Addition of the Probability of Mutually Exclusive Events [873]Archimedian Axiom [1339]Axiom of Associativity [668]Axiom of Commutativity [672]Axiom of Distributivity [682]Axiom of Empty Set [666]Axiom of Existence of an Identity [669]Axiom of Existence of Inverse Elements [670]Axiom of Extensionality [551]Axiom of Foundation [717]Axiom of Infinity [678]Axiom of Pairing [667]Law of Excluded Middle [705]Peano Axioms [504]Post. 1: Axiom of Straight Line Determined by Two Distinct Points [692]Post. 2: Axiom of Segment Extention [734]Post. 3: Axiom of Circle Determined by its Center and its Radius [694]Post. 4: Equality of all Right Angles [6424]Post. 5: Axiom of Drawing Exactly One Parallel Line to Another Line on a Point Not Lying on That Line (Parallel Postulate) [939]Principle of Relativity [6271]Principle of the Constancy of Light Speed [6272]Probability as a Non-Negative Number [872]Probability of the Certain Event [871]Zermelo-Fraenkel Axioms [1427]$$b$$-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 EquationsFirst-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]Continuously Differentiable Functions [6812]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 and Concave Functions [6785]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]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 Product of Groups [6822]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 of Sets [6841]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]Higher-Order Derivatives [6771]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 of Extended Real Numbers [6670]Infimum, Greatest Lower Bound [1755]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]Local Extremum [6775]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]One-sided Derivative, Right-Differentiability and Left-Differentiability [6762]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 and Total Maps (Functions) [592]Pentagon [6447]Periodic Functions [376]Permutations [188]Pi [6738]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 Parts of a Real-Valued Function [6798]Positive and Negative Real Numbers [585]Power Set [6831]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]Reciprocal Function [6756]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 Complement [6829]Set Difference [6830]Set Intersection [6828]Set of Binary Logical Values (True and False) [707]Set of Natural Numbers (Peano) [664]Set Union [6827]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 and Superset [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 of Extended Real Numbers [6669]Supremum, Least Upper Bound [1754]Surjective Function [770]Symmetric Bilinear Form [6336]Symmetric Matrix [1779]Syntax [709]Tangent Bundle [6326]Tangent of a Real Variable [6746]Tautology, Valid Boolean Functions [1318]Terms in Predicate Logic [6225]Topological Chart [6201]Topological Space, Topology [6189]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 SpacesBasis, 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 of a Function [6736]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]0 is less than 1 [6861]0 is unqual 1 [6858]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 Theorem of Tangent [6753]Additivity Theorems of 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]All Zeros of Cosine and Sine [6743]Alternating Sum of Binomial Coefficients [1407]An Upper Bound for the Product of General Powers [6787]Angles and Sides in a Triangle V [903]Angles of a Right And Isosceles Triangle [926]Angles of Right Triangle [930]Antiderivatives are Uniquely Defined Up to a Constant [6806]Any Positive Characteristic Is a Prime Number [882]Approximability of Continuous Real Functions On Closed Intervals By Step Functions [6619]Arguments for which Cosine and Sine are Equal to Each Other [6742]Associativity of Conjunction [6844]Associativity of Disjunction [6846]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]Cauchy–Schwarz Inequality [6791]Chain Rule [6769]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 Monotonic Functions via Derivatives [6783]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 Bijective Functions is Bijective [6866]Composition of Continuous Functions at a Single Point [1606]Composition of Injective Functions is Injective [6864]Composition of Relations (Sometimes) Preserves Their Left-Total Property [1312]Composition of Relations Preserves Their Right-Uniqueness Property [1310]Composition of Surjective Functions is Surjective [123]Composition of Total Functions [1314]Compositions of Continuous Functions on a Whole Domain [1608]ConjunctionCommutativity 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 SequencesCriteria [307]Convergent SeriesA General Criterion for the Convergence of Infinite Complex Series [308]Convex Functions on Open Intervals are Continuous [6793]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]Cosine and Sine are Periodic Functions [6740]Counting the Set's Elements Using Its Partition [982]Criteria for Subgroups [811]Criterion for Alternating Infinite Series [1266]Cyclic Groups are Abelian [813]Darboux's Theorem [6779]De Morgan's Laws (Logic) [6852]De Morgan's Laws (Sets) [6854]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]Defining Property of the Field of Real Numbers [6194]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]Derivate of Absolute Value Function Does Not Exist at $$0$$ [6761]Derivative of a Constant Function [1372]Derivative of a Linear Function $$ax+b$$ [1378]Derivative of an Invertible Function on Real Invervals [6765]Derivative of Cosine [6759]Derivative of General Powers of Positive Numbers [6770]Derivative of Sine [6760]Derivative of Tangent [6764]Derivative of the Exponential Function [6758]Derivative of the Inverse Sine [6767]Derivative of the Inverse Tangent [6768]Derivative of the n-th Power Function [6755]Derivative of the Natural Logarithm [6766]Derivative of the Reciprocal Function [6757]Derivatives of Even and Odd Functions [6773]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 and Tangent-Linear Approximation [6763]Differentiable Functions are Continuous [1374]Differential Equation of the Exponential Function [6782]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]DisjunctionCommutativity 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]EquivalenceCommutativity 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]Estimating the Growth of a Function with its Derivative [6780]Euler Characteristic for Planar Graphs [6374]Euler's Formula [1783]Euler's Identity [6744]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 (Oddness) of Polynomials [6774]Eveness of the Cosine of a Real Variable [1790]Ever Integer Is Either Even or Odd [6856]Every Bounded Real Sequence has a Convergent Subsequence [1152]Every Contraposition to a Proposition is a Tautology to this Proposition [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 (Archimedian Principle) [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]Fundamental Theorem of Calculus [6807]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]Generalized Product Rule [6772]Generating Co-Prime Numbers Knowing the Greatest Common Divisor [1289]Generating the Greatest Common Divisor Knowing Co-Prime Numbers [1291]Geometric