rational Cauchy Sequences
- Proof (related to "Distributivity Law For Rational Cauchy Sequences") [1507]
- Distributivity Law For Rational Numbers
- Proof (related to "Distributivity Law For Rational Numbers") [1492]
- Distributivity Law For Real Numbers
- Proof (related to "Distributivity Law For Real Numbers") [1582]
- Divergence of Harmonic Series
- Proof (related to "Divergence of Harmonic Series") [1334]
- Divisibility Laws
- Direct Proof (related to "Divisibility Laws") [514]
- Divisibility of Principal Ideals
- Proof (related to "Divisibility of Principal Ideals") [1067]
- Division with Quotient and Remainder
- Proof (related to "Division with Quotient and Remainder") [819]
- Divisors of a Product Of Many Factors, Co-Prime to All But One Factor, Divide This Factor
- Proof (related to "Divisors of a Product Of Many Factors, Co-Prime to All But One Factor, Divide This Factor") [1301]
- Divisors of a Product Of Two Factors, Co-Prime to One Factor Divide the Other Factor
- Proof (related to "Divisors of a Product Of Two Factors, Co-Prime to One Factor Divide the Other Factor") [1294]
- Divisors of Integers
- Proof (related to "Divisors of Integers") [1274]
- Double Summation
- Proof (related to "Double Summation") [1426]
- Dual Graph of a All Faces Contained in a Planar Hamiltonian Cycle is a Tree
- Proof (related to "Dual Graph of a All Faces Contained in a Planar Hamiltonian Cycle is a Tree") [6399]
- Equivalence of Set Inclusion and Element Inclusion of Ordinals
- Proof (related to "Equivalence of Set Inclusion and Element Inclusion of Ordinals") [776]
- Equivalency of Vectors in Vector Space If their Difference Forms a Subspace
- Proof (related to "Equivalency of Vectors in Vector Space If their Difference Forms a Subspace") [6329]
- Equivalent Knot Diagrams
- Proof (related to "Equivalent Knot Diagrams") [6361]
- Equivalent Statements Regarding Parallel Lines
- Proof (related to "Equivalent Statements Regarding Parallel Lines") [918]
- Estimate for the Remainder Term of Complex Exponential Function
- Proof (related to "Estimate for the Remainder Term of Complex Exponential Function") [1733]
- Estimate for the Remainder Term of Exponential Function
- Proof (related to "Estimate for the Remainder Term of Exponential Function") [1362]
- Euler's Formula
- Proof (related to "Euler's Formula") [1784]
- Even Number of Vertices with an Odd Degree in Finite Digraphs
- Elementary Proof (related to "Even Number of Vertices with an Odd Degree in Finite Digraphs") [569]
- Even Number of Vertices with an Odd Degree in Finite Graphs
- Proof (related to "Even Number of Vertices with an Odd Degree in Finite Graphs") [1176]
- Eveness of the Cosine of a Real Variable
- Proof (related to "Eveness of the Cosine of a Real Variable") [1791]
- Every Bounded Real Sequence has a Convergent Subsequence
- Proof (related to "Every Bounded Real Sequence has a Convergent Subsequence") [1154]
- Every Contraposition is a Tautology
- Proof (related to "Every Contraposition is a Tautology") [1329]
- Every Distance Is Positive Definite
- Direct Proof (related to "Every Distance Is Positive Definite") [616]
- Every Natural Number Is Greater or Equal Zero
- Proof (related to "Every Natural Number Is Greater or Equal Zero") [1557]
- Every Proposition Implies Itself
- Proof (related to "Every Proposition Implies Itself") [1324]
- Exchanging the Limit of Function Values with the Function Value of the Limit of Arguments
- Proof (related to "Exchanging the Limit of Function Values with the Function Value of the Limit of Arguments") [6711]
- Existence of Arbitrarily Small Positive Rational Numbers
- Proof (related to "Existence of Arbitrarily Small Positive Rational Numbers") [1847]
- Existence of Arbitrarily Small Powers
- Proof (related to "Existence of Arbitrarily Small Powers") [1351]
- Existence of Complex One (Neutral Element of Multiplication of Complex Numbers)
- Proof (related to "Existence of Complex One (Neutral Element of Multiplication of Complex Numbers)") [1674]
- Existence of Complex Zero (Neutral Element of Addition of Complex Numbers)
- Proof (related to "Existence of Complex Zero (Neutral Element of Addition of Complex Numbers)") [1663]
- Existence of Integer One (Neutral Element of Multiplication of Integers)
- Proof (related to "Existence of Integer One (Neutral Element of Multiplication of Integers)") [1459]
- Existence of Integer Zero (Neutral Element of Addition of Integers)
- Proof (related to "Existence of Integer Zero (Neutral Element of Addition of Integers)") [1453]
- Existence of Inverse Complex Numbers With Respect to Addition
- Proof (related to "Existence of Inverse Complex Numbers With Respect to Addition") [1665]
- Existence of Inverse Complex Numbers With Respect to Multiplication
- Proof (related to "Existence of Inverse Complex Numbers With Respect to Multiplication") [1676]
- Existence of Inverse Integers With Respect to Addition
- Proof (related to "Existence of Inverse Integers With Respect to Addition") [1513]
- Existence of Inverse Rational Cauchy Sequences With Respect to Addition
- Proof (related to "Existence of Inverse Rational Cauchy Sequences With Respect to Addition") [1510]
- Existence of Inverse Rational Numbers With Respect to Addition
- Proof (related to "Existence of Inverse Rational Numbers With Respect to Addition") [1512]
- Existence of Inverse Rational Numbers With Respect to Multiplication
- Proof (related to "Existence of Inverse Rational Numbers With Respect to Multiplication") [1677]
- Existence of Inverse Real Numbers With Respect to Addition
- Proof (related to "Existence of Inverse Real Numbers With Respect to Addition") [1587]
- Existence of Inverse Real Numbers With Respect to Multiplication
- Proof (related to "Existence of Inverse Real Numbers With Respect to Multiplication") [1651]
- Existence of Natural Numbers Exceeding Positive Real Numbers
- Proof (related to "Existence of Natural Numbers Exceeding Positive Real Numbers") [1341]
- Existence of Natural One (Neutral Element of Multiplication of Natural Numbers)
- Proof (related to "Existence of Natural One (Neutral Element of Multiplication of Natural Numbers)") [1458]
- Existence of Natural Zero (Neutral Element of Addition of Natural Numbers)
- Proof (related to "Existence of Natural Zero (Neutral Element of Addition of Natural Numbers)") [1456]
- Existence of Parallel Straight Lines
- Proof by Contradiction (related to "Existence of Parallel Straight Lines") [787]
- Existence of Powers Exceeding Any Positive Constant
- Proof (related to "Existence of Powers Exceeding Any Positive Constant") [1349]
- Existence of Rational Cauchy Sequence of Ones (Neutral Element of Multiplication of Rational Cauchy Sequences)
- Proof (related to "Existence of Rational Cauchy Sequence of Ones (Neutral Element of Multiplication of Rational Cauchy Sequences)") [1505]
- Existence of Rational Cauchy Sequence of Zeros (Neutral Element of Addition of Rational Cauchy Sequences)
- Proof (related to "Existence of Rational Cauchy Sequence of Zeros (Neutral Element of Addition of Rational Cauchy Sequences)") [1499]
- Existence of Rational One (Neutral Element of Multiplication of Rational Numbers)
- Proof (related to "Existence of Rational One (Neutral Element of Multiplication of Rational Numbers)") [1483]
- Existence of Rational Zero (Neutral Element of Addition of Rational Numbers)
- Proof (related to "Existence of Rational Zero (Neutral Element of Addition of Rational Numbers)") [1474]
- Existence of Real One (Neutral Element of Multiplication of Real Numbers)
- Proof (related to "Existence of Real One (Neutral Element of Multiplication of Real Numbers)") [1537]
- Existence of Real Zero (Neutral Element of Addition of Real Numbers)
- Proof (related to "Existence of Real Zero (Neutral Element of Addition of Real Numbers)") [1536]
- Exponential Function
- Proof (related to "Exponential Function") [1343]
- Exponential Function Is Strictly Monotonically Increasing
- Proof (related to "Exponential Function Is Strictly Monotonically Increasing") [1595]
- Exponential Function of General Base With Integer Exponents
- Proof (related to "Exponential Function of General Base With Integer Exponents") [1621]
- Exponential Function of General Base With Natural Exponents
- Proof by Induction (related to "Exponential Function of General Base With Natural Exponents") [1617]
- Factor Groups
- Proof (related to "Factor Groups") [1099]
- Factor Rings
- Proof (related to "Factor Rings") [1100]
- Factorial
- Proof (related to "Factorial") [1006]
- Factorials and Stirling Numbers of the First Kind
- Proof (related to "Factorials and Stirling Numbers of the First Kind") [1008]
- Fiber of Prime Ideals Under a Spectrum Function
- Proof (related to "Fiber of Prime Ideals Under a Spectrum Function") [6263]
- Finite Basis Theorem
- Proof (related to "Finite Basis Theorem") [1046]
- Finite Cardinal Numbers and Set Operations
- Proof (related to "Finite Cardinal Numbers and Set Operations") [989]
- Functional Equation of the Complex Exponential Function
- Proof (related to "Functional Equation of the Complex Exponential Function") [1737]
- Functional Equation of the Exponential Function
- Proof (related to "Functional Equation of the Exponential Function") [1416]
- Functional Equation of the Exponential Function of General Base
- Proof (related to "Functional Equation of the Exponential Function of General Base") [1613]
- Functional Equation of the Exponential Function of General Base (Revised)
- Proof (related to "Functional Equation of the Exponential Function of General Base (Revised)") [1631]
- Functional Equation of the Natural Logarithm
- Proof (related to "Functional Equation of the Natural Logarithm") [1602]
- Functions Continuous at a Point and Non-Zero at this Point are Non-Zero in a Neighborhood of This Point
- Proof (related to "Functions Continuous at a Point and Non-Zero at this Point are Non-Zero in a Neighborhood of This Point") [6699]
- Fundamental Lemma of Homogeneous Systems of Linear Equations
- Proof by Induction (related to "Fundamental Lemma of Homogeneous Systems of Linear Equations") [1047]
- General Associative Law
- Direct Proof (related to "General Associative Law") [545]
- General Associative Law of Multiplication
- Direct Proof (related to "General Associative Law of Multiplication") [547]
- General Commutative Law
- Direct Proof (related to "General Commutative Law") [546]
- General Commutative Law of Multiplication
- Direct Proof (related to "General Commutative Law of Multiplication") [548]
- General Powers of Positive Numbers
- Proof (related to "General Powers of Positive Numbers") [1627]
- Generalized Euclidean Lemma
- Proof (related to "Generalized Euclidean Lemma") [1299]
- Generating Co-Prime Numbers Knowing the Greatest Common Divisor
- Proof (related to "Generating Co-Prime Numbers Knowing the Greatest Common Divisor") [1290]
- Generating the Greatest Common Divisor Knowing Co-Prime Numbers
- Proof (related to "Generating the Greatest Common Divisor Knowing Co-Prime Numbers") [1292]
- Geometric Distribution
- Proof (related to "Geometric Distribution") [1827]
- Get All Components of a Graph
- Proof of Correctness (related to "Get All Components of a Graph") [1224]
- Proof of Time Complexity (related to "Get All Components of a Graph") [1225]
- Get the Component Induced by Vertices Connected to a Given Vertex
- Proof of Correctness (related to "Get the Component Induced by Vertices Connected to a Given Vertex") [1218]
- Proof of Time Complexity (related to "Get the Component Induced by Vertices Connected to a Given Vertex") [1217]
- Get the Cut Vertices and Biconnected Components of a Connected Graph
- Proof (related to "Get the Cut Vertices and Biconnected Components of a Connected Graph") [1241]
- Greatest Common Divisor (Euclid)
- Proof of Correctness (related to "Greatest Common Divisor (Euclid)") [1286]
- Greatest Common Divisor and Least Common Multiple of Ideals
- Proof (related to "Greatest Common Divisor and Least Common Multiple of Ideals") [1070]
- Greatest Common Divisors Of Integers and Prime Numbers
- Proof (related to "Greatest Common Divisors Of Integers and Prime Numbers") [1297]
- Group Homomorphisms and Normal Subgroups
- Proof (related to "Group Homomorphisms and Normal Subgroups") [835]
- Group Homomorphisms with Cyclic Groups
- Proof (related to "Group Homomorphisms with Cyclic Groups") [816]
- Handshaking Lemma for Finite Digraphs
- Combinatorial Proof (related to "Handshaking Lemma for Finite Digraphs") [567]
- Handshaking Lemma for Finite Graphs
- Proof (related to "Handshaking Lemma for Finite Graphs") [1174]
- Heine-Borel Theorem
- Proof (related to "Heine-Borel Theorem") [6597]
- Horner Scheme
- Proof of Correctness (related to "Horner Scheme") [1360]
- Proof of Time Complexity (related to "Horner Scheme") [1359]
- How Convergence Preserves the Order Relation of Sequence Members
- Proof (related to "How Convergence Preserves the Order Relation of Sequence Members") [1146]
- How Convergence Preserves Upper and Lower Bounds For Sequence Members
- Proof (related to "How Convergence Preserves Upper and Lower Bounds For Sequence Members") [1147]
- How the Boundary Changes the Property of a Set of Being Open
- Topological Proof (related to "How the Boundary Changes the Property of a Set of Being Open") [1204]
- Image of a Compact Set Under a Continuous Function
- Proof (related to "Image of a Compact Set Under a Continuous Function") [6601]
- Imaginary Unit
- Proof (related to "Imaginary Unit") [1693]
- Indefinite Integral, Antiderivative
- Proof (related to "Indefinite Integral, Antiderivative") [1774]
- Inequality of Natural Numbers and Their Successors
- Proof by Contraposition (related to "Inequality of Natural Numbers and Their Successors") [1541]
- Infinite Geometric Series
- Proof (related to "Infinite Geometric Series") [1354]
- Intermediate Root Value Theorem
- Proof (related to "Intermediate Root Value Theorem") [6695]
- Intermediate Value Theorem
- Proof (related to "Intermediate Value Theorem") [1263]
- Intersection of Convex Affine Sets
- Proof (related to "Intersection of Convex Affine Sets") [6290]
- Invertible Functions on Real Intervals
- Proof (related to "Invertible Functions on Real Intervals") [1382]
- Isometry is Injective
- Proof (related to "Isometry is Injective") [2780]
- It is true that something can be (either) true or false
- Proof (related to "It is true that something can be (either) true or false") [1325]
- Kernel and Image of a Group Homomorphism are Subgroups
- Proof (related to "Kernel and Image of a Group Homomorphism are Subgroups") [834]
- Kernel and Image of Group Homomorphism
- Proof (related to "Kernel and Image of Group Homomorphism") [810]
- Law of Total Probability