Construction of the Multiplication of Complex Numbers [6751]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]Hölder's inequality [6790]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]Integral of Cosine [6803]Integral of General Powers [6808]Integral of Inverse Sine [6816]Integral of Sine [6810]Integral of the Exponential Function [6811]Integral of the Inverse Tangent [6815]Integral of the Natural Logarithm [6814]Integral of the Reciprocal Function [6809]Integral p-Norm [6804]Integrals on Adjacent Intervals [6805]Integration by Substitution [6813]Intermediate Root Value Theorem [6692]Intermediate Value Theorem [1259]Intersection of a Set With Another Set is Subset of This Set [6834]Intersection of Convex Affine Sets [6289]Inverse Cosine of a Real Variable [6747]Inverse Hyperbolic Cosine [6723]Inverse Hyperbolic Sine [6722]Inverse Sine of a Real Variable [6748]Inverse Tangent and Complex Exponential Function [6754]Inverse Tangent of a Real Variable [6749]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]Legendre Polynomials and Legendre Differential Equations [6795]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]Maximum Norm as a Limit of p-Norms [6794]Mean Value Theorem For Riemann Integrals [1772]Metric Spaces and Empty Sets are Clopen [854]Metric Spaces are Hausdorff Spaces [850]Minkowski Inequality [6792]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]More Insight to Euler's Identity [6745]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]n-th Roots of Unity [6752]Natural Logarithm [1421]Necessary and Sufficient Condition for Convexity [6786]Negative Cosine and Sine vs Shifting the Argument [6741]Nested Closed Subset Theorem [127]Non-Cauchy Sequences are Not Convergent [6870]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]p-Norm, Taxicab Norm, Euclidean Norm, Maximum Norm [6789]Parallelogram - Defining Property II [938]Parallelogram - Defining Property I [937]Planarity of Subdivisions [6378]Polar Coordinates of a Complex Number [6750]Position of Minus Sign in Rational Numbers Representations [1592]Positive and Negative Parts of a Riemann-Integrable Functions are Riemann-Integrable [6799]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 Riemann-integrable Functions is Riemann-integrable [6800]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]Quadratic Formula [6825]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 Cauchy Sequences are Bounded [6872]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 Sum Converging To the Riemann Integral [6801]Riemann Upper and Riemann Lower Integrals for Bounded Real Functions [1761]Right-Distributivity Law For Natural Numbers [1436]Rolle's Theorem [6778]Rule of Combining Different Sets of Indices [1119]Rules for Exponentiation [676]Rules of Calculation with Inequalities [594]Second Law of Planetary Motion [6305]Set Intersection is Associative [6848]Set Intersection is Commutative [6840]Set Union is Associative [6849]Set Union is Commutative [6836]Sets are Subsets of Their Union [6832]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]Special Values for Real Sine, Real Cosine and Complex Exponential Function [6739]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 of a Non-Zero Element is Positive in Ordered Fields [6860]Square Roots [1161]Step Function on Closed Intervals are Riemann Integrable [6797]Step Functions as a Subspace of all Functions on a Closed Real Interval [6796]Subgroups of Cyclic Groups [817]Subgroups of Finite Cyclic Groups [825]Subset of Powers is Submonoid [6817]Subsets of Finite Sets [986]Subsets of Natural Numbers Relatively Prime to a Natural Number are Divisor-Closed [6407]Successor of Oridinal [774]Sufficient Condition for a Function to be Constant [6781]Sufficient Condition for a Local Extremum [6784]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 Cosines [6802]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, Contrapositive [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]Zero of Cosine [6737]Zero-Derivative as a Necessary Condition for a Local Extremum [6776]Addition (or Subtraction) of Two Registers [1192]Assignment of a Constant $$c$$ to the Register $$r_i$$ with a $$L O O P$$ -Program [1187]Conditional execution of $$L O O P$$ programs - IF-THEN construct [1191]Conditional execution of $$L O O P$$ programs - IF-THEN-ELSE construct [1190]Converting Decimal Numbers to Roman Numbers [6208]Converting Roman Numbers to Decimal Numbers [6209]Data StructuresGraph [244]Division with Quotient and Remainder [1195]Get All Components of a Graph [1220]Get the Component Induced by Vertices Connected to a Given Vertex [1216]Get the Cut Vertices and Biconnected Components of a Connected Graph [1240]getSubgraph [6218]Greatest Common Divisor (Euclid) [503]Horner Scheme [1358]Multiplication of Two Registers [1193]No-Operation Command (NOP) [1194]SearchingSequential Search [625]Semi-Numerical AlgorithmsComplex Numbers [360]Setting the value of a register to the value of another register plus or minus a constant [1189](Real) Exponential Function Is Always PositiveProof (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 ContinuityProof (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 SequencesProof (related to "$$b$$-Adic Fractions Are Real Cauchy Sequences") [1125]0 is less than 1Proof (related to "0 is less than 1") [6862]0 is unqual 1Proof by Contradiction (related to "0 is unqual 1") [6859]A Criterion for Isosceles TrianglesProof (related to "A Criterion for Isosceles Triangles") [750]A General Criterion for the Convergence of Infinite Complex SeriesProof (related to "A General Criterion for the Convergence of Infinite Complex Series") [1726]A General Criterion for the Convergence of Infinite SeriesProof (related to "A General Criterion for the Convergence of Infinite Series") [1149]A Necessary and a Sufficient Condition for Riemann Integrable FunctionsProof (related to "A Necessary and a Sufficient Condition for Riemann Integrable Functions") [1765]A Necessary But Not Sufficient Condition For Convergence Of Infinite SeriesProof (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 falseProof (related to "A proposition cannot be both, true and false") [1326]A proposition cannot be equivalent to its negationProof (related to "A proposition cannot be equivalent to its negation") [1327]Addition of Complex Numbers Is AssociativeProof (related to "Addition of Complex Numbers Is Associative") [1659]Addition of Complex Numbers Is CommutativeProof (related to "Addition of Complex Numbers Is Commutative") [1661]Addition of IntegersProof (related to "Addition of Integers") [1530]Addition of Integers Is AssociativeProof (related to "Addition of Integers Is Associative") [1444]Algebraic Proof (related to "Addition of Integers Is Associative") [1445]Addition of Integers Is CancellativeProof (related to "Addition of Integers Is Cancellative") [1463]Addition of Integers Is CommutativeProof (related to "Addition of Integers Is Commutative") [1461]Addition Of Natural NumbersProof (related to "Addition Of Natural Numbers") [1544]Addition Of Natural Numbers Is AssociativeProof (related to "Addition Of Natural Numbers Is Associative") [1429]Addition of Natural Numbers Is CancellativeProof (related to "Addition of Natural Numbers Is Cancellative") [1433]Addition of Natural Numbers Is Cancellative With Respect To InequalitiesProof (related to "Addition of Natural Numbers Is Cancellative With Respect To Inequalities") [1554]Addition of Natural Numbers Is CommutativeProof (related to "Addition of Natural Numbers Is Commutative") [1431]Addition of Rational Cauchy SequencesProof (related to "Addition of Rational Cauchy Sequences") [1487]Addition of Rational Cauchy Sequences Is AssociativeProof (related to "Addition of Rational Cauchy Sequences Is Associative") [1495]Addition of Rational Cauchy Sequences Is CancellativeProof (related to "Addition of Rational Cauchy Sequences Is Cancellative") [1570]Addition of Rational Cauchy Sequences Is CommutativeProof (related to "Addition of Rational Cauchy Sequences Is Commutative") [1497]Addition Of Rational NumbersProof (related to "Addition Of Rational Numbers") [1515]Addition of Rational Numbers Is AssociativeProof (related to "Addition of Rational Numbers Is Associative") [1468]Addition of Rational Numbers Is CancellativeProof (related to "Addition of Rational Numbers Is Cancellative") [1472]Addition of Rational Numbers Is CommutativeProof (related to "Addition of Rational Numbers Is Commutative") [1470]Addition of Real NumbersProof (related to "Addition of Real Numbers") [1526]Addition Of Real Numbers Is AssociativeProof (related to "Addition Of Real Numbers Is Associative") [1527]Addition of Real Numbers Is CancellativeProof (related to "Addition of Real Numbers Is Cancellative") [1576]Addition Of Real Numbers Is CommutativeProof (related to "Addition Of Real Numbers Is Commutative") [1528]Algebraic Structure of Complex Numbers Together with AdditionProof (related to "Algebraic Structure of Complex Numbers Together with Addition") [1667]Algebraic Structure of Complex Numbers Together with Addition and MultiplicationProof (related to "Algebraic Structure of Complex Numbers Together with Addition and Multiplication") [1691]Algebraic Structure of Integers Together with AdditionProof (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 MultiplicationProof (related to "Algebraic Structure of Integers Together with Addition and Multiplication") [1032]Algebraic Structure Of Natural Numbers Together With AdditionProof (related to "Algebraic Structure Of Natural Numbers Together With Addition") [843]Algebraic Structure Of Natural Numbers Together With MultiplicationProof (related to "Algebraic Structure Of Natural Numbers Together With Multiplication") [1442]Algebraic Structure of Non-Zero Complex Numbers Together with MultiplicationProof (related to "Algebraic Structure of Non-Zero Complex Numbers Together with Multiplication") [1689]Algebraic Structure of Non-Zero Rational Numbers Together with MultiplicationProof (related to "Algebraic Structure of Non-Zero Rational Numbers Together with Multiplication") [1650]Algebraic Structure of Non-Zero Real Numbers Together with MultiplicationProof (related to "Algebraic Structure of Non-Zero Real Numbers Together with Multiplication") [1642]Algebraic Structure of Rational Numbers Together with AdditionProof (related to "Algebraic Structure of Rational Numbers Together with Addition") [1648]Algebraic Structure of Rational Numbers Together with Addition and MultiplicationProof (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 AdditionProof (related to "Algebraic Structure of Real Numbers Together with Addition") [1641]Algebraic Structure of Real Numbers Together with Addition and MultiplicationProof (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 SequencesProof (related to "All Convergent Real Sequences Are Cauchy Sequences") [1395]Alternating Sum of Binomial CoefficientsProof (related to "Alternating Sum of Binomial Coefficients") [1408]Angles and Sides in a Triangle VProof (related to "Angles and Sides in a Triangle V") [904]Any Positive Characteristic Is a Prime NumberProof by Contradiction (related to "Any Positive Characteristic Is a Prime Number") [883]Approximability of Continuous Real Functions On Closed Intervals By Step FunctionsProof by Construction (related to "Approximability of Continuous Real Functions On Closed Intervals By Step Functions") [6620]Associativity of ConjunctionProof (related to "Associativity of Conjunction") [6845]Associativity of DisjunctionProof (related to "Associativity of Disjunction") [6847]Barycentric Coordinates, BarycenterProof (related to "Barycentric Coordinates, Barycenter") [6284]Basic Rules of Manipulating Finite SumsProof (related to "Basic Rules of Manipulating Finite Sums") [1115]Basis Arithmetic Operations Involving Differentiable Functions, Product Rule, Quotient RuleProof (related to "Basis Arithmetic Operations Involving Differentiable Functions, Product Rule, Quotient Rule") [1376]Bayes' TheoremProof (related to "Bayes' Theorem") [1833]Bernoulli's InequalityProof by Induction (related to "Bernoulli's Inequality") [1338]Biconnectivity is a Necessary Condition for a Hamiltonian GraphProof (related to "Biconnectivity is a Necessary Condition for a Hamiltonian Graph") [6397]Binomial DistributionProof (related to "Binomial Distribution") [1818]Binomial TheoremProof by Induction (related to "Binomial Theorem") [1398]Boolean FunctionProof (related to "Boolean Function") [1317]Bounds for the Minimal Tree DecomposabilityProof (related to "Bounds for the Minimal Tree Decomposability") [6395]Calculating the Number of Distinct Positive DivisorsProof (related to "Calculating the Number of Distinct Positive Divisors") [1303]Calculating with Complex ConjugatesProof (related to "Calculating with Complex Conjugates") [1252]Calculation Rules for General PowersProof (related to "Calculation Rules for General Powers") [1629]Calculation Rules for the Big O NotationProof (related to "Calculation Rules for the Big O Notation") [1168]Cancellation LawProof (related to "Cancellation Law") [824]Cardinal NumberProof (related to "Cardinal Number") [981]Cauchy Product of Absolutely Convergent SeriesProof (related to "Cauchy Product of Absolutely Convergent Series") [1396]Cauchy Product of Convergent Series Is Not Necessarily ConvergentProof (related to "Cauchy Product of Convergent Series Is Not Necessarily Convergent") [1393]Characteristic StringProof (related to "Characteristic String") [1002]Characterization of Bipartite GraphsProof (related to "Characterization of Bipartite Graphs") [6371]Characterization of Closed Sets by Limits of SequencesProof (related to "Characterization of Closed Sets by Limits of Sequences") [6586]Characterization of CutverticesProof (related to "Characterization of Cutvertices") [1239]Characterization of Eulerian GraphsProof (related to "Characterization of Eulerian Graphs") [6384]Characterization of Independent EventsProof (related to "Characterization of Independent Events") [1805]Characterization of Independent Events IIProof (related to "Characterization of Independent Events II") [1807]Characterization of Planar Hamiltonian GraphsProof (related to "Characterization of Planar Hamiltonian Graphs") [6401]Characterization of Semi-Eulerian GraphsProof (related to "Characterization of Semi-Eulerian Graphs") [6388]Closed Formula For Binomial CoefficientsCombinatorial Proof (related