- Proof (related to "Law of Total Probability") [1828]
- Least Common Multiple
- Proof (related to "Least Common Multiple") [1277]
- Lem. 10.016: Incommensurability of Sum of Incommensurable Magnitudes
- Proof (related to "Lem. 10.016: Incommensurability of Sum of Incommensurable Magnitudes") [6557]
- Lem. 10.021: Medial is Irrational
- Proof (related to "Lem. 10.021: Medial is Irrational") [6558]
- Lem. 10.028.1: Finding Two Squares With Sum Also Square
- Proof (related to "Lem. 10.028.1: Finding Two Squares With Sum Also Square") [6559]
- Lem. 10.028.2: Finding Two Squares With Sum Not Square
- Proof (related to "Lem. 10.028.2: Finding Two Squares With Sum Not Square") [6560]
- Lem. 10.032: Constructing Medial Commensurable in Square II
- Proof (related to "Lem. 10.032: Constructing Medial Commensurable in Square II") [6561]
- Lem. 10.041: Side of Sum of Medial Areas is Irrational
- Proof (related to "Lem. 10.041: Side of Sum of Medial Areas is Irrational") [6562]
- Lem. 10.053: Construction of Rectangle with Area in Mean Proportion to two Square Areas
- Proof (related to "Lem. 10.053: Construction of Rectangle with Area in Mean Proportion to two Square Areas") [6563]
- Lem. 10.059: Sum of Squares on Unequal Pieces of Segment Is Greater than Twice the Rectangle Contained by Them
- Proof (related to "Lem. 10.059: Sum of Squares on Unequal Pieces of Segment Is Greater than Twice the Rectangle Contained by Them") [6564]
- Lem. 10.13: Finding Pythagorean Magnitudes
- Proof (related to "Lem. 10.13: Finding Pythagorean Magnitudes") [2580]
- Lem. 11.23: Making a Square Area Equal to the Difference Of Areas of Two Other Incongruent Squares
- Proof (related to "Lem. 11.23: Making a Square Area Equal to the Difference Of Areas of Two Other Incongruent Squares") [2720]
- Lem. 12.02: Areas of Circles are as Squares on Diameters
- Proof (related to "Lem. 12.02: Areas of Circles are as Squares on Diameters") [6566]
- Lem. 12.04: Proportion of Sizes of Tetrahedra divided into Two Similar Tetrahedra and Two Equal Prisms
- Proof (related to "Lem. 12.04: Proportion of Sizes of Tetrahedra divided into Two Similar Tetrahedra and Two Equal Prisms") [6567]
- Lem. 13.02: Converse of Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio
- Proof (related to "Lem. 13.02: Converse of Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio") [6570]
- Lem. 13.13: Construction of Regular Tetrahedron within Given Sphere
- Proof (related to "Lem. 13.13: Construction of Regular Tetrahedron within Given Sphere") [6572]
- Lem. 13.18: Angle of the Pentagon
- Proof (related to "Lem. 13.18: Angle of the Pentagon") [2773]
- Limit of 1/n
- Proof (related to "Limit of 1/n") [6714]
- Limit of N-th Roots
- Proof (related to "Limit of N-th Roots") [1625]
- Limit of Nth Root
- Proof (related to "Limit of Nth Root") [6712]
- Linear Independence of the Imaginary Unit \(i\) and the Complex Number \(1\)
- Proof (related to "Linear Independence of the Imaginary Unit \(i\) and the Complex Number \(1\)") [1697]
- Linearity and Monotony of the Riemann Integral
- Proof (related to "Linearity and Monotony of the Riemann Integral") [1771]
- Linearity and Monotony of the Riemann Integral for Step Functions
- Proof (related to "Linearity and Monotony of the Riemann Integral for Step Functions") [1760]
- LOOP-Computable Functions are Total
- Proof (related to "LOOP-Computable Functions are Total") [1186]
- Lower Bound of Leaves in a Tree
- Proof (related to "Lower Bound of Leaves in a Tree") [6368]
- Magnitude of Divisors
- Proof (related to "Magnitude of Divisors") [1279]
- Mean Value Theorem For Riemann Integrals
- Proof (related to "Mean Value Theorem For Riemann Integrals") [1773]
- Metric Spaces and Empty Sets are Clopen
- Proof (related to "Metric Spaces and Empty Sets are Clopen") [855]
- Metric Spaces are Hausdorff Spaces
- Topological Proof (related to "Metric Spaces are Hausdorff Spaces") [851]
- Monotone Convergence
- Proof (related to "Monotone Convergence") [1157]
- Monotonic Real Functions on Closed Intervals are Riemann-Integrable
- Proof by Construction (related to "Monotonic Real Functions on Closed Intervals are Riemann-Integrable") [6622]
- Monotonically Increasing Property of Probability Distributions
- Proof (related to "Monotonically Increasing Property of Probability Distributions") [1817]
- Multinomial Coefficient
- Proof (related to "Multinomial Coefficient") [1820]
- Multinomial Distribution
- Proof (related to "Multinomial Distribution") [1826]
- Multinomial Theorem
- Proof by Induction (related to "Multinomial Theorem") [1823]
- Multiplication of Complex Numbers Is Associative
- Proof (related to "Multiplication of Complex Numbers Is Associative") [1670]
- Multiplication of Complex Numbers Is Commutative
- Proof (related to "Multiplication of Complex Numbers Is Commutative") [1672]
- Multiplication of Integers
- Proof (related to "Multiplication of Integers") [1531]
- Multiplication of Integers Is Associative
- Proof (related to "Multiplication of Integers Is Associative") [1451]
- Multiplication of Integers Is Cancellative
- Proof (related to "Multiplication of Integers Is Cancellative") [1465]
- Multiplication of Integers Is Commutative
- Proof (related to "Multiplication of Integers Is Commutative") [1449]
- Multiplication of Natural Numbers Is Associative
- Proof (related to "Multiplication of Natural Numbers Is Associative") [1439]
- Multiplication of Natural Numbers Is Cancellative
- Proof (related to "Multiplication of Natural Numbers Is Cancellative") [1441]
- Multiplication of Natural Numbers Is Cancellative With Respect to the Order Relation
- Proof (related to "Multiplication of Natural Numbers Is Cancellative With Respect to the Order Relation") [1584]
- Multiplication of Natural Numbers is Commutative
- Proof (related to "Multiplication of Natural Numbers is Commutative") [1438]
- Multiplication Of Rational Cauchy Sequences
- Proof (related to "Multiplication Of Rational Cauchy Sequences") [1493]
- Multiplication of Rational Cauchy Sequences Is Associative
- Proof (related to "Multiplication of Rational Cauchy Sequences Is Associative") [1501]
- Multiplication of Rational Cauchy Sequences Is Cancellative
- Proof (related to "Multiplication of Rational Cauchy Sequences Is Cancellative") [1573]
- Multiplication of Rational Cauchy Sequences Is Commutative
- Proof (related to "Multiplication of Rational Cauchy Sequences Is Commutative") [1503]
- Multiplication Of Rational Numbers
- Proof (related to "Multiplication Of Rational Numbers") [1529]
- Multiplication of Rational Numbers Is Associative
- Proof (related to "Multiplication of Rational Numbers Is Associative") [1477]
- Multiplication Of Rational Numbers Is Cancellative
- Proof (related to "Multiplication Of Rational Numbers Is Cancellative") [1481]
- Multiplication Of Rational Numbers Is Commutative
- Proof (related to "Multiplication Of Rational Numbers Is Commutative") [1479]
- Multiplication of Real Numbers
- Proof (related to "Multiplication of Real Numbers") [1533]
- Multiplication of Real Numbers Is Associative
- Proof (related to "Multiplication of Real Numbers Is Associative") [1534]
- Multiplication of Real Numbers Is Cancellative
- Proof (related to "Multiplication of Real Numbers Is Cancellative") [1577]
- Multiplication of Real Numbers Is Commutative
- Proof (related to "Multiplication of Real Numbers Is Commutative") [1535]
- Multiplying Negative and Positive Integers
- Proof (related to "Multiplying Negative and Positive Integers") [1590]
- Multiplying Negative and Positive Rational Numbers
- Proof (related to "Multiplying Negative and Positive Rational Numbers") [1597]
- Multiplying Negative and Positive Real Numbers
- Proof (related to "Multiplying Negative and Positive Real Numbers") [1599]
- Natural Logarithm
- Proof (related to "Natural Logarithm") [1600]
- Nested Closed Subset Theorem
- Proof (related to "Nested Closed Subset Theorem") [6587]
- Not all Cauchy sequences converge in the set of rational numbers.
- Proof (related to "Not all Cauchy sequences converge in the set of rational numbers.") [1098]
- Nth Powers
- Proof (related to "Nth Powers") [1619]
- Nth Roots of Positive Numbers
- Proof (related to "Nth Roots of Positive Numbers") [1383]
- Number of Relations on a Finite Set
- Proof (related to "Number of Relations on a Finite Set") [1000]
- Number of Strings With a Fixed Length Over an Alphabet with k Letters
- Proof (related to "Number of Strings With a Fixed Length Over an Alphabet with k Letters") [997]
- Number of Subsets of a Finite Set
- Proof (related to "Number of Subsets of a Finite Set") [1003]
- Proof by Induction (related to "Number of Subsets of a Finite Set") [999]
- Oddness of the Sine of a Real Variable
- Proof (related to "Oddness of the Sine of a Real Variable") [1793]
- Open and Closed Subsets of a Zariski Topology
- Proof (related to "Open and Closed Subsets of a Zariski Topology") [6327]
- Order Relation for Natural Numbers, Revised
- Proof (related to "Order Relation for Natural Numbers, Revised") [1558]
- Ordinals Are Downward Closed
- Proof (related to "Ordinals Are Downward Closed") [728]
- Planarity of Subdivisions
- Proof (related to "Planarity of Subdivisions") [6379]
- Position of Minus Sign in Rational Numbers Representations
- Proof (related to "Position of Minus Sign in Rational Numbers Representations") [1593]
- Preservation of Continuity with Arithmetic Operations on Continuous Functions
- Proof (related to "Preservation of Continuity with Arithmetic Operations on Continuous Functions") [1262]
- Preservation of Continuity with Arithmetic Operations on Continuous Functions on a Whole Domain
- Proof (related to "Preservation of Continuity with Arithmetic Operations on Continuous Functions on a Whole Domain") [1605]
- Probability of Event Difference
- Proof (related to "Probability of Event Difference") [870]
- Probability of Event Union
- Proof (related to "Probability of Event Union") [869]
- Probability of Included Event
- Proof (related to "Probability of Included Event") [866]
- Probability of Joint Events
- Proof (related to "Probability of Joint Events") [1803]
- Probability of Laplace Experiments
- Proof (related to "Probability of Laplace Experiments") [976]
- Probability of the Complement Event
- Proof (related to "Probability of the Complement Event") [863]
- Probability of the Impossible Event
- Proof (related to "Probability of the Impossible Event") [864]
- Product of a Complex Number and a Convergent Complex Sequence
- Proof (related to "Product of a Complex Number and a Convergent Complex Sequence") [1720]
- Product of a Real Number and a Convergent Real Sequence
- Proof (related to "Product of a Real Number and a Convergent Real Sequence") [1141]
- Product of a Real Number and a Convergent Real Series
- Proof (related to "Product of a Real Number and a Convergent Real Series") [6648]
- Product of Convegent Complex Sequences
- Proof (related to "Product of Convegent Complex Sequences") [1718]
- Product of Convegent Real Sequences
- Proof (related to "Product of Convegent Real Sequences") [1139]
- Prop. 1.01: Constructing an Equilateral Triangle
- Proof (related to "Prop. 1.01: Constructing an Equilateral Triangle") [6484]
- Geometric Proof (Euclid) (related to "Prop. 1.01: Constructing an Equilateral Triangle") [695]
- Prop. 1.02: Constructing a Segment Equal to an Arbitrary Segment
- Proof (related to "Prop. 1.02: Constructing a Segment Equal to an Arbitrary Segment") [6485]
- Geometric Proof (related to "Prop. 1.02: Constructing a Segment Equal to an Arbitrary Segment") [735]
- Prop. 1.03: Cutting a Segment at a Given Size
- Proof (related to "Prop. 1.03: Cutting a Segment at a Given Size") [6486]
- Geometric Proof (related to "Prop. 1.03: Cutting a Segment at a Given Size") [737]
- Prop. 1.04: "Side-Angle-Side" Theorem for the Congruence of Triangle
- Proof (related to "Prop. 1.04: "Side-Angle-Side" Theorem for the Congruence of Triangle") [6487]
- Geometric Proof (related to "Prop. 1.04: "Side-Angle-Side" Theorem for the Congruence of Triangle") [739]
- Prop. 1.05: Isosceles Triagles I
- Proof (related to "Prop. 1.05: Isosceles Triagles I") [6488]
- Geometric Proof (related to "Prop. 1.05: Isosceles Triagles I") [741]
- Prop. 1.06: Isosceles Triagles II
- Proof (related to "Prop. 1.06: Isosceles Triagles II") [6489]
- Proof by Contradiction (related to "Prop. 1.06: Isosceles Triagles II") [746]
- Prop. 1.07: Uniqueness of Triangles
- Proof (related to "Prop. 1.07: Uniqueness of Triangles") [6490]
- Proof by Contradiction (related to "Prop. 1.07: Uniqueness of Triangles") [752]
- Prop. 1.08: "Side-Side-Side" Theorem for the Congruence of Triangles
- Proof (related to "Prop. 1.08: "Side-Side-Side" Theorem for the Congruence of Triangles") [6491]
- Proof by Contradiction (related to "Prop. 1.08: "Side-Side-Side" Theorem for the Congruence of Triangles") [754]
- Prop. 1.09: Bisecting an Angle
- Proof (related to "Prop. 1.09: Bisecting an Angle") [6492]
- Geometric Proof (related to "Prop. 1.09: Bisecting an Angle") [756]
- Prop. 1.10: Bisecting a Segment
- Proof (related to "Prop. 1.10: Bisecting a Segment") [6493]
- Geometric Proof (related to "Prop. 1.10: Bisecting a Segment") [758]
- Prop. 1.11: Constructing a Perpendicular Segment to a Straight Line From a Given Point On the Stright Line
- Proof (related to "Prop. 1.11: Constructing a Perpendicular Segment to a Straight Line From a Given Point On the Stright Line") [6494]
- Geometric Proof (related to "Prop. 1.11: Constructing a Perpendicular Segment to a Straight Line From a Given Point On the Stright Line") [761]
- Prop. 1.12: Constructing a Perpendicular Segment to a Straight Line From a Given Point Not On the Straight Line
- Proof (related to "Prop. 1.12: Constructing a Perpendicular Segment to a Straight Line From a Given Point Not On the Straight Line") [6495]
- Geometric Proof (related to "Prop. 1.12: Constructing a Perpendicular Segment to a Straight Line From a Given Point Not On the Straight Line") [762]
- Prop. 1.13: Angles at Intersections of Straight Lines
- Proof (related to "Prop. 1.13: Angles at Intersections of Straight Lines") [6496]
- Geometric Proof (related to "Prop. 1.13: Angles at Intersections of Straight Lines") [764]
- Prop. 1.14: Combining Rays to Straight Lines
- Proof (related to "Prop. 1.14: Combining Rays to Straight Lines") [6497]
- Proof by Contradiction (related to "Prop. 1.14: Combining Rays to Straight Lines") [768]
- Prop. 1.15: Opposite Angles on Intersecting Straight Lines
- Proof (related to "Prop. 1.15: Opposite Angles on Intersecting Straight Lines") [6498]
- Geometric Proof (related to "Prop. 1.15: Opposite Angles on Intersecting Straight Lines") [783]