to "Closed Formula For Binomial Coefficients") [1401]Closed n-Dimensional Cuboids Are CompactProof (related to "Closed n-Dimensional Cuboids Are Compact") [6588]Closed Real Intervals Are CompactProof (related to "Closed Real Intervals Are Compact") [6584]Closed Subsets of Compact Sets are CompactProof (related to "Closed Subsets of Compact Sets are Compact") [6595]Commutativity of ConjunctionProof (related to "Commutativity of Conjunction") [6837]Commutativity of DisjunctionProof (related to "Commutativity of Disjunction") [6838]Commutativity of EquivalenceProof (related to "Commutativity of Equivalence") [6839]Compact Subset of Real Numbers Contains its Maximum and its MinimumProof (related to "Compact Subset of Real Numbers Contains its Maximum and its Minimum") [6599]Compact Subsets of Metric Spaces Are Bounded and ClosedProof (related to "Compact Subsets of Metric Spaces Are Bounded and Closed") [6590]Comparing Natural Numbers Using the Concept of AdditionProof (related to "Comparing Natural Numbers Using the Concept of Addition") [1548]Completeness Principle For Complex NumbersProof (related to "Completeness Principle For Complex Numbers") [1710]Complex Cauchy Sequences Vs. Real Cauchy SequencesProof (related to "Complex Cauchy Sequences Vs. Real Cauchy Sequences") [1706]Complex Conjugate of Complex Exponential FunctionProof (related to "Complex Conjugate of Complex Exponential Function") [1748]Complex Exponential FunctionProof (related to "Complex Exponential Function") [1731]Complex Numbers are a Field Extension of Real NumbersProof (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 BasisProof (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 NumbersProof (related to "Complex Numbers as a Vector Space Over the Field of Real Numbers") [1695]Composition of Bijective Functions is BijectiveProof (related to "Composition of Bijective Functions is Bijective") [6867]Composition of Continuous Functions at a Single PointProof (related to "Composition of Continuous Functions at a Single Point") [1607]Composition of Injective Functions is InjectiveProof (related to "Composition of Injective Functions is Injective") [6865]Composition of Relations (Sometimes) Preserves Their Left-Total PropertyProof (related to "Composition of Relations (Sometimes) Preserves Their Left-Total Property") [1313]Composition of Relations Preserves Their Right-Uniqueness PropertyProof (related to "Composition of Relations Preserves Their Right-Uniqueness Property") [1311]Composition of Surjective Functions is SurjectiveProof (related to "Composition of Surjective Functions is Surjective") [6863]Composition of Total FunctionsProof (related to "Composition of Total Functions") [1315]Compositions of Continuous Functions on a Whole DomainProof (related to "Compositions of Continuous Functions on a Whole Domain") [1609]Connection between Quotient, Remainder, Modulo and Floor FunctionProof (related to "Connection between Quotient, Remainder, Modulo and Floor Function") [1285]Connectivity Is an Equivalence Relation - Components Are a Partition of a GraphProof (related to "Connectivity Is an Equivalence Relation - Components Are a Partition of a Graph") [1222]Construction of a Light ClockProof (related to "Construction of a Light Clock") [6276]Construction of Fields from Integral DomainsProof (related to "Construction of Fields from Integral Domains") [889]Construction of Groups from Commutative and Cancellative SemigroupsProof (related to "Construction of Groups from Commutative and Cancellative Semigroups") [840]Continuity of Complex Exponential FunctionProof (related to "Continuity of Complex Exponential Function") [1744]Continuity of Cosine and SineProof (related to "Continuity of Cosine and Sine") [1785]Continuity of Exponential FunctionProof (related to "Continuity of Exponential Function") [1425]Continuity of Exponential Function of General BaseProof (related to "Continuity of Exponential Function of General Base") [1611]Continuous Functions Mapping Compact Domains to Real Numbers are BoundedProof (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 DomainsProof (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 ContinuousProof (related to "Continuous Functions on Compact Domains are Uniformly Continuous") [6615]Continuous Real Functions on Closed Intervals are Riemann-IntegrableProof (related to "Continuous Real Functions on Closed Intervals are Riemann-Integrable") [6621]Continuous Real Functions on Closed Intervals are Uniformly ContinuousProof (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 IntegersProof by Contraposition (related to "Contraposition of Cancellative Law for Adding Integers") [1562]Contraposition of Cancellative Law for Adding Natural NumbersProof by Contraposition (related to "Contraposition of Cancellative Law for Adding Natural Numbers") [1546]Contraposition of Cancellative Law for Adding Rational NumbersProof by Contraposition (related to "Contraposition of Cancellative Law for Adding Rational Numbers") [1566]Contraposition of Cancellative Law for Adding Real NumbersProof (related to "Contraposition of Cancellative Law for Adding Real Numbers") [1579]Contraposition of Cancellative Law for Multiplying IntegersProof by Contraposition (related to "Contraposition of Cancellative Law for Multiplying Integers") [1564]Contraposition of Cancellative Law for Multiplying Natural NumbersProof by Contraposition (related to "Contraposition of Cancellative Law for Multiplying Natural Numbers") [1560]Contraposition of Cancellative Law for Multiplying Rational NumbersProof by Contraposition (related to "Contraposition of Cancellative Law for Multiplying Rational Numbers") [1568]Contraposition of Cancellative Law of for Multiplying Real NumbersProof (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 SeriesProof (related to "Convergence Behaviour of Absolutely Convergent Series") [1269]Convergence of Alternating Harmonic SeriesProof (related to "Convergence of Alternating Harmonic Series") [1368]Convergence of Complex Conjugate SequenceProof (related to "Convergence of Complex Conjugate Sequence") [1708]Convergence of Infinite Series with Non-Negative TermsProof (related to "Convergence of Infinite Series with Non-Negative Terms") [1159]Convergent Complex Sequences Are BoundedProof (related to "Convergent Complex Sequences Are Bounded") [1717]Convergent Complex Sequences Vs. Convergent Real SequencesProof (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 AdditionProof (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 SequencesProof (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 SequencesProof (related to "Convergent Rational Sequences With Limit $$0$$ Are Rational Cauchy Sequences") [1517]Convergent Sequence together with Limit is a Compact Subset of Metric SpaceProof (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 SpaceProof by Explicit Counterexample (related to "Convergent Sequence without Limit Is Not a Compact Subset of Metric Space") [6580]Convergent Sequences are BoundedProof (related to "Convergent Sequences are Bounded") [6593]Proof (related to "Convergent Sequences are Bounded") [1138]Convergent Sequences are Cauchy SequencesProof (related to "Convergent Sequences are Cauchy Sequences") [1074]Cor. 10.111: Thirteen Irrational Straight-Lines of Different OrderProof (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 SidesProof (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 SpheresProof (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 ProportionProof (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 DivisorProof (related to "Cor. 7.02: Any Divisor Dividing Two Numbers Divides Their Greatest Common Divisor") [6415]Corollaries From the Group AxiomsProof (related to "Corollaries From the Group Axioms") [556]Counting the Set's Elements Using Its PartitionProof (related to "Counting the Set's Elements Using Its Partition") [991]Criteria for SubgroupsProof (related to "Criteria for Subgroups") [812]Criterion for Alternating Infinite SeriesProof (related to "Criterion for Alternating Infinite Series") [1267]Cyclic Groups are AbelianProof (related to "Cyclic Groups are Abelian") [814]De Morgan's Laws (Logic)Proof (related to "De Morgan's Laws (Logic)") [6853]De Morgan's Laws (Sets)Proof of Equality of Sets (related to "De Morgan's Laws (Sets)") [6855]Def. 7.08: Even-Times-Even NumberProof (related to "Def. 7.08: Even-Times-Even Number") [6409]Def. 7.09: Even-Times-Odd NumberProof (related to "Def. 7.09: Even-Times-Odd Number") [6410]Def. 7.10: Odd-Times-Odd NumberProof (related to "Def. 7.10: Odd-Times-Odd Number") [6411]Definition of Continuity Using Open SetsProof (related to "Definition of Continuity Using Open Sets") [6600]Definition of IntegersProof (related to "Definition of Integers") [845]Definition of Rational NumbersProof (related to "Definition of Rational Numbers") [1034]Definition of Real NumbersProof (related to "Definition of Real Numbers") [1106]Definition of the Metric Space $$\mathbb R^n$$, Euclidean NormProof (related to "Definition of the Metric Space $$\mathbb R^n$$, Euclidean Norm") [1207]Derivative of a Constant FunctionProof (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 SequencesProof (related to "Difference of Convergent Complex Sequences") [1721]Difference of Convergent Real SequencesProof (related to "Difference of Convergent Real Sequences") [1134]Difference of Convergent Real SeriesProof (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 SeriesProof by Contradiction (related to "Direct Comparison Test For Divergence Series") [1336]Discovery of Irrational NumbersGeometric Proof (related to "Discovery of Irrational Numbers") [1097]Distance in Normed Vector SpacesProof (related to "Distance in Normed Vector Spaces") [848]Distributivity Law for Complex NumbersProof (related to "Distributivity Law for Complex Numbers") [1679]Distributivity Law For IntegersProof (related to "Distributivity Law For Integers") [1467]Distributivity Law For Natural NumbersProof by Induction (related to "Distributivity Law For Natural Numbers") [1031]Distributivity Law For Rational Cauchy SequencesProof (related to "Distributivity Law For Rational Cauchy Sequences") [1507]Distributivity Law For Rational NumbersProof (related to "Distributivity Law For Rational Numbers") [1492]Distributivity Law For Real NumbersProof (related to "Distributivity Law For Real Numbers") [1582]Divergence of Harmonic SeriesProof (related to "Divergence of Harmonic Series") [1334]Divisibility LawsDirect Proof (related to "Divisibility Laws") [514]Divisibility of Principal IdealsProof (related to "Divisibility of Principal Ideals") [1067]Division with Quotient and RemainderProof (related to "Division with Quotient and Remainder") [819]Divisors of a Product Of Many Factors, Co-Prime to All But One Factor, Divide This FactorProof (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 FactorProof (related to "Divisors of a Product Of Two Factors, Co-Prime to One Factor Divide the Other Factor") [1294]Divisors of IntegersProof (related to "Divisors of Integers") [1274]Double SummationProof (related to "Double Summation") [1426]Dual Graph of a All Faces Contained in a Planar Hamiltonian Cycle is a TreeProof (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 OrdinalsProof (related to "Equivalence of Set Inclusion and Element Inclusion of Ordinals") [776]Equivalency of Vectors in Vector Space If their Difference Forms a SubspaceProof (related to "Equivalency of Vectors in Vector Space If their Difference Forms a Subspace") [6329]Equivalent Knot DiagramsProof (related to "Equivalent Knot Diagrams") [6361]Equivalent Statements Regarding Parallel LinesProof (related to "Equivalent Statements Regarding Parallel Lines") [918]Estimate for the Remainder Term of Complex Exponential FunctionProof (related to "Estimate for the Remainder Term of Complex Exponential Function") [1733]Estimate for the Remainder Term of Exponential FunctionProof (related to "Estimate for the Remainder Term of Exponential Function") [1362]Euler's FormulaProof (related to "Euler's Formula") [1784]Even Number of Vertices with an Odd Degree in Finite DigraphsElementary 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 GraphsProof (related to "Even Number of Vertices with an Odd Degree in Finite Graphs") [1176]Eveness of the Cosine of a Real VariableProof (related to "Eveness of the Cosine of a Real Variable") [1791]Ever Integer Is Either Even or OddProof (related to "Ever Integer Is Either Even or Odd") [6857]Every Bounded Real Sequence has a Convergent SubsequenceProof (related to "Every Bounded Real Sequence has a Convergent Subsequence") [1154]Every Contraposition to a Proposition is a Tautology to this PropositionProof (related to "Every Contraposition to a Proposition is a Tautology to this Proposition") [1329]Every Distance Is Positive DefiniteDirect Proof (related to "Every Distance Is Positive Definite") [616]Every Natural Number Is Greater or Equal ZeroProof (related to "Every Natural Number Is Greater or Equal Zero") [1557]Every Proposition Implies ItselfProof (related to "Every Proposition Implies Itself") [1324]Exchanging the Limit of Function Values with the Function Value of the Limit of ArgumentsProof (related to "Exchanging the Limit of Function Values with the Function Value of the Limit of Arguments") [6711]Existence of Arbitrarily Small Positive Rational NumbersProof (related to "Existence of Arbitrarily Small Positive Rational Numbers") [1847]Existence of Arbitrarily Small PowersProof (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 AdditionProof (related to "Existence of Inverse Complex Numbers With Respect to Addition") [1665]Existence of Inverse Complex Numbers With Respect to MultiplicationProof (related to "Existence of Inverse Complex Numbers With Respect to Multiplication") [1676]Existence of Inverse Integers With Respect to AdditionProof (related to "Existence of Inverse Integers With Respect to Addition") [1513]Existence of Inverse Rational Cauchy Sequences With Respect to AdditionProof (related to "Existence of Inverse Rational Cauchy Sequences With Respect to Addition") [1510]Existence of Inverse Rational Numbers With Respect to AdditionProof (related to "Existence of Inverse Rational Numbers With Respect to Addition") [1512]Existence of Inverse Rational Numbers With Respect to MultiplicationProof (related to "Existence of Inverse Rational Numbers With Respect to Multiplication") [1677]Existence