- Prop. 1.16: The Exterior Angle Is Greater Than Either of the Non-Adjacent Interior Angles
- Proof (related to "Prop. 1.16: The Exterior Angle Is Greater Than Either of the Non-Adjacent Interior Angles") [6499]
- Geometric Proof (related to "Prop. 1.16: The Exterior Angle Is Greater Than Either of the Non-Adjacent Interior Angles") [785]
- Prop. 1.17: The Sum of Two Angles of a Triangle
- Proof (related to "Prop. 1.17: The Sum of Two Angles of a Triangle") [6500]
- Geometric Proof (related to "Prop. 1.17: The Sum of Two Angles of a Triangle") [790]
- Prop. 1.18: Angles and Sides in a Triangle I
- Proof (related to "Prop. 1.18: Angles and Sides in a Triangle I") [6501]
- Geometric Proof (related to "Prop. 1.18: Angles and Sides in a Triangle I") [792]
- Prop. 1.19: Angles and Sides in a Triangle II
- Proof (related to "Prop. 1.19: Angles and Sides in a Triangle II") [6502]
- Proof by Contradiction (related to "Prop. 1.19: Angles and Sides in a Triangle II") [794]
- Prop. 1.20: The Sum of the Lengths of Any Pair of Sides of a Triangle (Triangle Inequality)
- Proof (related to "Prop. 1.20: The Sum of the Lengths of Any Pair of Sides of a Triangle (Triangle Inequality)") [6503]
- Geometric Proof (related to "Prop. 1.20: The Sum of the Lengths of Any Pair of Sides of a Triangle (Triangle Inequality)") [878]
- Prop. 1.21: Triangles within Triangles
- Proof (related to "Prop. 1.21: Triangles within Triangles") [6504]
- Geometric Proof (related to "Prop. 1.21: Triangles within Triangles") [894]
- Prop. 1.22: Construction of Triangles From Arbitrary Segments
- Proof (related to "Prop. 1.22: Construction of Triangles From Arbitrary Segments") [6505]
- Geometric Proof (related to "Prop. 1.22: Construction of Triangles From Arbitrary Segments") [896]
- Prop. 1.23: Constructing an Angle Equal to an Arbitrary Rectilinear Angle
- Proof (related to "Prop. 1.23: Constructing an Angle Equal to an Arbitrary Rectilinear Angle") [6506]
- Geometric Proof (related to "Prop. 1.23: Constructing an Angle Equal to an Arbitrary Rectilinear Angle") [898]
- Prop. 1.24: Angles and Sides in a Triangle III
- Proof (related to "Prop. 1.24: Angles and Sides in a Triangle III") [6507]
- Geometric Proof (related to "Prop. 1.24: Angles and Sides in a Triangle III") [900]
- Prop. 1.25: Angles and Sides in a Triangle IV
- Proof (related to "Prop. 1.25: Angles and Sides in a Triangle IV") [6508]
- Geometric Proof (related to "Prop. 1.25: Angles and Sides in a Triangle IV") [902]
- Prop. 1.26: "Angle-Side-Angle" and "Angle-Angle-Side" Theorems for the Congruence of Triangles
- Proof (related to "Prop. 1.26: "Angle-Side-Angle" and "Angle-Angle-Side" Theorems for the Congruence of Triangles") [6509]
- Geometric Proof (related to "Prop. 1.26: "Angle-Side-Angle" and "Angle-Angle-Side" Theorems for the Congruence of Triangles") [906]
- Prop. 1.27: Parallel Lines I
- Proof (related to "Prop. 1.27: Parallel Lines I") [6510]
- Geometric Proof (related to "Prop. 1.27: Parallel Lines I") [912]
- Prop. 1.28: Parallel Lines II
- Proof (related to "Prop. 1.28: Parallel Lines II") [6511]
- Geometric Proof (related to "Prop. 1.28: Parallel Lines II") [914]
- Prop. 1.29: Parallel Lines III
- Proof (related to "Prop. 1.29: Parallel Lines III") [6512]
- Geometric Proof (related to "Prop. 1.29: Parallel Lines III") [916]
- Prop. 1.30: Transitivity of Parallel Lines
- Proof (related to "Prop. 1.30: Transitivity of Parallel Lines") [920]
- Proof (related to "Prop. 1.30: Transitivity of Parallel Lines") [6513]
- Prop. 1.31: Constructing a Parallel Line from a Line and a Point
- Proof (related to "Prop. 1.31: Constructing a Parallel Line from a Line and a Point") [6514]
- Geometric Proof (related to "Prop. 1.31: Constructing a Parallel Line from a Line and a Point") [922]
- Prop. 1.32: Sum Of Angles in a Triangle and Exterior Angle
- Proof (related to "Prop. 1.32: Sum Of Angles in a Triangle and Exterior Angle") [6515]
- Geometric Proof (related to "Prop. 1.32: Sum Of Angles in a Triangle and Exterior Angle") [925]
- Prop. 1.33: Parallel Equal Segments Determine a Parallelogram
- Proof (related to "Prop. 1.33: Parallel Equal Segments Determine a Parallelogram") [6516]
- Geometric Proof (related to "Prop. 1.33: Parallel Equal Segments Determine a Parallelogram") [932]
- Prop. 1.34: Opposite Sides and Opposite Angles of Parallelograms
- Proof (related to "Prop. 1.34: Opposite Sides and Opposite Angles of Parallelograms") [6517]
- Geometric Proof (related to "Prop. 1.34: Opposite Sides and Opposite Angles of Parallelograms") [934]
- Prop. 1.35: Parallelograms On the Same Base and On the Same Parallels
- Proof (related to "Prop. 1.35: Parallelograms On the Same Base and On the Same Parallels") [6518]
- Geometric Proof (related to "Prop. 1.35: Parallelograms On the Same Base and On the Same Parallels") [944]
- Prop. 1.36: Parallelograms on Equal Bases and on the Same Parallels
- Proof (related to "Prop. 1.36: Parallelograms on Equal Bases and on the Same Parallels") [6519]
- Geometric Proof (related to "Prop. 1.36: Parallelograms on Equal Bases and on the Same Parallels") [946]
- Prop. 1.37: Triangles of Equal Area I
- Proof (related to "Prop. 1.37: Triangles of Equal Area I") [6520]
- Geometric Proof (related to "Prop. 1.37: Triangles of Equal Area I") [948]
- Prop. 1.38: Triangles of Equal Area II
- Proof (related to "Prop. 1.38: Triangles of Equal Area II") [6521]
- Geometric Proof (related to "Prop. 1.38: Triangles of Equal Area II") [950]
- Prop. 1.39: Triangles of Equal Area III
- Proof (related to "Prop. 1.39: Triangles of Equal Area III") [6522]
- Geometric Proof (related to "Prop. 1.39: Triangles of Equal Area III") [952]
- Prop. 1.40: Triangles of Equal Area IV
- Proof (related to "Prop. 1.40: Triangles of Equal Area IV") [6523]
- Geometric Proof (related to "Prop. 1.40: Triangles of Equal Area IV") [954]
- Prop. 1.41: Parallelograms and Triagles
- Proof (related to "Prop. 1.41: Parallelograms and Triagles") [6524]
- Geometric Proof (related to "Prop. 1.41: Parallelograms and Triagles") [956]
- Prop. 1.42: Construction of Parallelograms I
- Proof (related to "Prop. 1.42: Construction of Parallelograms I") [6525]
- Geometric Proof (related to "Prop. 1.42: Construction of Parallelograms I") [958]
- Prop. 1.43: Complementary Segments of Parallelograms
- Proof (related to "Prop. 1.43: Complementary Segments of Parallelograms") [6526]
- Geometric Proof (related to "Prop. 1.43: Complementary Segments of Parallelograms") [960]
- Prop. 1.44: Construction of Parallelograms II
- Proof (related to "Prop. 1.44: Construction of Parallelograms II") [6527]
- Geometric Proof (related to "Prop. 1.44: Construction of Parallelograms II") [962]
- Prop. 1.45: Construction of Parallelograms III
- Proof (related to "Prop. 1.45: Construction of Parallelograms III") [6528]
- Geometric Proof (related to "Prop. 1.45: Construction of Parallelograms III") [964]
- Prop. 1.46: Construction of a Square I
- Proof (related to "Prop. 1.46: Construction of a Square I") [6529]
- Geometric Proof (related to "Prop. 1.46: Construction of a Square I") [967]
- Prop. 1.47: Pythagorean Theorem
- Proof (related to "Prop. 1.47: Pythagorean Theorem") [6530]
- Geometric Proof (related to "Prop. 1.47: Pythagorean Theorem") [970]
- Geometric Proof (related to "Prop. 1.47: Pythagorean Theorem") [969]
- Prop. 1.48: The Converse of the Pythagorean Theorem
- Proof (related to "Prop. 1.48: The Converse of the Pythagorean Theorem") [6531]
- Geometric Proof (related to "Prop. 1.48: The Converse of the Pythagorean Theorem") [972]
- Prop. 10.001: Existence of Fraction of Number Smaller than Given Number
- Proof (related to "Prop. 10.001: Existence of Fraction of Number Smaller than Given Number") [2582]
- Prop. 10.002: Incommensurable Magnitudes do not Terminate in Euclidean Algorithm
- Proof (related to "Prop. 10.002: Incommensurable Magnitudes do not Terminate in Euclidean Algorithm") [2583]
- Prop. 10.003: Greatest Common Measure of Commensurable Magnitudes
- Proof (related to "Prop. 10.003: Greatest Common Measure of Commensurable Magnitudes") [2584]
- Prop. 10.004: Greatest Common Measure of Three Commensurable Magnitudes
- Proof (related to "Prop. 10.004: Greatest Common Measure of Three Commensurable Magnitudes") [2585]
- Prop. 10.005: Ratio of Commensurable Magnitudes
- Proof (related to "Prop. 10.005: Ratio of Commensurable Magnitudes") [2586]
- Prop. 10.006: Magnitudes with Rational Ratio are Commensurable
- Proof (related to "Prop. 10.006: Magnitudes with Rational Ratio are Commensurable") [2587]
- Prop. 10.007: Incommensurable Magnitudes Have Irrational Ratio
- Proof (related to "Prop. 10.007: Incommensurable Magnitudes Have Irrational Ratio") [2588]
- Prop. 10.008: Magnitudes with Irrational Ratio are Incommensurable
- Proof (related to "Prop. 10.008: Magnitudes with Irrational Ratio are Incommensurable") [2589]
- Prop. 10.009: Commensurability of Squares
- Proof (related to "Prop. 10.009: Commensurability of Squares") [2590]
- Prop. 10.010: Construction of Incommensurable Lines
- Proof (related to "Prop. 10.010: Construction of Incommensurable Lines") [2591]
- Prop. 10.011: Commensurability of Elements of Proportional Magnitudes
- Proof (related to "Prop. 10.011: Commensurability of Elements of Proportional Magnitudes") [2592]
- Prop. 10.012: Commensurability is Transitive Relation
- Proof (related to "Prop. 10.012: Commensurability is Transitive Relation") [2593]
- Prop. 10.013: Commensurable Magnitudes are Incommensurable with Same Magnitude
- Proof (related to "Prop. 10.013: Commensurable Magnitudes are Incommensurable with Same Magnitude") [2594]
- Prop. 10.014: Commensurability of Squares on Proportional Straight Lines
- Proof (related to "Prop. 10.014: Commensurability of Squares on Proportional Straight Lines") [2595]
- Prop. 10.015: Commensurability of Sum of Commensurable Magnitudes
- Proof (related to "Prop. 10.015: Commensurability of Sum of Commensurable Magnitudes") [2596]
- Prop. 10.016: Incommensurability of Sum of Incommensurable Magnitudes
- Proof (related to "Prop. 10.016: Incommensurability of Sum of Incommensurable Magnitudes") [2597]
- Prop. 10.017: Condition for Commensurability of Roots of Quadratic Equation
- Proof (related to "Prop. 10.017: Condition for Commensurability of Roots of Quadratic Equation") [2598]
- Prop. 10.018: Condition for Incommensurability of Roots of Quadratic Equation
- Proof (related to "Prop. 10.018: Condition for Incommensurability of Roots of Quadratic Equation") [2599]
- Prop. 10.019: Product of Rational Numbers is Rational
- Proof (related to "Prop. 10.019: Product of Rational Numbers is Rational") [2600]
- Prop. 10.020: Quotient of Rational Numbers is Rational
- Proof (related to "Prop. 10.020: Quotient of Rational Numbers is Rational") [2601]
- Prop. 10.021: Medial is Irrational
- Proof (related to "Prop. 10.021: Medial is Irrational") [2602]
- Prop. 10.022: Square on Medial Straight Line
- Proof (related to "Prop. 10.022: Square on Medial Straight Line") [2603]
- Prop. 10.023: Segment Commensurable with Medial Segment is Medial
- Proof (related to "Prop. 10.023: Segment Commensurable with Medial Segment is Medial") [2604]
- Prop. 10.024: Rectangle Contained by Medial Straight Lines Commensurable in Length is Medial
- Proof (related to "Prop. 10.024: Rectangle Contained by Medial Straight Lines Commensurable in Length is Medial") [2605]
- Prop. 10.025: Rationality of Rectangle Contained by Medial Straight Lines Commensurable in Square
- Proof (related to "Prop. 10.025: Rationality of Rectangle Contained by Medial Straight Lines Commensurable in Square") [2606]
- Prop. 10.026: Medial Area not greater than Medial Area by Rational Area
- Proof (related to "Prop. 10.026: Medial Area not greater than Medial Area by Rational Area") [2607]
- Prop. 10.027: Construction of Components of First Bimedial
- Proof (related to "Prop. 10.027: Construction of Components of First Bimedial") [2608]
- Prop. 10.028: Construction of Components of Second Bimedial
- Proof (related to "Prop. 10.028: Construction of Components of Second Bimedial") [2609]
- Prop. 10.029: Construction of Rational Straight Lines Commensurable in Square Only whose Square Differences Commensurable with Gre
- Proof (related to "Prop. 10.029: Construction of Rational Straight Lines Commensurable in Square Only whose Square Differences Commensurable with Gre") [2610]
- Prop. 10.030: Construction of Rational Straight Lines Commensurable in Square Only whose Square Differences Incommensurable with G
- Proof (related to "Prop. 10.030: Construction of Rational Straight Lines Commensurable in Square Only whose Square Differences Incommensurable with G") [2611]
- Prop. 10.031: Constructing Medial Commensurable in Square I
- Proof (related to "Prop. 10.031: Constructing Medial Commensurable in Square I") [2612]
- Prop. 10.032: Constructing Medial Commensurable in Square II
- Proof (related to "Prop. 10.032: Constructing Medial Commensurable in Square II") [2613]
- Prop. 10.033: Construction of Components of Major
- Proof (related to "Prop. 10.033: Construction of Components of Major") [2614]
- Prop. 10.034: Construction of Components of Side of Rational plus Medial Area
- Proof (related to "Prop. 10.034: Construction of Components of Side of Rational plus Medial Area") [2615]
- Prop. 10.035: Construction of Components of Side of Sum of Medial Areas
- Proof (related to "Prop. 10.035: Construction of Components of Side of Sum of Medial Areas") [2616]
- Prop. 10.036: Binomial is Irrational
- Proof (related to "Prop. 10.036: Binomial is Irrational") [2617]
- Prop. 10.037: First Bimedial is Irrational
- Proof (related to "Prop. 10.037: First Bimedial is Irrational") [2618]
- Prop. 10.038: Second Bimedial is Irrational
- Proof (related to "Prop. 10.038: Second Bimedial is Irrational") [2619]
- Prop. 10.039: Major is Irrational
- Proof (related to "Prop. 10.039: Major is Irrational") [2620]
- Prop. 10.040: Side of Rational plus Medial Area is Irrational
- Proof (related to "Prop. 10.040: Side of Rational plus Medial Area is Irrational") [2621]
- Prop. 10.041: Side of Sum of Medial Areas is Irrational
- Proof (related to "Prop. 10.041: Side of Sum of Medial Areas is Irrational") [2622]
- Prop. 10.042: Binomial Straight Line is Divisible into Terms Uniquely
- Proof (related to "Prop. 10.042: Binomial Straight Line is Divisible into Terms Uniquely") [2623]
- Prop. 10.043: First Bimedial Straight Line is Divisible Uniquely
- Proof (related to "Prop. 10.043: First Bimedial Straight Line is Divisible Uniquely") [2624]
- Prop. 10.044: Second Bimedial Straight Line is Divisible Uniquely
- Proof (related to "Prop. 10.044: Second Bimedial Straight Line is Divisible Uniquely") [2625]
- Prop. 10.045: Major Straight Line is Divisible Uniquely