of Inverse Real Numbers With Respect to AdditionProof (related to "Existence of Inverse Real Numbers With Respect to Addition") [1587]Existence of Inverse Real Numbers With Respect to MultiplicationProof (related to "Existence of Inverse Real Numbers With Respect to Multiplication") [1651]Existence of Natural Numbers Exceeding Positive Real Numbers (Archimedian Principle)Proof (related to "Existence of Natural Numbers Exceeding Positive Real Numbers (Archimedian Principle)") [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 LinesProof by Contradiction (related to "Existence of Parallel Straight Lines") [787]Existence of Powers Exceeding Any Positive ConstantProof (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 FunctionProof (related to "Exponential Function") [1343]Exponential Function Is Strictly Monotonically IncreasingProof (related to "Exponential Function Is Strictly Monotonically Increasing") [1595]Exponential Function of General Base With Integer ExponentsProof (related to "Exponential Function of General Base With Integer Exponents") [1621]Exponential Function of General Base With Natural ExponentsProof by Induction (related to "Exponential Function of General Base With Natural Exponents") [1617]Factor GroupsProof (related to "Factor Groups") [1099]Factor RingsProof (related to "Factor Rings") [1100]FactorialProof (related to "Factorial") [1006]Factorials and Stirling Numbers of the First KindProof (related to "Factorials and Stirling Numbers of the First Kind") [1008]Fiber of Prime Ideals Under a Spectrum FunctionProof (related to "Fiber of Prime Ideals Under a Spectrum Function") [6263]Finite Basis TheoremProof (related to "Finite Basis Theorem") [1046]Finite Cardinal Numbers and Set OperationsProof (related to "Finite Cardinal Numbers and Set Operations") [989]Functional Equation of the Complex Exponential FunctionProof (related to "Functional Equation of the Complex Exponential Function") [1737]Functional Equation of the Exponential FunctionProof (related to "Functional Equation of the Exponential Function") [1416]Functional Equation of the Exponential Function of General BaseProof (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 LogarithmProof (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 PointProof (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 EquationsProof by Induction (related to "Fundamental Lemma of Homogeneous Systems of Linear Equations") [1047]General Associative LawDirect Proof (related to "General Associative Law") [545]General Associative Law of MultiplicationDirect Proof (related to "General Associative Law of Multiplication") [547]General Commutative LawDirect Proof (related to "General Commutative Law") [546]General Commutative Law of MultiplicationDirect Proof (related to "General Commutative Law of Multiplication") [548]General Powers of Positive NumbersProof (related to "General Powers of Positive Numbers") [1627]Generalized Euclidean LemmaProof (related to "Generalized Euclidean Lemma") [1299]Generating Co-Prime Numbers Knowing the Greatest Common DivisorProof (related to "Generating Co-Prime Numbers Knowing the Greatest Common Divisor") [1290]Generating the Greatest Common Divisor Knowing Co-Prime NumbersProof (related to "Generating the Greatest Common Divisor Knowing Co-Prime Numbers") [1292]Geometric DistributionProof (related to "Geometric Distribution") [1827]Get All Components of a GraphProof 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 VertexProof 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 GraphProof (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 IdealsProof (related to "Greatest Common Divisor and Least Common Multiple of Ideals") [1070]Greatest Common Divisors Of Integers and Prime NumbersProof (related to "Greatest Common Divisors Of Integers and Prime Numbers") [1297]Group Homomorphisms and Normal SubgroupsProof (related to "Group Homomorphisms and Normal Subgroups") [835]Group Homomorphisms with Cyclic GroupsProof (related to "Group Homomorphisms with Cyclic Groups") [816]Handshaking Lemma for Finite DigraphsCombinatorial Proof (related to "Handshaking Lemma for Finite Digraphs") [567]Handshaking Lemma for Finite GraphsProof (related to "Handshaking Lemma for Finite Graphs") [1174]Heine-Borel TheoremProof (related to "Heine-Borel Theorem") [6597]Horner SchemeProof of Correctness (related to "Horner Scheme") [1360]Proof of Time Complexity (related to "Horner Scheme") [1359]How Convergence Preserves the Order Relation of Sequence MembersProof (related to "How Convergence Preserves the Order Relation of Sequence Members") [1146]How Convergence Preserves Upper and Lower Bounds For Sequence MembersProof (related to "How Convergence Preserves Upper and Lower Bounds For Sequence Members") [1147]How the Boundary Changes the Property of a Set of Being OpenTopological Proof (related to "How the Boundary Changes the Property of a Set of Being Open") [1204]Image of a Compact Set Under a Continuous FunctionProof (related to "Image of a Compact Set Under a Continuous Function") [6601]Imaginary UnitProof (related to "Imaginary Unit") [1693]Indefinite Integral, AntiderivativeProof (related to "Indefinite Integral, Antiderivative") [1774]Inequality of Natural Numbers and Their SuccessorsProof by Contraposition (related to "Inequality of Natural Numbers and Their Successors") [1541]Infinite Geometric SeriesProof (related to "Infinite Geometric Series") [1354]Intermediate Root Value TheoremProof (related to "Intermediate Root Value Theorem") [6695]Intermediate Value TheoremProof (related to "Intermediate Value Theorem") [1263]Intersection of a Set With Another Set is Subset of This SetProof (related to "Intersection of a Set With Another Set is Subset of This Set") [6835]Intersection of Convex Affine SetsProof (related to "Intersection of Convex Affine Sets") [6290]Invertible Functions on Real IntervalsProof (related to "Invertible Functions on Real Intervals") [1382]Isometry is InjectiveProof (related to "Isometry is Injective") [2780]It is true that something can be (either) true or falseProof (related to "It is true that something can be (either) true or false") [1325]Kernel and Image of a Group Homomorphism are SubgroupsProof (related to "Kernel and Image of a Group Homomorphism are Subgroups") [834]Kernel and Image of Group HomomorphismProof (related to "Kernel and Image of Group Homomorphism") [810]Law of Total ProbabilityProof (related to "Law of Total Probability") [1828]Least Common MultipleProof (related to "Least Common Multiple") [1277]Lem. 10.016: Incommensurability of Sum of Incommensurable MagnitudesProof (related to "Lem. 10.016: Incommensurability of Sum of Incommensurable Magnitudes") [6557]Lem. 10.021: Medial is IrrationalProof (related to "Lem. 10.021: Medial is Irrational") [6558]Lem. 10.028.1: Finding Two Squares With Sum Also SquareProof (related to "Lem. 10.028.1: Finding Two Squares With Sum Also Square") [6559]Lem. 10.028.2: Finding Two Squares With Sum Not SquareProof (related to "Lem. 10.028.2: Finding Two Squares With Sum Not Square") [6560]Lem. 10.032: Constructing Medial Commensurable in Square IIProof (related to "Lem. 10.032: Constructing Medial Commensurable in Square II") [6561]Lem. 10.041: Side of Sum of Medial Areas is IrrationalProof (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 AreasProof (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 ThemProof (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 MagnitudesProof (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 SquaresProof (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 DiametersProof (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 PrismsProof (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 RatioProof (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 SphereProof (related to "Lem. 13.13: Construction of Regular Tetrahedron within Given Sphere") [6572]Lem. 13.18: Angle of the PentagonProof (related to "Lem. 13.18: Angle of the Pentagon") [2773]Limit of 1/nProof (related to "Limit of 1/n") [6714]Limit of N-th RootsProof (related to "Limit of N-th Roots") [1625]Limit of Nth RootProof (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 IntegralProof (related to "Linearity and Monotony of the Riemann Integral") [1771]Linearity and Monotony of the Riemann Integral for Step FunctionsProof (related to "Linearity and Monotony of the Riemann Integral for Step Functions") [1760]LOOP-Computable Functions are TotalProof (related to "LOOP-Computable Functions are Total") [1186]Lower Bound of Leaves in a TreeProof (related to "Lower Bound of Leaves in a Tree") [6368]Magnitude of DivisorsProof (related to "Magnitude of Divisors") [1279]Mean Value Theorem For Riemann IntegralsProof (related to "Mean Value Theorem For Riemann Integrals") [1773]Metric Spaces and Empty Sets are ClopenProof (related to "Metric Spaces and Empty Sets are Clopen") [855]Metric Spaces are Hausdorff SpacesTopological Proof (related to "Metric Spaces are Hausdorff Spaces") [851]Monotone ConvergenceProof (related to "Monotone Convergence") [1157]Monotonic Real Functions on Closed Intervals are Riemann-IntegrableProof by Construction (related to "Monotonic Real Functions on Closed Intervals are Riemann-Integrable") [6622]Monotonically Increasing Property of Probability DistributionsProof (related to "Monotonically Increasing Property of Probability Distributions") [1817]Multinomial CoefficientProof (related to "Multinomial Coefficient") [1820]Multinomial DistributionProof (related to "Multinomial Distribution") [1826]Multinomial TheoremProof by Induction (related to "Multinomial Theorem") [1823]Multiplication of Complex Numbers Is AssociativeProof (related to "Multiplication of Complex Numbers Is Associative") [1670]Multiplication of Complex Numbers Is CommutativeProof (related to "Multiplication of Complex Numbers Is Commutative") [1672]Multiplication of IntegersProof (related to "Multiplication of Integers") [1531]Multiplication of Integers Is AssociativeProof (related to "Multiplication of Integers Is Associative") [1451]Multiplication of Integers Is CancellativeProof (related to "Multiplication of Integers Is Cancellative") [1465]Multiplication of Integers Is CommutativeProof (related to "Multiplication of Integers Is Commutative") [1449]Multiplication of Natural Numbers Is AssociativeProof (related to "Multiplication of Natural Numbers Is Associative") [1439]Multiplication of Natural Numbers Is CancellativeProof (related to "Multiplication of Natural Numbers Is Cancellative") [1441]Multiplication of Natural Numbers Is Cancellative With Respect to the Order RelationProof (related to "Multiplication of Natural Numbers Is Cancellative With Respect to the Order Relation") [1584]Multiplication of Natural Numbers is CommutativeProof (related to "Multiplication of Natural Numbers is Commutative") [1438]Multiplication Of Rational Cauchy SequencesProof (related to "Multiplication Of Rational Cauchy Sequences") [1493]Multiplication of Rational Cauchy Sequences Is AssociativeProof (related to "Multiplication of Rational Cauchy Sequences Is Associative") [1501]Multiplication of Rational Cauchy Sequences Is CancellativeProof (related to "Multiplication of Rational Cauchy Sequences Is Cancellative") [1573]Multiplication of Rational Cauchy Sequences Is CommutativeProof (related to "Multiplication of Rational Cauchy Sequences Is Commutative") [1503]Multiplication Of Rational NumbersProof (related to "Multiplication Of Rational Numbers") [1529]Multiplication of Rational Numbers Is AssociativeProof (related to "Multiplication of Rational Numbers Is Associative") [1477]Multiplication Of Rational Numbers Is CancellativeProof (related to "Multiplication Of Rational Numbers Is Cancellative") [1481]Multiplication Of Rational Numbers Is CommutativeProof (related to "Multiplication Of Rational Numbers Is Commutative") [1479]Multiplication of Real NumbersProof (related to "Multiplication of Real Numbers") [1533]Multiplication of Real Numbers Is AssociativeProof (related to "Multiplication of Real Numbers Is Associative") [1534]Multiplication of Real Numbers Is CancellativeProof (related to "Multiplication of Real Numbers Is Cancellative") [1577]Multiplication of Real Numbers Is CommutativeProof (related to "Multiplication of Real Numbers Is Commutative") [1535]Multiplying Negative and Positive IntegersProof (related to "Multiplying Negative and Positive Integers") [1590]Multiplying Negative and Positive Rational NumbersProof (related to "Multiplying Negative and Positive Rational Numbers") [1597]Multiplying Negative and Positive Real NumbersProof (related to "Multiplying Negative and Positive Real Numbers") [1599]Natural LogarithmProof (related to "Natural Logarithm") [1600]Nested Closed Subset TheoremProof (related to "Nested Closed Subset Theorem") [6587]Non-Cauchy Sequences are Not ConvergentProof by Contraposition (related to "Non-Cauchy Sequences are Not Convergent") [6871]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 PowersProof (related to "Nth Powers") [1619]Nth Roots of Positive NumbersProof (related to "Nth Roots of Positive Numbers") [1383]Number of Relations on a Finite SetProof (related to "Number of Relations on a Finite Set") [1000]Number of Strings With a Fixed Length Over an Alphabet with k LettersProof (related to "Number of Strings With a Fixed Length Over an Alphabet with k Letters") [997]Number of Subsets of a Finite SetProof (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 VariableProof (related to "Oddness of the Sine of a Real Variable") [1793]Open and Closed Subsets of a Zariski TopologyProof (related to "Open and Closed Subsets of a Zariski Topology") [6327]Order Relation for Natural Numbers, RevisedProof (related to "Order Relation for Natural Numbers, Revised") [1558]Ordinals Are Downward ClosedProof (related to "Ordinals Are Downward Closed") [728]Planarity of SubdivisionsProof (related to "Planarity of Subdivisions") [6379]Position of Minus Sign in Rational Numbers RepresentationsProof (related to "Position of Minus Sign in Rational Numbers Representations") [1593]Preservation of Continuity with Arithmetic Operations on Continuous FunctionsProof (related to "Preservation of Continuity with Arithmetic Operations on Continuous Functions") [1262]Preservation of Continuity with Arithmetic Operations on Continuous Functions on a Whole DomainProof (related to "Preservation of Continuity with Arithmetic Operations on