- Proof (related to "Prop. 10.045: Major Straight Line is Divisible Uniquely") [2626]
- Prop. 10.046: Side of Rational Plus Medial Area is Divisible Uniquely
- Proof (related to "Prop. 10.046: Side of Rational Plus Medial Area is Divisible Uniquely") [2627]
- Prop. 10.047: Side of Sum of Two Medial Areas is Divisible Uniquely
- Proof (related to "Prop. 10.047: Side of Sum of Two Medial Areas is Divisible Uniquely") [2628]
- Prop. 10.048: Construction of First Binomial Straight Line
- Proof (related to "Prop. 10.048: Construction of First Binomial Straight Line") [2629]
- Prop. 10.049: Construction of Second Binomial Straight Line
- Proof (related to "Prop. 10.049: Construction of Second Binomial Straight Line") [2630]
- Prop. 10.050: Construction of Third Binomial Straight Line
- Proof (related to "Prop. 10.050: Construction of Third Binomial Straight Line") [2631]
- Prop. 10.051: Construction of Fourth Binomial Straight Line
- Proof (related to "Prop. 10.051: Construction of Fourth Binomial Straight Line") [2632]
- Prop. 10.052: Construction of Fifth Binomial Straight Line
- Proof (related to "Prop. 10.052: Construction of Fifth Binomial Straight Line") [2633]
- Prop. 10.053: Construction of Sixth Binomial Straight Line
- Proof (related to "Prop. 10.053: Construction of Sixth Binomial Straight Line") [2634]
- Prop. 10.054: Root of Area contained by Rational Straight Line and First Binomial
- Proof (related to "Prop. 10.054: Root of Area contained by Rational Straight Line and First Binomial") [2635]
- Prop. 10.055: Root of Area contained by Rational Straight Line and Second Binomial
- Proof (related to "Prop. 10.055: Root of Area contained by Rational Straight Line and Second Binomial") [2636]
- Prop. 10.056: Root of Area contained by Rational Straight Line and Third Binomial
- Proof (related to "Prop. 10.056: Root of Area contained by Rational Straight Line and Third Binomial") [2637]
- Prop. 10.057: Root of Area contained by Rational Straight Line and Fourth Binomial
- Proof (related to "Prop. 10.057: Root of Area contained by Rational Straight Line and Fourth Binomial") [2638]
- Prop. 10.058: Root of Area contained by Rational Straight Line and Fifth Binomial
- Proof (related to "Prop. 10.058: Root of Area contained by Rational Straight Line and Fifth Binomial") [2639]
- Prop. 10.059: Root of Area contained by Rational Straight Line and Sixth Binomial
- Proof (related to "Prop. 10.059: Root of Area contained by Rational Straight Line and Sixth Binomial") [2640]
- Prop. 10.060: Square on Binomial Straight Line applied to Rational Straight Line
- Proof (related to "Prop. 10.060: Square on Binomial Straight Line applied to Rational Straight Line") [2641]
- Prop. 10.061: Square on First Bimedial Straight Line applied to Rational Straight Line
- Proof (related to "Prop. 10.061: Square on First Bimedial Straight Line applied to Rational Straight Line") [2642]
- Prop. 10.062: Square on Second Bimedial Straight Line applied to Rational Straight Line
- Proof (related to "Prop. 10.062: Square on Second Bimedial Straight Line applied to Rational Straight Line") [2643]
- Prop. 10.063: Square on Major Straight Line applied to Rational Straight Line
- Proof (related to "Prop. 10.063: Square on Major Straight Line applied to Rational Straight Line") [2644]
- Prop. 10.064: Square on Side of Rational plus Medial Area applied to Rational Straight Line
- Proof (related to "Prop. 10.064: Square on Side of Rational plus Medial Area applied to Rational Straight Line") [2645]
- Prop. 10.065: Square on Side of Sum of two Medial Area applied to Rational Straight Line
- Proof (related to "Prop. 10.065: Square on Side of Sum of two Medial Area applied to Rational Straight Line") [2646]
- Prop. 10.066: Straight Line Commensurable with Binomial Straight Line is Binomial and of Same Order
- Proof (related to "Prop. 10.066: Straight Line Commensurable with Binomial Straight Line is Binomial and of Same Order") [2647]
- Prop. 10.067: Straight Line Commensurable with Bimedial Straight Line is Bimedial and of Same Order
- Proof (related to "Prop. 10.067: Straight Line Commensurable with Bimedial Straight Line is Bimedial and of Same Order") [2648]
- Prop. 10.068: Straight Line Commensurable with Major Straight Line is Major
- Proof (related to "Prop. 10.068: Straight Line Commensurable with Major Straight Line is Major") [2649]
- Prop. 10.069: Straight Line Commensurable with Side of Rational plus Medial Area
- Proof (related to "Prop. 10.069: Straight Line Commensurable with Side of Rational plus Medial Area") [2650]
- Prop. 10.070: Straight Line Commensurable with Side of Sum of two Medial Areas
- Proof (related to "Prop. 10.070: Straight Line Commensurable with Side of Sum of two Medial Areas") [2651]
- Prop. 10.071: Sum of Rational Area and Medial Area gives rise to four Irrational Straight Lines
- Proof (related to "Prop. 10.071: Sum of Rational Area and Medial Area gives rise to four Irrational Straight Lines") [2652]
- Prop. 10.072: Sum of two Incommensurable Medial Areas give rise to two Irrational Straight Lines
- Proof (related to "Prop. 10.072: Sum of two Incommensurable Medial Areas give rise to two Irrational Straight Lines") [2653]
- Prop. 10.073: Apotome is Irrational
- Proof (related to "Prop. 10.073: Apotome is Irrational") [2654]
- Prop. 10.074: First Apotome of Medial is Irrational
- Proof (related to "Prop. 10.074: First Apotome of Medial is Irrational") [2655]
- Prop. 10.075: Second Apotome of Medial is Irrational
- Proof (related to "Prop. 10.075: Second Apotome of Medial is Irrational") [2656]
- Prop. 10.076: Minor is Irrational
- Proof (related to "Prop. 10.076: Minor is Irrational") [2657]
- Prop. 10.077: That which produces Medial Whole with Rational Area is Irrational
- Proof (related to "Prop. 10.077: That which produces Medial Whole with Rational Area is Irrational") [2658]
- Prop. 10.078: That which produces Medial Whole with Medial Area is Irrational
- Proof (related to "Prop. 10.078: That which produces Medial Whole with Medial Area is Irrational") [2659]
- Prop. 10.079: Construction of Apotome is Unique
- Proof (related to "Prop. 10.079: Construction of Apotome is Unique") [2660]
- Prop. 10.080: Construction of First Apotome of Medial is Unique
- Proof (related to "Prop. 10.080: Construction of First Apotome of Medial is Unique") [2661]
- Prop. 10.081: Construction of Second Apotome of Medial is Unique
- Proof (related to "Prop. 10.081: Construction of Second Apotome of Medial is Unique") [2662]
- Prop. 10.082: Construction of Minor is Unique
- Proof (related to "Prop. 10.082: Construction of Minor is Unique") [2663]
- Prop. 10.083: Construction of that which produces Medial Whole with Rational Area is Unique
- Proof (related to "Prop. 10.083: Construction of that which produces Medial Whole with Rational Area is Unique") [2664]
- Prop. 10.084: Construction of that which produces Medial Whole with Medial Area is Unique
- Proof (related to "Prop. 10.084: Construction of that which produces Medial Whole with Medial Area is Unique") [2665]
- Prop. 10.085: Construction of First Apotome
- Proof (related to "Prop. 10.085: Construction of First Apotome") [2666]
- Prop. 10.086: Construction of Second Apotome
- Proof (related to "Prop. 10.086: Construction of Second Apotome") [2667]
- Prop. 10.087: Construction of Third Apotome
- Proof (related to "Prop. 10.087: Construction of Third Apotome") [2668]
- Prop. 10.088: Construction of Fourth Apotome
- Proof (related to "Prop. 10.088: Construction of Fourth Apotome") [2669]
- Prop. 10.089: Construction of Fifth Apotome
- Proof (related to "Prop. 10.089: Construction of Fifth Apotome") [2670]
- Prop. 10.090: Construction of Sixth Apotome
- Proof (related to "Prop. 10.090: Construction of Sixth Apotome") [2671]
- Prop. 10.091: Side of Area Contained by Rational Straight Line and First Apotome
- Proof (related to "Prop. 10.091: Side of Area Contained by Rational Straight Line and First Apotome") [2672]
- Prop. 10.092: Side of Area Contained by Rational Straight Line and Second Apotome
- Proof (related to "Prop. 10.092: Side of Area Contained by Rational Straight Line and Second Apotome") [2673]
- Prop. 10.093: Side of Area Contained by Rational Straight Line and Third Apotome
- Proof (related to "Prop. 10.093: Side of Area Contained by Rational Straight Line and Third Apotome") [2674]
- Prop. 10.094: Side of Area Contained by Rational Straight Line and Fourth Apotome
- Proof (related to "Prop. 10.094: Side of Area Contained by Rational Straight Line and Fourth Apotome") [2675]
- Prop. 10.095: Side of Area Contained by Rational Straight Line and Fifth Apotome
- Proof (related to "Prop. 10.095: Side of Area Contained by Rational Straight Line and Fifth Apotome") [2676]
- Prop. 10.096: Side of Area Contained by Rational Straight Line and Sixth Apotome
- Proof (related to "Prop. 10.096: Side of Area Contained by Rational Straight Line and Sixth Apotome") [2677]
- Prop. 10.097: Square on Apotome applied to Rational Straight Line
- Proof (related to "Prop. 10.097: Square on Apotome applied to Rational Straight Line") [2678]
- Prop. 10.098: Square on First Apotome of Medial Straight Line applied to Rational Straight Line
- Proof (related to "Prop. 10.098: Square on First Apotome of Medial Straight Line applied to Rational Straight Line") [2679]
- Prop. 10.099: Square on Second Apotome of Medial Straight Line applied to Rational Straight Line
- Proof (related to "Prop. 10.099: Square on Second Apotome of Medial Straight Line applied to Rational Straight Line") [2680]
- Prop. 10.100: Square on Minor Straight Line applied to Rational Straight Line
- Proof (related to "Prop. 10.100: Square on Minor Straight Line applied to Rational Straight Line") [2681]
- Prop. 10.101: Square on Straight Line which produces Medial Whole with Rational Area applied to Rational Straight Line
- Proof (related to "Prop. 10.101: Square on Straight Line which produces Medial Whole with Rational Area applied to Rational Straight Line") [2682]
- Prop. 10.102: Square on Straight Line which produces Medial Whole with Medial Area applied to Rational Straight Line
- Proof (related to "Prop. 10.102: Square on Straight Line which produces Medial Whole with Medial Area applied to Rational Straight Line") [2683]
- Prop. 10.103: Straight Line Commensurable with Apotome
- Proof (related to "Prop. 10.103: Straight Line Commensurable with Apotome") [2684]
- Prop. 10.104: Straight Line Commensurable with Apotome of Medial Straight Line
- Proof (related to "Prop. 10.104: Straight Line Commensurable with Apotome of Medial Straight Line") [2685]
- Prop. 10.105: Straight Line Commensurable with Minor Straight Line
- Proof (related to "Prop. 10.105: Straight Line Commensurable with Minor Straight Line") [2686]
- Prop. 10.106: Straight Line Commensurable with that which produces Medial Whole with Rational Area
- Proof (related to "Prop. 10.106: Straight Line Commensurable with that which produces Medial Whole with Rational Area") [2687]
- Prop. 10.107: Straight Line Commensurable With That Which Produces Medial Whole With Medial Area
- Proof (related to "Prop. 10.107: Straight Line Commensurable With That Which Produces Medial Whole With Medial Area") [2688]
- Prop. 10.108: Side of Remaining Area from Rational Area from which Medial Area Subtracted
- Proof (related to "Prop. 10.108: Side of Remaining Area from Rational Area from which Medial Area Subtracted") [2689]
- Prop. 10.109: Two Irrational Straight Lines arising from Medial Area from which Rational Area Subtracted
- Proof (related to "Prop. 10.109: Two Irrational Straight Lines arising from Medial Area from which Rational Area Subtracted") [2690]
- Prop. 10.110: Two Irrational Straight Lines arising from Medial Area from which Medial Area Subtracted
- Proof (related to "Prop. 10.110: Two Irrational Straight Lines arising from Medial Area from which Medial Area Subtracted") [2691]
- Prop. 10.111: Apotome not same with Binomial Straight Line
- Proof (related to "Prop. 10.111: Apotome not same with Binomial Straight Line") [2692]
- Prop. 10.112: Square on Rational Straight Line applied to Binomial Straight Line
- Proof (related to "Prop. 10.112: Square on Rational Straight Line applied to Binomial Straight Line") [2693]
- Prop. 10.113: Square on Rational Straight Line applied to Apotome
- Proof (related to "Prop. 10.113: Square on Rational Straight Line applied to Apotome") [2694]
- Prop. 10.114: Area contained by Apotome and Binomial Straight Line Commensurable with Terms of Apotome and in same Ratio
- Proof (related to "Prop. 10.114: Area contained by Apotome and Binomial Straight Line Commensurable with Terms of Apotome and in same Ratio") [2695]
- Prop. 10.115: From Medial Straight Line arises Infinite Number of Irrational Straight Lines
- Proof (related to "Prop. 10.115: From Medial Straight Line arises Infinite Number of Irrational Straight Lines") [2696]
- Prop. 11.01: Straight Line cannot be in Two Planes
- Proof (related to "Prop. 11.01: Straight Line cannot be in Two Planes") [2697]
- Prop. 11.02: Two Intersecting Straight Lines are in One Plane
- Proof (related to "Prop. 11.02: Two Intersecting Straight Lines are in One Plane") [2698]
- Prop. 11.03: Common Section of Two Planes is Straight Line
- Proof (related to "Prop. 11.03: Common Section of Two Planes is Straight Line") [2699]
- Prop. 11.04: Line Perpendicular to Two Intersecting Lines is Perpendicular to their Plane
- Proof (related to "Prop. 11.04: Line Perpendicular to Two Intersecting Lines is Perpendicular to their Plane") [2700]
- Prop. 11.05: Three Intersecting Lines Perpendicular to Another Line are in One Plane
- Proof (related to "Prop. 11.05: Three Intersecting Lines Perpendicular to Another Line are in One Plane") [2701]
- Prop. 11.06: Two Lines Perpendicular to Same Plane are Parallel
- Proof (related to "Prop. 11.06: Two Lines Perpendicular to Same Plane are Parallel") [2702]
- Prop. 11.07: Line joining Points on Parallel Lines is in Same Plane
- Proof (related to "Prop. 11.07: Line joining Points on Parallel Lines is in Same Plane") [2703]
- Prop. 11.08: Line Parallel to Perpendicular Line to Plane is Perpendicular to Same Plane
- Proof (related to "Prop. 11.08: Line Parallel to Perpendicular Line to Plane is Perpendicular to Same Plane") [2704]
- Prop. 11.09: Lines Parallel to Same Line not in Same Plane are Parallel to each other
- Proof (related to "Prop. 11.09: Lines Parallel to Same Line not in Same Plane are Parallel to each other") [2705]
- Prop. 11.10: Two Lines Meeting which are Parallel to Two Other Lines Meeting contain Equal Angles
- Proof (related to "Prop. 11.10: Two Lines Meeting which are Parallel to Two Other Lines Meeting contain Equal Angles") [2706]
- Prop. 11.11: Construction of Straight Line Perpendicular to Plane from point not on Plane
- Proof (related to "Prop. 11.11: Construction of Straight Line Perpendicular to Plane from point not on Plane") [2707]
- Prop. 11.12: Construction of Straight Line Perpendicular to Plane from point on Plane
- Proof (related to "Prop. 11.12: Construction of Straight Line Perpendicular to Plane from point on Plane") [2708]
- Prop. 11.13: Straight Line Perpendicular to Plane from Point is Unique
- Proof (related to "Prop. 11.13: Straight Line Perpendicular to Plane from Point is Unique") [2709]
- Prop. 11.14: Planes Perpendicular to same Straight Line are Parallel
- Proof (related to "Prop. 11.14: Planes Perpendicular to same Straight Line are Parallel") [2710]
- Prop. 11.15: Planes through Parallel Pairs of Meeting Lines are Parallel
- Proof (related to "Prop. 11.15: Planes through Parallel Pairs of Meeting Lines are Parallel") [2711]
- Prop. 11.16: Common Sections of Parallel Planes with other Plane are Parallel
- Proof (related to "Prop. 11.16: Common Sections of Parallel Planes with other Plane are Parallel") [2712]
- Prop. 11.17: Straight Lines cut in Same Ratio by Parallel Planes
- Proof (related to "Prop. 11.17: Straight Lines cut in Same Ratio by Parallel Planes") [2713]
- Prop. 11.18: Plane through Straight Line Perpendicular to other Plane is Perpendicular to that Plane
- Proof (related to "Prop. 11.18: Plane through Straight Line Perpendicular to other Plane is Perpendicular to that Plane") [2714]
- Prop. 11.19: Common Section of Planes Perpendicular to other Plane is Perpendicular to that Plane
- Proof (related to "Prop. 11.19: Common Section of Planes Perpendicular to other Plane is Perpendicular to that Plane") [2715]
- Prop. 11.20: Sum of Two Angles of Three containing Solid Angle is Greater than Other Angle
- Proof (related to "Prop. 11.20: Sum of Two Angles of Three containing Solid Angle is Greater than Other Angle") [2716]
- Prop. 11.21: Solid Angle contained by Plane Angles is Less than Four Right Angles
- Proof (related to "Prop. 11.21: Solid Angle contained by Plane Angles is Less than Four Right Angles") [2717]
- Prop. 11.22: Extremities of Line Segments containing three Plane Angles any Two of which are Greater than Other form Triangle
- Proof (related to "Prop. 11.22: Extremities of Line Segments containing three Plane Angles any Two of which are Greater than Other form Triangle") [2718]
- Prop. 11.23: Sum of Plane Angles Used to Construct a Solid Angle is Less Than Four Right Angles
- Proof (related to "Prop. 11.23: Sum of Plane Angles Used to Construct a Solid Angle is Less Than Four Right Angles") [2719]
- Prop. 11.24: Opposite Planes of Solid contained by Parallel Planes are Equal Parallelograms
- Proof (related to "Prop. 11.24: Opposite Planes of Solid contained by Parallel Planes are Equal Parallelograms") [2721]
- Prop. 11.25: Parallelepiped cut by Plane Parallel to Opposite Planes
- Proof (related to "Prop. 11.25: Parallelepiped cut by Plane Parallel to Opposite Planes") [2722]
- Prop. 11.26: Construction of Solid Angle equal to Given Solid Angle
- Proof (related to "Prop. 11.26: Construction of Solid Angle equal to Given Solid Angle") [2723]
- Prop. 11.27: Construction of Parallelepiped Similar to Given Parallelepiped
- Proof (related to "Prop. 11.27: Construction of Parallelepiped Similar to Given Parallelepiped") [2724]
- Prop. 11.28: Parallelepiped cut by Plane through Diagonals of Opposite Planes is Bisected
- Proof (related to "Prop. 11.28: Parallelepiped cut by Plane through Diagonals of Opposite Planes is Bisected") [2725]
- Prop. 11.29: Parallelepipeds on Same Base and Same Height whose Extremities are on Same Lines are Equal in Volume
- Proof (related to "Prop. 11.29: Parallelepipeds on Same Base and Same Height whose Extremities are on Same Lines are Equal in Volume") [2726]
- Prop. 11.30: Parallelepipeds on Same Base and Same Height whose Extremities are not on Same Lines are Equal in Volume
- Proof (related to "Prop. 11.30: Parallelepipeds on Same Base and Same Height whose Extremities are not on Same Lines are Equal in Volume") [2727]
- Prop. 11.31: Parallelepipeds on Equal Bases and Same Height are Equal in Volume
- Proof (related to "Prop. 11.31: Parallelepipeds on Equal Bases and Same Height are Equal in Volume") [2728]
- Prop. 11.32: Parallelepipeds of Same Height have Volume Proportional to Bases
- Proof (related to "Prop. 11.32: Parallelepipeds of Same Height have Volume Proportional to Bases") [2729]
- Prop. 11.33: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides
- Proof (related to "Prop. 11.33: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides") [2730]
- Prop. 11.34: Parallelepipeds are of Equal Volume iff Bases are in Reciprocal Proportion to Heights
- Proof (related to "Prop. 11.34: Parallelepipeds are of Equal Volume iff Bases are in Reciprocal Proportion to Heights") [2731]
- Prop. 11.35: Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles
- Proof (related to "Prop. 11.35: Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles") [2732]
- Prop. 11.36: Parallelepiped formed from Three Proportional Lines equal to Equilateral Parallelepiped with Equal Angles to it forme
- Proof (related to "Prop. 11.36: Parallelepiped formed from Three Proportional Lines equal to Equilateral Parallelepiped with Equal Angles to it forme") [2733]
- Prop. 11.37: Four Straight Lines are Proportional iff Similar Parallelepipeds formed on them are Proportional
- Proof (related to "Prop. 11.37: Four Straight Lines are Proportional iff Similar Parallelepipeds formed on them are Proportional") [2734]
- Prop. 11.38: Common Section of Bisecting Planes of Cube Bisect and are Bisected by Diagonal of Cube
- Proof (related to "Prop. 11.38: Common Section of Bisecting Planes of Cube Bisect and are Bisected by Diagonal of Cube") [2735]
- Prop. 11.39: Prisms of Equal Height with Parallelogram and Triangle as Base
- Proof (related to "Prop. 11.39: Prisms of Equal Height with Parallelogram and Triangle as Base") [2736]
- Prop. 12.01: Areas of Similar Polygons Inscribed in Circles are as Squares on Diameters
- Proof (related to "Prop. 12.01: Areas of Similar Polygons Inscribed in Circles are as Squares on Diameters") [2737]
- Prop. 12.02: Areas of Circles are as Squares on Diameters
- Proof (related to "Prop. 12.02: Areas of Circles are as Squares on Diameters") [2738]
- Prop. 12.03: Tetrahedron divided into Two Similar Tetrahedra and Two Equal Prisms
- Proof (related to "Prop. 12.03: Tetrahedron divided into Two Similar Tetrahedra and Two Equal Prisms") [2739]
- Prop. 12.04: Proportion of Sizes of Tetrahedra divided into Two Similar Tetrahedra and Two Equal Prisms
- Proof (related to "Prop. 12.04: Proportion of Sizes of Tetrahedra divided into Two Similar Tetrahedra and Two Equal Prisms") [2740]
- Prop. 12.05: Sizes of Tetrahedra of Same Height are as Bases
- Proof (related to "Prop. 12.05: Sizes of Tetrahedra of Same Height are as Bases") [2741]
- Prop. 12.06: Sizes of Pyramids of Same Height with Polygonal Bases are as Bases
- Proof (related to "Prop. 12.06: Sizes of Pyramids of Same Height with Polygonal Bases are as Bases") [2742]
- Prop. 12.07: Prism on Triangular Base divided into Three Equal Tetrahedra
- Proof (related to "Prop. 12.07: Prism on Triangular Base divided into Three Equal Tetrahedra") [2743]
- Prop. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides
- Proof (related to "Prop. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides") [2744]
- Prop. 12.09: Tetrahedra are Equal iff Bases are Reciprocally Proportional to Heights
- Proof (related to "Prop. 12.09: Tetrahedra are Equal iff Bases are Reciprocally Proportional to Heights") [2745]
- Prop. 12.10: Volume of Cone is Third of Cylinder on Same Base and of Same Height
- Proof (related to "Prop. 12.10: Volume of Cone is Third of Cylinder on Same Base and of Same Height") [2746]
- Prop. 12.11: Volume of Cones or Cylinders of Same Height are in Same Ratio as Bases
- Proof (related to "Prop. 12.11: Volume of Cones or Cylinders of Same Height are in Same Ratio as Bases") [2747]
- Prop. 12.12: Volumes of Similar Cones and Cylinders are in Triplicate Ratio of Diameters of Bases
- Proof (related to "Prop. 12.12: Volumes of Similar Cones and Cylinders are in Triplicate Ratio of Diameters of Bases") [2748]
- Prop. 12.13: Volumes of Parts of Cylinder cut by Plane Parallel to Opposite Planes are as Parts of Axis
- Proof (related to "Prop. 12.13: Volumes of Parts of Cylinder cut by Plane Parallel to Opposite Planes are as Parts of Axis") [2749]
- Prop. 12.14: Volumes of Cones or Cylinders on Equal Bases are in Same Ratio as Heights
- Proof (related to "Prop. 12.14: Volumes of Cones or Cylinders on Equal Bases are in Same Ratio as Heights") [2750]
- Prop. 12.15: Cones or Cylinders are Equal iff Bases are Reciprocally Proportional to Heights
- Proof (related to "Prop. 12.15: Cones or Cylinders are Equal iff Bases are Reciprocally Proportional to Heights") [2751]
- Prop. 12.16: Construction of Equilateral Polygon with Even Number of Sides in Outer of Concentric Circles
- Proof (related to "Prop. 12.16: Construction of Equilateral Polygon with Even Number of Sides in Outer of Concentric Circles") [2752]
- Prop. 12.17: Construction of Polyhedron in Outer of Concentric Spheres
- Proof (related to "Prop. 12.17: Construction of Polyhedron in Outer of Concentric Spheres") [2753]
- Prop. 12.18: Volumes of Spheres are in Triplicate Ratio of Diameters
- Proof (related to "Prop. 12.18: Volumes of Spheres are in Triplicate Ratio of Diameters") [2754]
- Prop. 13.01: Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio
- Proof (related to "Prop. 13.01: Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio") [2755]
- Prop. 13.02: Converse of Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio
- Proof (related to "Prop. 13.02: Converse of Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio") [2756]
- Prop. 13.03: Area of Square on Lesser Segment of Straight Line cut in Extreme and Mean Ratio
- Proof (related to "Prop. 13.03: Area of Square on Lesser Segment of Straight Line cut in Extreme and Mean Ratio") [2757]
- Prop. 13.04: Area of Squares on Whole and Lesser Segment of Straight Line cut in Extreme and Mean Ratio
- Proof (related to "Prop. 13.04: Area of Squares on Whole and Lesser Segment of Straight Line cut in Extreme and Mean Ratio") [2758]
- Prop. 13.05: Straight Line cut in Extreme and Mean Ratio plus its Greater Segment
- Proof (related to "Prop. 13.05: Straight Line cut in Extreme and Mean Ratio plus its Greater Segment") [2759]
- Prop. 13.06: Segments of Rational Straight Line cut in Extreme and Mean Ratio are Apotome
- Proof (related to "Prop. 13.06: Segments of Rational Straight Line cut in Extreme and Mean Ratio are Apotome") [2760]
- Prop. 13.07: Equilateral Pentagon is Equiangular if Three Angles are Equal
- Proof (related to "Prop. 13.07: Equilateral Pentagon is Equiangular if Three Angles are Equal") [2761]
- Prop. 13.08: Straight Lines Subtending Two Consecutive Angles in Regular Pentagon cut in Extreme and Mean Ratio
- Proof (related to "Prop. 13.08: Straight Lines Subtending Two Consecutive Angles in Regular Pentagon cut in Extreme and Mean Ratio") [2762]
- Prop. 13.09: Sides Appended of Hexagon and Decagon inscribed in same Circle are cut in Extreme and Mean Ratio
- Proof (related to "Prop. 13.09: Sides Appended of Hexagon and Decagon inscribed in same Circle are cut in Extreme and Mean Ratio") [2763]
- Prop. 13.10: Square on Side of Regular Pentagon inscribed in Circle equals Squares on Sides of Hexagon and Decagon inscribed in sa
- Proof (related to "Prop. 13.10: Square on Side of Regular Pentagon inscribed in Circle equals Squares on Sides of Hexagon and Decagon inscribed in sa") [2764]
- Prop. 13.11: Side of Regular Pentagon inscribed in Circle with Rational Diameter is Minor
- Proof (related to "Prop. 13.11: Side of Regular Pentagon inscribed in Circle with Rational Diameter is Minor") [2765]
- Prop. 13.12: Square on Side of Equilateral Triangle inscribed in Circle is Triple Square on Radius of Circle
- Proof (related to "Prop. 13.12: Square on Side of Equilateral Triangle inscribed in Circle is Triple Square on Radius of Circle") [2766]
- Prop. 13.13: Construction of Regular Tetrahedron within Given Sphere
- Proof (related to "Prop. 13.13: Construction of Regular Tetrahedron within Given Sphere") [2767]
- Prop. 13.14: Construction of Regular Octahedron within Given Sphere
- Proof (related to "Prop. 13.14: Construction of Regular Octahedron within Given Sphere") [2768]
- Prop. 13.15: Construction of Cube within Given Sphere
- Proof (related to "Prop. 13.15: Construction of Cube within Given Sphere") [2769]
- Prop. 13.16: Construction of Regular Icosahedron within Given Sphere
- Proof (related to "Prop. 13.16: Construction of Regular Icosahedron within Given Sphere") [2770]
- Prop. 13.17: Construction of Regular Dodecahedron within Given Sphere
- Proof (related to "Prop. 13.17: Construction of Regular Dodecahedron within Given Sphere") [2771]
- Prop. 13.18: Comparison of Sides of Platonic Figures - There are only Five Platonic Solids
- Proof (related to "Prop. 13.18: Comparison of Sides of Platonic Figures - There are only Five Platonic Solids") [2772]
- Prop. 2.01: Summing Areas or Rectangles
- Proof (related to "Prop. 2.01: Summing Areas or Rectangles") [1016]
- Proof (related to "Prop. 2.01: Summing Areas or Rectangles") [6532]
- Prop. 2.02: Square is Sum of Two Rectangles
- Proof (related to "Prop. 2.02: Square is Sum of Two Rectangles") [2511]
- Prop. 2.03: Rectangle is Sum of Square and Rectangle
- Proof (related to "Prop. 2.03: Rectangle is Sum of Square and Rectangle") [2560]
- Prop. 2.04: Square of Sum
- Proof (related to "Prop. 2.04: Square of Sum") [6533]
- Geometric Proof (related to "Prop. 2.04: Square of Sum") [1018]
- Prop. 2.05: Rectangle is Difference of Two Squares
- Proof (related to "Prop. 2.05: Rectangle is Difference of Two Squares") [6534]
- Prop. 2.06: Square of Sum with One Halved Summand
- Proof (related to "Prop. 2.06: Square of Sum with One Halved Summand") [6535]
- Prop. 2.07: Sum of Squares
- Proof (related to "Prop. 2.07: Sum of Squares") [6536]
- Prop. 2.08: Square of Sum with One Doubled Summand
- Proof (related to "Prop. 2.08: Square of Sum with One Doubled Summand") [6537]
- Prop. 2.09: Sum of Squares of Sum and Difference
- Proof (related to "Prop. 2.09: Sum of Squares of Sum and Difference") [6538]
- Prop. 2.10: Sum of Squares (II)
- Proof (related to "Prop. 2.10: Sum of Squares (II)") [6539]
- Prop. 2.11: Constructing the Golden Ratio of a Segment
- Proof (related to "Prop. 2.11: Constructing the Golden Ratio of a Segment") [6540]
- Prop. 2.12: Law of Cosines (for Obtuse Angles)
- Proof (related to "Prop. 2.12: Law of Cosines (for Obtuse Angles)") [6541]
- Prop. 2.13: Law of Cosines (for Acute Angles)
- Proof (related to "Prop. 2.13: Law of Cosines (for Acute Angles)") [6542]
- Prop. 2.14: Constructing a Square from a Rectilinear Figure
- Proof (related to "Prop. 2.14: Constructing a Square from a Rectilinear Figure") [6543]
- Geometric Proof (related to "Prop. 2.14: Constructing a Square from a Rectilinear Figure") [1029]
- Prop. 3.01: Finding the Centre of a given Circle
- Proof (related to "Prop. 3.01: Finding the Centre of a given Circle") [6544]
- Geometric Proof (related to "Prop. 3.01: Finding the Centre of a given Circle") [1059]
- Prop. 3.02: Chord Lies Inside its Circle
- Proof (related to "Prop. 3.02: Chord Lies Inside its Circle") [6545]
- Prop. 3.03: Conditions for Diameter to be Perpendicular Bisector
- Proof (related to "Prop. 3.03: Conditions for Diameter to be Perpendicular Bisector") [2373]
- Prop. 3.04: Chords do not Bisect Each Other
- Proof (related to "Prop. 3.04: Chords do not Bisect Each Other") [2374]
- Prop. 3.05: Intersecting Circles have Different Centers
- Proof (related to "Prop. 3.05: Intersecting Circles have Different Centers") [2375]
- Prop. 3.06: Touching Circles have Different Centers
- Proof (related to "Prop. 3.06: Touching Circles have Different Centers") [2376]
- Prop. 3.07: Relative Lengths of Lines Inside Circle
- Proof (related to "Prop. 3.07: Relative Lengths of Lines Inside Circle") [2377]
- Prop. 3.08: Relative Lengths of Lines Outside Circle
- Proof (related to "Prop. 3.08: Relative Lengths of Lines Outside Circle") [2378]
- Prop. 3.09: Condition for Point to be Center of Circle
- Proof (related to "Prop. 3.09: Condition for Point to be Center of Circle") [2379]
- Prop. 3.10: Two Circles have at most Two Points of Intersection
- Proof (related to "Prop. 3.10: Two Circles have at most Two Points of Intersection") [2380]
- Prop. 3.11: Line Joining Centers of Two Circles Touching Internally
- Proof (related to "Prop. 3.11: Line Joining Centers of Two Circles Touching Internally") [2381]
- Prop. 3.12: Line Joining Centers of Two Circles Touching Externally
- Proof (related to "Prop. 3.12: Line Joining Centers of Two Circles Touching Externally") [2382]
- Prop. 3.13: Circles Touch at One Point at Most
- Proof (related to "Prop. 3.13: Circles Touch at One Point at Most") [2383]
- Prop. 3.14: Equal Chords in Circle
- Proof (related to "Prop. 3.14: Equal Chords in Circle") [2384]
- Prop. 3.15: Relative Lengths of Chords of Circles
- Proof (related to "Prop. 3.15: Relative Lengths of Chords of Circles") [2385]
- Prop. 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle
- Proof (related to "Prop. 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle") [2386]
- Prop. 3.17: Construction of Tangent from Point to Circle
- Proof (related to "Prop. 3.17: Construction of Tangent from Point to Circle") [2387]
- Prop. 3.18: Radius at Right Angle to Tangent
- Proof (related to "Prop. 3.18: Radius at Right Angle to Tangent") [2388]
- Prop. 3.19: Right Angle to Tangent of Circle goes through Center
- Proof (related to "Prop. 3.19: Right Angle to Tangent of Circle goes through Center") [2389]
- Prop. 3.20: Inscribed Angle Theorem
- Proof (related to "Prop. 3.20: Inscribed Angle Theorem") [2390]
- Prop. 3.21: Angles in Same Segment of Circle are Equal
- Proof (related to "Prop. 3.21: Angles in Same Segment of Circle are Equal") [2391]
- Prop. 3.22: Opposite Angles of Cyclic Quadrilateral
- Proof (related to "Prop. 3.22: Opposite Angles of Cyclic Quadrilateral") [2392]
- Prop. 3.23: Segment on Given Base Unique
- Proof (related to "Prop. 3.23: Segment on Given Base Unique") [2393]
- Prop. 3.24: Similar Segments on Equal Bases are Equal
- Proof (related to "Prop. 3.24: Similar Segments on Equal Bases are Equal") [2394]
- Prop. 3.25: Construction of Circle from Segment
- Proof (related to "Prop. 3.25: Construction of Circle from Segment") [2395]
- Prop. 3.26: Equal Angles in Equal Circles
- Proof (related to "Prop. 3.26: Equal Angles in Equal Circles") [2396]
- Prop. 3.27: Angles on Equal Arcs are Equal
- Proof (related to "Prop. 3.27: Angles on Equal Arcs are Equal") [2397]
- Prop. 3.28: Straight Lines Cut Off Equal Arcs in Equal Circles
- Proof (related to "Prop. 3.28: Straight Lines Cut Off Equal Arcs in Equal Circles") [2398]
- Prop. 3.29: Equal Arcs of Circles Subtended by Equal Straight Lines
- Proof (related to "Prop. 3.29: Equal Arcs of Circles Subtended by Equal Straight Lines") [2399]
- Prop. 3.30: Bisection of Arc
- Proof (related to "Prop. 3.30: Bisection of Arc") [2400]
- Prop. 3.31: Relative Sizes of Angles in Segments
- Proof (related to "Prop. 3.31: Relative Sizes of Angles in Segments") [2401]
- Prop. 3.32: Angles made by Chord with Tangent
- Proof (related to "Prop. 3.32: Angles made by Chord with Tangent") [2402]
- Prop. 3.33: Construction of Segment on Given Line Admitting Given Angle
- Proof (related to "Prop. 3.33: Construction of Segment on Given Line Admitting Given Angle") [2403]
- Prop. 3.34: Construction of Segment on Given Circle Admitting Given Angle
- Proof (related to "Prop. 3.34: Construction of Segment on Given Circle Admitting Given Angle") [2404]
- Prop. 3.35: Intersecting Chord Theorem
- Proof (related to "Prop. 3.35: Intersecting Chord Theorem") [2405]
- Prop. 3.36: Tangent Secant Theorem
- Proof (related to "Prop. 3.36: Tangent Secant Theorem") [2406]
- Prop. 3.37: Converse of Tangent Secant Theorem
- Proof (related to "Prop. 3.37: Converse of Tangent Secant Theorem") [2407]
- Prop. 4.01: Fitting Chord Into Circle
- Proof (related to "Prop. 4.01: Fitting Chord Into Circle") [2408]
- Prop. 4.02: Inscribing in Circle Triangle Equiangular with Given
- Proof (related to "Prop. 4.02: Inscribing in Circle Triangle Equiangular with Given") [2409]
- Prop. 4.03: Circumscribing about Circle Triangle Equiangular with Given
- Proof (related to "Prop. 4.03: Circumscribing about Circle Triangle Equiangular with Given") [2410]
- Prop. 4.04: Inscribing Circle in Triangle
- Proof (related to "Prop. 4.04: Inscribing Circle in Triangle") [2411]
- Prop. 4.05: Circumscribing Circle about Triangle
- Proof (related to "Prop. 4.05: Circumscribing Circle about Triangle") [2412]
- Prop. 4.06: Inscribing Square in Circle
- Proof (related to "Prop. 4.06: Inscribing Square in Circle") [2413]
- Prop. 4.07: Circumscribing Square about Circle
- Proof (related to "Prop. 4.07: Circumscribing Square about Circle") [2414]
- Prop. 4.08: Inscribing Circle in Square
- Proof (related to "Prop. 4.08: Inscribing Circle in Square") [2415]
- Prop. 4.09: Circumscribing Circle about Square
- Proof (related to "Prop. 4.09: Circumscribing Circle about Square") [2416]
- Prop. 4.10: Construction of Isosceles Triangle whose Base Angle is Twice Apex
- Proof (related to "Prop. 4.10: Construction of Isosceles Triangle whose Base Angle is Twice Apex") [2417]
- Prop. 4.11: Inscribing Regular Pentagon in Circle
- Proof (related to "Prop. 4.11: Inscribing Regular Pentagon in Circle") [2418]
- Prop. 4.12: Circumscribing Regular Pentagon about Circle
- Proof (related to "Prop. 4.12: Circumscribing Regular Pentagon about Circle") [2419]
- Prop. 4.13: Inscribing Circle in Regular Pentagon
- Proof (related to "Prop. 4.13: Inscribing Circle in Regular Pentagon") [2420]
- Prop. 4.14: Circumscribing Circle about Regular Pentagon
- Proof (related to "Prop. 4.14: Circumscribing Circle about Regular Pentagon") [2421]
- Prop. 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle
- Proof (related to "Prop. 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle") [2422]
- Prop. 4.16: Inscribing Regular 15-gon in Circle
- Proof (related to "Prop. 4.16: Inscribing Regular 15-gon in Circle") [2423]
- Prop. 5.01: Multiplication of Numbers is Left Distributive over Addition
- Proof (related to "Prop. 5.01: Multiplication of Numbers is Left Distributive over Addition") [2786]
- Proof (related to "Prop. 5.01: Multiplication of Numbers is Left Distributive over Addition") [6546]
- Geometric Proof (related to "Prop. 5.01: Multiplication of Numbers is Left Distributive over Addition") [2424]
- Prop. 5.02: Multiplication of Numbers is Right Distributive over Addition
- Proof (related to "Prop. 5.02: Multiplication of Numbers is Right Distributive over Addition") [2425]
- Proof (related to "Prop. 5.02: Multiplication of Numbers is Right Distributive over Addition") [6547]
- Geometric Proof (related to "Prop. 5.02: Multiplication of Numbers is Right Distributive over Addition") [2787]
- Prop. 5.03: Multiplication of Numbers is Associative
- Proof (related to "Prop. 5.03: Multiplication of Numbers is Associative") [2426]
- Proof (related to "Prop. 5.03: Multiplication of Numbers is Associative") [6548]
- Geometric Proof (related to "Prop. 5.03: Multiplication of Numbers is Associative") [2788]
- Prop. 5.04: Multiples of Terms in Equal Ratios
- Proof (related to "Prop. 5.04: Multiples of Terms in Equal Ratios") [2427]
- Proof (related to "Prop. 5.04: Multiples of Terms in Equal Ratios") [6549]
- Prop. 5.05: Multiplication of Real Numbers is Left Distributive over Subtraction
- Proof (related to "Prop. 5.05: Multiplication of Real Numbers is Left Distributive over Subtraction") [2428]
- Prop. 5.06: Multiplication of Real Numbers is Right Distributive over Subtraction
- Proof (related to "Prop. 5.06: Multiplication of Real Numbers is Right Distributive over Subtraction") [2429]
- Prop. 5.07: Ratios of Equal Magnitudes
- Proof (related to "Prop. 5.07: Ratios of Equal Magnitudes") [2430]
- Prop. 5.08: Relative Sizes of Ratios on Unequal Magnitudes
- Proof (related to "Prop. 5.08: Relative Sizes of Ratios on Unequal Magnitudes") [2431]
- Prop. 5.09: Magnitudes with Same Ratios are Equal
- Proof (related to "Prop. 5.09: Magnitudes with Same Ratios are Equal") [2432]
- Prop. 5.10: Relative Sizes of Magnitudes on Unequal Ratios
- Proof (related to "Prop. 5.10: Relative Sizes of Magnitudes on Unequal Ratios") [2433]
- Prop. 5.11: Equality of Ratios is Transitive
- Proof (related to "Prop. 5.11: Equality of Ratios is Transitive") [2434]
- Prop. 5.12: Sum of Components of Equal Ratios
- Proof (related to "Prop. 5.12: Sum of Components of Equal Ratios") [2435]
- Prop. 5.13: Relative Sizes of Proportional Magnitudes
- Proof (related to "Prop. 5.13: Relative Sizes of Proportional Magnitudes") [2436]
- Prop. 5.14: Relative Sizes of Components of Ratios
- Proof (related to "Prop. 5.14: Relative Sizes of Components of Ratios") [2437]
- Prop. 5.15: Ratio Equals its Multiples
- Proof (related to "Prop. 5.15: Ratio Equals its Multiples") [2438]
- Prop. 5.16: Proportional Magnitudes are Proportional Alternately
- Proof (related to "Prop. 5.16: Proportional Magnitudes are Proportional Alternately") [2439]
- Prop. 5.17: Magnitudes Proportional Compounded are Proportional Separated
- Proof (related to "Prop. 5.17: Magnitudes Proportional Compounded are Proportional Separated") [2440]
- Prop. 5.18: Magnitudes Proportional Separated are Proportional Compounded
- Proof (related to "Prop. 5.18: Magnitudes Proportional Separated are Proportional Compounded") [2441]
- Prop. 5.19: Proportional Magnitudes have Proportional Remainders
- Proof (related to "Prop. 5.19: Proportional Magnitudes have Proportional Remainders") [2442]
- Prop. 5.20: Relative Sizes of Successive Ratios
- Proof (related to "Prop. 5.20: Relative Sizes of Successive Ratios") [2443]
- Prop. 5.21: Relative Sizes of Elements in Perturbed Proportion
- Proof (related to "Prop. 5.21: Relative Sizes of Elements in Perturbed Proportion") [2444]
- Prop. 5.22: Equality of Ratios Ex Aequali
- Proof (related to "Prop. 5.22: Equality of Ratios Ex Aequali") [2445]
- Prop. 5.23: Equality of Ratios in Perturbed Proportion
- Proof (related to "Prop. 5.23: Equality of Ratios in Perturbed Proportion") [2446]
- Prop. 5.24: Sum of Antecedents of Proportion
- Proof (related to "Prop. 5.24: Sum of Antecedents of Proportion") [2447]
- Prop. 5.25: Sum of Antecedent and Consequent of Proportion
- Proof (related to "Prop. 5.25: Sum of Antecedent and Consequent of Proportion") [2448]
- Prop. 6.01: Areas of Triangles and Parallelograms Proportional to Base
- Proof (related to "Prop. 6.01: Areas of Triangles and Parallelograms Proportional to Base") [2449]
- Prop. 6.02: Parallel Line in Triangle Cuts Sides Proportionally
- Proof (related to "Prop. 6.02: Parallel Line in Triangle Cuts Sides Proportionally") [2450]
- Prop. 6.03: Angle Bisector Theorem
- Proof (related to "Prop. 6.03: Angle Bisector Theorem") [2451]
- Prop. 6.04: Equiangular Triangles are Similar
- Proof (related to "Prop. 6.04: Equiangular Triangles are Similar") [2452]
- Prop. 6.05: Triangles with Proportional Sides are Similar
- Proof (related to "Prop. 6.05: Triangles with Proportional Sides are Similar") [2453]
- Prop. 6.06: Triangles with One Equal Angle and Two Sides Proportional are Similar
- Proof (related to "Prop. 6.06: Triangles with One Equal Angle and Two Sides Proportional are Similar") [2454]
- Prop. 6.07: Triangles with One Equal Angle and Two Other Sides Proportional are Similar
- Proof (related to "Prop. 6.07: Triangles with One Equal Angle and Two Other Sides Proportional are Similar") [2455]
- Prop. 6.08: Perpendicular in Right-Angled Triangle makes two Similar Triangles
- Proof (related to "Prop. 6.08: Perpendicular in Right-Angled Triangle makes two Similar Triangles") [2456]
- Prop. 6.09: Construction of Part of Line
- Proof (related to "Prop. 6.09: Construction of Part of Line") [2457]
- Prop. 6.10: Construction of Similarly Cut Straight Line
- Proof (related to "Prop. 6.10: Construction of Similarly Cut Straight Line") [2458]
- Prop. 6.11: Construction of Third Proportional Straight Line
- Proof (related to "Prop. 6.11: Construction of Third Proportional Straight Line") [2459]
- Prop. 6.12: Construction of Fourth Proportional Straight Line
- Proof (related to "Prop. 6.12: Construction of Fourth Proportional Straight Line") [2460]
- Prop. 6.13: Construction of Mean Proportional
- Proof (related to "Prop. 6.13: Construction of Mean Proportional") [2461]
- Prop. 6.14: Sides of Equal and Equiangular Parallelograms are Reciprocally Proportional
- Proof (related to "Prop. 6.14: Sides of Equal and Equiangular Parallelograms are Reciprocally Proportional") [2462]
- Prop. 6.15: Sides of Equiangular Triangles are Reciprocally Proportional
- Proof (related to "Prop. 6.15: Sides of Equiangular Triangles are Reciprocally Proportional") [2463]
- Prop. 6.16: Rectangles Contained by Proportional Straight Lines
- Proof (related to "Prop. 6.16: Rectangles Contained by Proportional Straight Lines") [2464]
- Prop. 6.17: Rectangles Contained by Three Proportional Straight Lines
- Proof (related to "Prop. 6.17: Rectangles Contained by Three Proportional Straight Lines") [2465]
- Prop. 6.18: Construction of Similar Polygon
- Proof (related to "Prop. 6.18: Construction of Similar Polygon") [2466]
- Prop. 6.19: Ratio of Areas of Similar Triangles
- Proof (related to "Prop. 6.19: Ratio of Areas of Similar Triangles") [2467]
- Prop. 6.20: Similar Polygons are Composed of Similar Triangles
- Proof (related to "Prop. 6.20: Similar Polygons are Composed of Similar Triangles") [2468]
- Prop. 6.21: Similarity of Polygons is Equivalence Relation
- Proof (related to "Prop. 6.21: Similarity of Polygons is Equivalence Relation") [2469]
- Prop. 6.22: Similar Figures on Proportional Straight Lines
- Proof (related to "Prop. 6.22: Similar Figures on Proportional Straight Lines") [2470]
- Prop. 6.23: Ratio of Areas of Equiangular Parallelograms
- Proof (related to "Prop. 6.23: Ratio of Areas of Equiangular Parallelograms") [2471]
- Prop. 6.24: Parallelograms About Diameter are Similar
- Proof (related to "Prop. 6.24: Parallelograms About Diameter are Similar") [2472]
- Prop. 6.25: Construction of Figure Similar to One and Equal to Another
- Proof (related to "Prop. 6.25: Construction of Figure Similar to One and Equal to Another") [2473]
- Prop. 6.26: Parallelogram Similar and in Same Angle has Same Diameter
- Proof (related to "Prop. 6.26: Parallelogram Similar and in Same Angle has Same Diameter") [2474]
- Prop. 6.27: Similar Parallelogram on Half a Straight Line
- Proof (related to "Prop. 6.27: Similar Parallelogram on Half a Straight Line") [2475]
- Prop. 6.28: Construction of Parallelogram Equal to Given Figure Less a Parallelogram
- Proof (related to "Prop. 6.28: Construction of Parallelogram Equal to Given Figure Less a Parallelogram") [2476]
- Prop. 6.29: Construction of Parallelogram Equal to Given Figure Exceeding a Parallelogram
- Proof (related to "Prop. 6.29: Construction of Parallelogram Equal to Given Figure Exceeding a Parallelogram") [2477]
- Prop. 6.30: Construction of Golden Section
- Proof (related to "Prop. 6.30: Construction of Golden Section") [2478]
- Prop. 6.31: Similar Figures on Sides of Right-Angled Triangle
- Proof (related to "Prop. 6.31: Similar Figures on Sides of Right-Angled Triangle") [2479]
- Prop. 6.32: Triangles with Two Sides Parallel and Equal
- Proof (related to "Prop. 6.32: Triangles with Two Sides Parallel and Equal") [2480]
- Prop. 6.33: Angles in Circles have Same Ratio as Arcs
- Proof (related to "Prop. 6.33: Angles in Circles have Same Ratio as Arcs") [2481]
- Prop. 7.01: Sufficient Condition for Coprimality
- Proof (related to "Prop. 7.01: Sufficient Condition for Coprimality") [2482]
- Proof (related to "Prop. 7.01: Sufficient Condition for Coprimality") [6550]
- Prop. 7.02: Greatest Common Divisor of Two Numbers - Euclidean Algorithm
- Proof (related to "Prop. 7.02: Greatest Common Divisor of Two Numbers - Euclidean Algorithm") [2483]
- Proof (related to "Prop. 7.02: Greatest Common Divisor of Two Numbers - Euclidean Algorithm") [6551]
- Prop. 7.03: Greatest Common Divisor of Three Numbers
- Proof (related to "Prop. 7.03: Greatest Common Divisor of Three Numbers") [2484]
- Prop. 7.04: Natural Number Divisor or Multiple of Divisor of Another
- Proof (related to "Prop. 7.04: Natural Number Divisor or Multiple of Divisor of Another") [2485]
- Prop. 7.05: Divisors obey Distributive Law
- Proof (related to "Prop. 7.05: Divisors obey Distributive Law") [2486]
- Prop. 7.06: Multiples of Divisors Obey Distributive Law
- Proof (related to "Prop. 7.06: Multiples of Divisors Obey Distributive Law") [2487]
- Prop. 7.07: Subtraction of Divisors Obeys Distributive Law
- Proof (related to "Prop. 7.07: Subtraction of Divisors Obeys Distributive Law") [2488]
- Prop. 7.08: Subtraction of Multiples of Divisors Obeys Distributive Law
- Proof (related to "Prop. 7.08: Subtraction of Multiples of Divisors Obeys Distributive Law") [2489]
- Prop. 7.09: Alternate Ratios of Equal Fractions
- Proof (related to "Prop. 7.09: Alternate Ratios of Equal Fractions") [2490]
- Prop. 7.10: Multiples of Alternate Ratios of Equal Fractions
- Proof (related to "Prop. 7.10: Multiples of Alternate Ratios of Equal Fractions") [2491]
- Prop. 7.11: Proportional Numbers have Proportional Differences
- Proof (related to "Prop. 7.11: Proportional Numbers have Proportional Differences") [2492]
- Prop. 7.12: Ratios of Numbers is Distributive over Addition
- Proof (related to "Prop. 7.12: Ratios of Numbers is Distributive over Addition") [2493]
- Prop. 7.13: Proportional Numbers are Proportional Alternately
- Proof (related to "Prop. 7.13: Proportional Numbers are Proportional Alternately") [2494]
- Prop. 7.14: Proportion of Numbers is Transitive
- Proof (related to "Prop. 7.14: Proportion of Numbers is Transitive") [2495]
- Prop. 7.15: Alternate Ratios of Multiples
- Proof (related to "Prop. 7.15: Alternate Ratios of Multiples") [2496]
- Prop. 7.16: Natural Number Multiplication is Commutative
- Proof (related to "Prop. 7.16: Natural Number Multiplication is Commutative") [2497]
- Prop. 7.17: Multiples of Ratios of Numbers
- Proof (related to "Prop. 7.17: Multiples of Ratios of Numbers") [2498]
- Prop. 7.18: Ratios of Multiples of Numbers
- Proof (related to "Prop. 7.18: Ratios of Multiples of Numbers") [2499]
- Prop. 7.19: Relation of Ratios to Products
- Proof (related to "Prop. 7.19: Relation of Ratios to Products") [2500]
- Prop. 7.20: Ratios of Fractions in Lowest Terms
- Proof (related to "Prop. 7.20: Ratios of Fractions in Lowest Terms") [2501]
- Prop. 7.21: Coprime Numbers form Fraction in Lowest Terms
- Proof (related to "Prop. 7.21: Coprime Numbers form Fraction in Lowest Terms") [2502]
- Prop. 7.22: Numbers forming Fraction in Lowest Terms are Coprime
- Proof (related to "Prop. 7.22: Numbers forming Fraction in Lowest Terms are Coprime") [2503]
- Prop. 7.23: Divisor of One of Coprime Numbers is Coprime to Other
- Proof (related to "Prop. 7.23: Divisor of One of Coprime Numbers is Coprime to Other") [2504]
- Prop. 7.24: Integer Coprime to all Factors is Coprime to Whole
- Proof (related to "Prop. 7.24: Integer Coprime to all Factors is Coprime to Whole") [2505]
- Prop. 7.25: Square of Coprime Number is Coprime
- Proof (related to "Prop. 7.25: Square of Coprime Number is Coprime") [2506]
- Prop. 7.26: Product of Coprime Pairs is Coprime
- Proof (related to "Prop. 7.26: Product of Coprime Pairs is Coprime") [2507]
- Prop. 7.27: Powers of Coprime Numbers are Coprime
- Proof (related to "Prop. 7.27: Powers of Coprime Numbers are Coprime") [2774]
- Prop. 7.28: Numbers are Coprime iff Sum is Coprime to Both
- Proof (related to "Prop. 7.28: Numbers are Coprime iff Sum is Coprime to Both") [2508]
- Prop. 7.29: Prime not Divisor implies Coprime
- Proof (related to "Prop. 7.29: Prime not Divisor implies Coprime") [2509]
- Prop. 7.30: Euclidean Lemma
- Proof (related to "Prop. 7.30: Euclidean Lemma") [806]
- Proof (related to "Prop. 7.30: Euclidean Lemma") [1300]
- Proof (related to "Prop. 7.30: Euclidean Lemma") [6416]
- Prop. 7.31: Existence of Prime Divisors
- Proof (related to "Prop. 7.31: Existence of Prime Divisors") [799]
- Proof (related to "Prop. 7.31: Existence of Prime Divisors") [6417]
- Prop. 7.32: Natural Number is Prime or has Prime Factor
- Proof (related to "Prop. 7.32: Natural Number is Prime or has Prime Factor") [2512]
- Prop. 7.33: Least Ratio of Numbers
- Proof (related to "Prop. 7.33: Least Ratio of Numbers") [2513]
- Prop. 7.34: Existence of Lowest Common Multiple
- Proof (related to "Prop. 7.34: Existence of Lowest Common Multiple") [2514]
- Prop. 7.35: Least Common Multiple Divides Common Multiple
- Proof (related to "Prop. 7.35: Least Common Multiple Divides Common Multiple") [2515]
- Prop. 7.36: Least Common Multiple of Three Numbers
- Proof (related to "Prop. 7.36: Least Common Multiple of Three Numbers") [2516]
- Prop. 7.37: Integer Divided by Divisor is Integer
- Proof (related to "Prop. 7.37: Integer Divided by Divisor is Integer") [2517]
- Prop. 7.38: Divisor is Reciprocal of Divisor of Integer
- Proof (related to "Prop. 7.38: Divisor is Reciprocal of Divisor of Integer") [2518]
- Prop. 7.39: Least Number with Three Given Fractions
- Proof (related to "Prop. 7.39: Least Number with Three Given Fractions") [2519]
- Prop. 8.01: Geometric Progression with Coprime Extremes is in Lowest Terms
- Proof (related to "Prop. 8.01: Geometric Progression with Coprime Extremes is in Lowest Terms") [2520]
- Prop. 8.02: Construction of Geometric Progression in Lowest Terms
- Proof (related to "Prop. 8.02: Construction of Geometric Progression in Lowest Terms") [2521]
- Prop. 8.03: Geometric Progression in Lowest Terms has Coprime Extremes
- Proof (related to "Prop. 8.03: Geometric Progression in Lowest Terms has Coprime Extremes") [2522]
- Prop. 8.04: Construction of Sequence of Numbers with Given Ratios
- Proof (related to "Prop. 8.04: Construction of Sequence of Numbers with Given Ratios") [2523]
- Prop. 8.05: Ratio of Products of Sides of Plane Numbers
- Proof (related to "Prop. 8.05: Ratio of Products of Sides of Plane Numbers") [2524]
- Prop. 8.06: First Element of Geometric Progression not dividing Second
- Proof (related to "Prop. 8.06: First Element of Geometric Progression not dividing Second") [2525]
- Prop. 8.07: First Element of Geometric Progression that divides Last also divides Second
- Proof (related to "Prop. 8.07: First Element of Geometric Progression that divides Last also divides Second") [2526]
- Prop. 8.08: Geometric Progressions in Proportion have Same Number of Elements
- Proof (related to "Prop. 8.08: Geometric Progressions in Proportion have Same Number of Elements") [2527]
- Prop. 8.09: Elements of Geometric Progression between Coprime Numbers
- Proof (related to "Prop. 8.09: Elements of Geometric Progression between Coprime Numbers") [2528]
- Prop. 8.10: Product of Geometric Progressions from One
- Proof (related to "Prop. 8.10: Product of Geometric Progressions from One") [2529]
- Prop. 8.11: Between two Squares exists one Mean Proportional
- Proof (related to "Prop. 8.11: Between two Squares exists one Mean Proportional") [2530]
- Prop. 8.12: Between two Cubes exist two Mean Proportionals
- Proof (related to "Prop. 8.12: Between two Cubes exist two Mean Proportionals") [2531]
- Prop. 8.13: Powers of Elements of Geometric Progression are in Geometric Progression
- Proof (related to "Prop. 8.13: Powers of Elements of Geometric Progression are in Geometric Progression") [2532]
- Prop. 8.14: Number divides Number iff Square divides Square
- Proof (related to "Prop. 8.14: Number divides Number iff Square divides Square") [2533]
- Prop. 8.15: Number divides Number iff Cube divides Cube
- Proof (related to "Prop. 8.15: Number divides Number iff Cube divides Cube") [2534]
- Prop. 8.16: Number does not divide Number iff Square does not divide Square
- Proof (related to "Prop. 8.16: Number does not divide Number iff Square does not divide Square") [2535]
- Prop. 8.17: Number does not divide Number iff Cube does not divide Cube
- Proof (related to "Prop. 8.17: Number does not divide Number iff Cube does not divide Cube") [2536]
- Prop. 8.18: Between two Similar Plane Numbers exists one Mean Proportional
- Proof (related to "Prop. 8.18: Between two Similar Plane Numbers exists one Mean Proportional") [2537]
- Prop. 8.19: Between two Similar Solid Numbers exist two Mean Proportionals
- Proof (related to "Prop. 8.19: Between two Similar Solid Numbers exist two Mean Proportionals") [2538]
- Prop. 8.20: Numbers between which exists one Mean Proportional are Similar Plane
- Proof (related to "Prop. 8.20: Numbers between which exists one Mean Proportional are Similar Plane") [2539]
- Prop. 8.21: Numbers between which exist two Mean Proportionals are Similar Solid
- Proof (related to "Prop. 8.21: Numbers between which exist two Mean Proportionals are Similar Solid") [2540]
- Prop. 8.22: If First of Three Numbers in Geometric Progression is Square then Third is Square
- Proof (related to "Prop. 8.22: If First of Three Numbers in Geometric Progression is Square then Third is Square") [2541]
- Prop. 8.23: If First of Four Numbers in Geometric Progression is Cube then Fourth is Cube
- Proof (related to "Prop. 8.23: If First of Four Numbers in Geometric Progression is Cube then Fourth is Cube") [2542]
- Prop. 8.24: If Ratio of Square to Number is as between Two Squares then Number is Square
- Proof (related to "Prop. 8.24: If Ratio of Square to Number is as between Two Squares then Number is Square") [2543]
- Prop. 8.25: If Ratio of Cube to Number is as between Two Cubes then Number is Cube
- Proof (related to "Prop. 8.25: If Ratio of Cube to Number is as between Two Cubes then Number is Cube") [2544]
- Prop. 8.26: Similar Plane Numbers have Same Ratio as between Two Squares
- Proof (related to "Prop. 8.26: Similar Plane Numbers have Same Ratio as between Two Squares") [2545]
- Prop. 8.27: Similar Solid Numbers have Same Ratio as between Two Cubes
- Proof (related to "Prop. 8.27: Similar Solid Numbers have Same Ratio as between Two Cubes") [2546]
- Prop. 9.01: Product of Similar Plane Numbers is Square
- Proof (related to "Prop. 9.01: Product of Similar Plane Numbers is Square") [2547]
- Prop. 9.02: Numbers whose Product is Square are Similar Plane Numbers
- Proof (related to "Prop. 9.02: Numbers whose Product is Square are Similar Plane Numbers") [2548]
- Prop. 9.03: Square of Cube Number is Cube
- Proof (related to "Prop. 9.03: Square of Cube Number is Cube") [2549]
- Prop. 9.04: Cube Number multiplied by Cube Number is Cube
- Proof (related to "Prop. 9.04: Cube Number multiplied by Cube Number is Cube") [2550]
- Prop. 9.05: Number multiplied by Cube Number making Cube is itself Cube
- Proof (related to "Prop. 9.05: Number multiplied by Cube Number making Cube is itself Cube") [2551]
- Prop. 9.06: Number Squared making Cube is itself Cube
- Proof (related to "Prop. 9.06: Number Squared making Cube is itself Cube") [2552]
- Prop. 9.07: Product of Composite Number with Number is Solid Number
- Proof (related to "Prop. 9.07: Product of Composite Number with Number is Solid Number") [2553]
- Prop. 9.08: Elements of Geometric Progression from One which are Powers of Number
- Proof (related to "Prop. 9.08: Elements of Geometric Progression from One which are Powers of Number") [2554]
- Prop. 9.09: Elements of Geometric Progression from One where First Element is Power of Number
- Proof (related to "Prop. 9.09: Elements of Geometric Progression from One where First Element is Power of Number") [2555]
- Prop. 9.10: Elements of Geometric Progression from One where First Element is not Power of Number
- Proof (related to "Prop. 9.10: Elements of Geometric Progression from One where First Element is not Power of Number") [2556]
- Prop. 9.11: Elements of Geometric Progression from One which Divide Later Elements
- Proof (related to "Prop. 9.11: Elements of Geometric Progression from One which Divide Later Elements") [2557]
- Prop. 9.12: Elements of Geometric Progression from One Divisible by Prime
- Proof (related to "Prop. 9.12: Elements of Geometric Progression from One Divisible by Prime") [2558]
- Prop. 9.13: Divisibility of Elements of Geometric Progression from One where First Element is Prime
- Proof (related to "Prop. 9.13: Divisibility of Elements of Geometric Progression from One where First Element is Prime") [2559]
- Prop. 9.14: Fundamental Theorem of Arithmetic
- Proof (related to "Prop. 9.14: Fundamental Theorem of Arithmetic") [802]
- Proof (related to "Prop. 9.14: Fundamental Theorem of Arithmetic") [6554]
- Prop. 9.15: Sum of Pair of Elements of Geometric Progression with Three Elements in Lowest Terms is Coprime to other Element
- Proof (related to "Prop. 9.15: Sum of Pair of Elements of Geometric Progression with Three Elements in Lowest Terms is Coprime to other Element") [2561]
- Prop. 9.16: Two Coprime Integers have no Third Integer Proportional
- Proof (related to "Prop. 9.16: Two Coprime Integers have no Third Integer Proportional") [2562]
- Prop. 9.17: Last Element of Geometric Progression with Coprime Extremes has no Integer Proportional as First to Second
- Proof (related to "Prop. 9.17: Last Element of Geometric Progression with Coprime Extremes has no Integer Proportional as First to Second") [2563]
- Prop. 9.18: Condition for Existence of Third Number Proportional to Two Numbers
- Proof (related to "Prop. 9.18: Condition for Existence of Third Number Proportional to Two Numbers") [2564]
- Prop. 9.19: Condition for Existence of Fourth Number Proportional to Three Numbers
- Proof (related to "Prop. 9.19: Condition for Existence of Fourth Number Proportional to Three Numbers") [2565]
- Prop. 9.20: Infinite Number of Primes
- Proof (related to "Prop. 9.20: Infinite Number of Primes") [6555]
- Analytic Proof (Erdös 1938) (related to "Prop. 9.20: Infinite Number of Primes") [510]
- Proof by Contradiction (Euclid) (related to "Prop. 9.20: Infinite Number of Primes") [509]
- Proof by Contradiction (Kummer) (related to "Prop. 9.20: Infinite Number of Primes") [515]
- Prop. 9.21: Sum of Even Numbers is Even
- Proof (related to "Prop. 9.21: Sum of Even Numbers is Even") [2566]
- Prop. 9.22: Sum of Even Number of Odd Numbers is Even
- Proof (related to "Prop. 9.22: Sum of Even Number of Odd Numbers is Even") [2567]
- Prop. 9.23: Sum of Odd Number of Odd Numbers is Odd
- Proof (related to "Prop. 9.23: Sum of Odd Number of Odd Numbers is Odd") [2568]
- Prop. 9.24: Even Number minus Even Number is Even
- Proof (related to "Prop. 9.24: Even Number minus Even Number is Even") [2569]
- Prop. 9.25: Even Number minus Odd Number is Odd
- Proof (related to "Prop. 9.25: Even Number minus Odd Number is Odd") [2570]
- Prop. 9.26: Odd Number minus Odd Number is Even
- Proof (related to "Prop. 9.26: Odd Number minus Odd Number is Even") [2571]
- Prop. 9.27: Odd Number minus Even Number is Odd
- Proof (related to "Prop. 9.27: Odd Number minus Even Number is Odd") [2572]
- Prop. 9.28: Odd Number multiplied by Even Number is Even
- Proof (related to "Prop. 9.28: Odd Number multiplied by Even Number is Even") [2573]
- Prop. 9.29: Odd Number multiplied by Odd Number is Odd
- Proof (related to "Prop. 9.29: Odd Number multiplied by Odd Number is Odd") [2574]
- Prop. 9.30: Odd Divisor of Even Number Also Divides Its Half
- Proof (related to "Prop. 9.30: Odd Divisor of Even Number Also Divides Its Half") [2575]
- Prop. 9.31: Odd Number Coprime to Number is also Coprime to its Double
- Proof (related to "Prop. 9.31: Odd Number Coprime to Number is also Coprime to its Double") [2576]
- Prop. 9.32: Power of Two is Even-Times Even Only
- Proof (related to "Prop. 9.32: Power of Two is Even-Times Even Only") [2577]
- Prop. 9.33: Number whose Half is Odd is Even-Times Odd
- Proof (related to "Prop. 9.33: Number whose Half is Odd is Even-Times Odd") [2578]
- Prop. 9.34: Number neither whose Half is Odd nor Power of Two is both Even-Times Even and Even-Times Odd
- Proof (related to "Prop. 9.34: Number neither whose Half is Odd nor Power of Two is both Even-Times Even and Even-Times Odd") [2579]
- Prop. 9.35: Sum of Geometric Progression
- Proof (related to "Prop. 9.35: Sum of Geometric Progression") [1124]
- Proof (related to "Prop. 9.35: Sum of Geometric Progression") [6556]
- Prop. 9.36: Theorem of Even Perfect Numbers (first part)
- Proof (related to "Prop. 9.36: Theorem of Even Perfect Numbers (first part)") [2581]
- Properties of a Group Homomorphism
- Direct Proof (related to "Properties of a Group Homomorphism") [681]
- Properties of Cosets
- Proof (related to "Properties of Cosets") [830]
- Properties of Ordinal Numbers
- Proof (related to "Properties of Ordinal Numbers") [725]
- Properties of the Absolute Value
- Proof (related to "Properties of the Absolute Value") [1089]
- Properties of Transitive Sets
- Proof (related to "Properties of Transitive Sets") [722]
- Pythagorean Identity
- Proof (related to "Pythagorean Identity") [1795]
- Quotient of Convergent Complex Sequences
- Proof (related to "Quotient of Convergent Complex Sequences") [1723]
- Quotient of Convergent Real Sequences
- Proof (related to "Quotient of Convergent Real Sequences") [1143]
- Quotient Space
- Proof (related to "Quotient Space") [6331]
- Ratio Test For Absolutely Convergent Complex Series
- Proof (related to "Ratio Test For Absolutely Convergent Complex Series") [1730]
- Ratio Test For Absolutely Convergent Series
- Proof (related to "Ratio Test For Absolutely Convergent Series") [1355]
- Rational Cauchy Sequence Members Are Bounded
- Proof (related to "Rational Cauchy Sequence Members Are Bounded") [1490]
- Rational Cauchy Sequences Build a Commutative Group With Respect To Addition
- Proof (related to "Rational Cauchy Sequences Build a Commutative Group With Respect To Addition") [1519]
- Rational Cauchy Sequences Build a Commutative Monoid With Respect To Multiplication
- Proof (related to "Rational Cauchy Sequences Build a Commutative Monoid With Respect To Multiplication") [1521]
- Rational Powers of Positive Numbers
- Proof (related to "Rational Powers of Positive Numbers") [1623]
- Rearrangement of Absolutely Convergent Series
- Proof (related to "Rearrangement of Absolutely Convergent Series") [1365]
- Reciprocity Law of Falling And Rising Factorial Powers
- Proof (related to "Reciprocity Law of Falling And Rising Factorial Powers") [1413]
- Reciprocity of Complex Exponential Function, Non-Zero Property
- Proof (related to "Reciprocity of Complex Exponential Function, Non-Zero Property") [1741]
- Reciprocity of Exponential Function of General Base, Non-Zero Property
- Proof (related to "Reciprocity of Exponential Function of General Base, Non-Zero Property") [1615]
- Reciprocity of Exponential Function, Non-Zero Property
- Proof (related to "Reciprocity of Exponential Function, Non-Zero Property") [1418]
- Recursive Formula for Binomial Coefficients
- Proof (related to "Recursive Formula for Binomial Coefficients") [995]
- Relationship Between the Greatest Common Divisor and the Least Common Multiple
- Proof (related to "Relationship Between the Greatest Common Divisor and the Least Common Multiple") [1282]
- Replacing Mutually Independent Events by Their Complements
- Proof (related to "Replacing Mutually Independent Events by Their Complements") [1811]
- Representing Real Cosine by Complex Exponential Function
- Proof (related to "Representing Real Cosine by Complex Exponential Function") [1787]
- Representing Real Sine by Complex Exponential Function
- Proof (related to "Representing Real Sine by Complex Exponential Function") [1789]
- Riemann Integral for Step Functions
- Proof (related to "Riemann Integral for Step Functions") [1753]
- Riemann Upper and Riemann Lower Integrals for Bounded Real Functions
- Proof (related to "Riemann Upper and Riemann Lower Integrals for Bounded Real Functions") [1762]
- Right-Distributivity Law For Natural Numbers
- Proof (related to "Right-Distributivity Law For Natural Numbers") [1437]
- Rule of Combining Different Sets of Indices
- Proof (related to "Rule of Combining Different Sets of Indices") [1120]
- Rules of Calculation with Inequalities
- Elementary Proof (related to "Rules of Calculation with Inequalities") [595]
- Simple Binomial Identities
- Proof (related to "Simple Binomial Identities") [1840]
- Simulating LOOP Programs Using WHILE Programs
- Proof (related to "Simulating LOOP Programs Using WHILE Programs") [1200]
- Size of an \(r\)-Regular Graph with \(n\) Vertices
- Proof (related to "Size of an \(r\)-Regular Graph with \(n\) Vertices") [6356]
- Splitting a Graph with Even Degree Vertices into Cycles
- Proof (related to "Splitting a Graph with Even Degree Vertices into Cycles") [6383]
- Square Roots
- Proof (related to "Square Roots") [1162]
- Subgroups of Cyclic Groups
- Proof (related to "Subgroups of Cyclic Groups") [820]
- Subgroups of Finite Cyclic Groups
- Proof (related to "Subgroups of Finite Cyclic Groups") [826]
- Subsets of Finite Sets
- Proof (related to "Subsets of Finite Sets") [987]
- Subsets of Natural Numbers Relatively Prime to a Natural Number are Divisor-Closed
- Proof (related to "Subsets of Natural Numbers Relatively Prime to a Natural Number are Divisor-Closed") [6408]
- Successor of Oridinal
- Proof (related to "Successor of Oridinal") [775]
- Sum of Arithmetic Progression
- Proof (related to "Sum of Arithmetic Progression") [1118]
- Sum of Binomial Coefficients
- Proof (related to "Sum of Binomial Coefficients") [1406]
- Sum of Binomial Coefficients I
- Proof (related to "Sum of Binomial Coefficients I") [1842]
- Sum of Binomial Coefficients II
- Proof (related to "Sum of Binomial Coefficients II") [1844]
- Sum of Convergent Complex Sequences
- Proof (related to "Sum of Convergent Complex Sequences") [1712]
- Sum of Convergent Real Sequences
- Proof (related to "Sum of Convergent Real Sequences") [1132]
- Sum of Convergent Real Series
- Proof (related to "Sum of Convergent Real Series") [6644]
- Supremum Property, Infimum Property
- Proof of Existence (related to "Supremum Property, Infimum Property") [1757]
- The absolute value makes the set of rational numbers a metric space.
- Direct Proof (related to "The absolute value makes the set of rational numbers a metric space.") [1091]
- The distance of complex numbers makes complex numbers a metric space.
- Proof (related to "The distance of complex numbers makes complex numbers a metric space.") [1734]
- The distance of real numbers makes real numbers a metric space.
- Direct Proof (related to "The distance of real numbers makes real numbers a metric space.") [620]
- The Fundamental Counting Principle
- Proof (related to "The Fundamental Counting Principle") [992]
- The General Perturbation Method
- Proof (related to "The General Perturbation Method") [1122]
- The Proving Principle by Contradiction
- Proof (related to "The Proving Principle by Contradiction") [745]
- The Proving Principle By Contraposition
- Proof (related to "The Proving Principle By Contraposition") [1331]
- The Proving Principle of Complete Induction (Variant 1)
- Direct Proof (related to "The Proving Principle of Complete Induction (Variant 1)") [658]
- The set of WHILE-computable functions is included in the set of partially WHILE-computable functions
- Proof (related to "The set of WHILE-computable functions is included in the set of partially WHILE-computable functions") [1198]
- The supplemental angle of a right angle is another right angle.
- Direct Proof (related to "The supplemental angle of a right angle is another right angle.") [655]
- Theorem of Bolzano-Weierstrass
- Proof (related to "Theorem of Bolzano-Weierstrass") [6609]
- Theorem of Large Numbers for Relative Frequencies
- Proof (related to "Theorem of Large Numbers for Relative Frequencies") [1848]
- Time Dilation, Lorentz Factor
- Proof (related to "Time Dilation, Lorentz Factor") [6298]
- Transitivity of the Order Relation of Natural Numbers
- Proof (related to "Transitivity of the Order Relation of Natural Numbers") [1550]
- Triangle Inequality
- Proof (related to "Triangle Inequality") [1088]
- Trichotomy of Ordinals
- Proof (related to "Trichotomy of Ordinals") [731]
- Trichotomy of the Order Relation for Natural Numbers
- Proof (related to "Trichotomy of the Order Relation for Natural Numbers") [1553]
- Union of Countable Many Countable Sets
- Proof (related to "Union of Countable Many Countable Sets") [797]
- Unique Solvability of \(a+x=b\)
- Elementary Proof (related to "Unique Solvability of \(a+x=b\)") [518]
- Unique Solvability of \(ax=b\)
- Elementary Proof (related to "Unique Solvability of \(ax=b\)") [519]
- Uniqueness Lemma of a Finite Basis
- Proof by Contradiction (related to "Uniqueness Lemma of a Finite Basis") [1040]
- Uniqueness of 1
- Elementary Proof of Uniqueness (related to "Uniqueness of 1") [49]
- Uniqueness of Complex Zero
- Proof (related to "Uniqueness of Complex Zero") [1687]
- Uniqueness of Integer Zero
- Proof (related to "Uniqueness of Integer Zero") [1683]
- Uniqueness of Natural Zero
- Proof (related to "Uniqueness of Natural Zero") [1681]
- Uniqueness of Negative Numbers
- Elementary Proof of Uniqueness (related to "Uniqueness of Negative Numbers") [60]
- Uniqueness Of Predecessors Of Natural Numbers
- Proof (related to "Uniqueness Of Predecessors Of Natural Numbers") [1543]
- Uniqueness of Rational Zero
- Proof (related to "Uniqueness of Rational Zero") [1685]
- Uniqueness of Real Zero
- Elementary Proof of Uniqueness (related to "Uniqueness of Real Zero") [44]
- Uniqueness of Reciprocal Numbers
- Elementary Proof of Uniqueness (related to "Uniqueness of Reciprocal Numbers") [61]
- Uniqueness of the Limit of a Sequence
- Proof (related to "Uniqueness of the Limit of a Sequence") [1130]
- Unit Circle
- Proof (related to "Unit Circle") [1750]
- Unit Ring of All Rational Cauchy Sequences
- Proof (related to "Unit Ring of All Rational Cauchy Sequences") [1104]
- Urn Model With Replacement
- Proof (related to "Urn Model With Replacement") [1800]
- Urn Model Without Replacement
- Proof (related to "Urn Model Without Replacement") [1798]
- Well-Ordering Principle
- Proof (related to "Well-Ordering Principle") [699]
- When is it possible to find a separating cycle in a biconnected graph, given a non-separating cycle?
- Proof (related to "When is it possible to find a separating cycle in a biconnected graph, given a non-separating cycle?") [1234]
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