Continuous Functions on a Whole Domain") [1605]Probability of Event DifferenceProof (related to "Probability of Event Difference") [870]Probability of Event UnionProof (related to "Probability of Event Union") [869]Probability of Included EventProof (related to "Probability of Included Event") [866]Probability of Joint EventsProof (related to "Probability of Joint Events") [1803]Probability of Laplace ExperimentsProof (related to "Probability of Laplace Experiments") [976]Probability of the Complement EventProof (related to "Probability of the Complement Event") [863]Probability of the Impossible EventProof (related to "Probability of the Impossible Event") [864]Product of a Complex Number and a Convergent Complex SequenceProof (related to "Product of a Complex Number and a Convergent Complex Sequence") [1720]Product of a Real Number and a Convergent Real SequenceProof (related to "Product of a Real Number and a Convergent Real Sequence") [1141]Product of a Real Number and a Convergent Real SeriesProof (related to "Product of a Real Number and a Convergent Real Series") [6648]Product of Convegent Complex SequencesProof (related to "Product of Convegent Complex Sequences") [1718]Product of Convegent Real SequencesProof (related to "Product of Convegent Real Sequences") [1139]Prop. 1.01: Constructing an Equilateral TriangleProof (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 SegmentProof (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 SizeProof (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 TriangleProof (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 IProof (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 IIProof (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 TrianglesProof (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 TrianglesProof (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 AngleProof (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 SegmentProof (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 LineProof (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 LineProof (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 LinesProof (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 LinesProof (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 LinesProof (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 AnglesProof (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 TriangleProof (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 IProof (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 IIProof (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 TrianglesProof (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 SegmentsProof (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 AngleProof (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 IIIProof (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 IVProof (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 TrianglesProof (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 IProof (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 IIProof (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 IIIProof (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 LinesProof (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 PointProof (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 AngleProof (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 ParallelogramProof (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 ParallelogramsProof (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 ParallelsProof (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 ParallelsProof (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 IProof (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 IIProof (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 IIIProof (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 IVProof (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 TriaglesProof (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 IProof (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 ParallelogramsProof (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 IIProof (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 IIIProof (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 IProof (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 TheoremProof (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 TheoremProof (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 NumberProof (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 AlgorithmProof (related to "Prop. 10.002: Incommensurable Magnitudes do not Terminate in Euclidean Algorithm") [2583]Prop. 10.003: Greatest Common Measure of Commensurable MagnitudesProof (related to "Prop. 10.003: Greatest Common Measure of Commensurable Magnitudes") [2584]Prop. 10.004: Greatest Common Measure of Three Commensurable MagnitudesProof (related to "Prop. 10.004: Greatest Common Measure of Three Commensurable Magnitudes") [2585]Prop. 10.005: Ratio of Commensurable MagnitudesProof (related to "Prop. 10.005: Ratio of Commensurable Magnitudes") [2586]Prop. 10.006: Magnitudes with Rational Ratio are CommensurableProof (related to "Prop. 10.006: Magnitudes with Rational Ratio are Commensurable") [2587]Prop. 10.007: Incommensurable Magnitudes Have Irrational RatioProof (related to "Prop. 10.007: Incommensurable Magnitudes Have Irrational Ratio") [2588]Prop. 10.008: Magnitudes with Irrational Ratio are IncommensurableProof (related to "Prop. 10.008: Magnitudes with Irrational Ratio are Incommensurable") [2589]Prop. 10.009: Commensurability of SquaresProof (related to "Prop. 10.009: Commensurability of Squares") [2590]Prop. 10.010: Construction of Incommensurable LinesProof (related to "Prop. 10.010: Construction of Incommensurable Lines") [2591]Prop. 10.011: Commensurability of Elements of Proportional MagnitudesProof (related to "Prop. 10.011: Commensurability of Elements of Proportional Magnitudes") [2592]Prop. 10.012: Commensurability is Transitive RelationProof (related to "Prop. 10.012: Commensurability is Transitive Relation") [2593]Prop. 10.013: Commensurable Magnitudes are Incommensurable with Same MagnitudeProof (related to "Prop. 10.013: Commensurable Magnitudes are Incommensurable with Same Magnitude") [2594]Prop. 10.014: Commensurability of Squares on Proportional Straight LinesProof (related to "Prop. 10.014: Commensurability of Squares on Proportional Straight Lines") [2595]Prop. 10.015: Commensurability of Sum of Commensurable MagnitudesProof (related to "Prop. 10.015: Commensurability of Sum of Commensurable Magnitudes") [2596]Prop. 10.016: Incommensurability of Sum of Incommensurable MagnitudesProof (related to "Prop. 10.016: Incommensurability of Sum of Incommensurable Magnitudes") [2597]Prop. 10.017: Condition for Commensurability of Roots of Quadratic EquationProof (related to "Prop. 10.017: Condition for Commensurability of Roots of Quadratic Equation") [2598]Prop. 10.018: Condition for Incommensurability of Roots of Quadratic EquationProof (related to "Prop. 10.018: Condition for Incommensurability of Roots of Quadratic Equation") [2599]Prop. 10.019: Product of Rational Numbers is RationalProof (related to "Prop. 10.019: Product of Rational Numbers is Rational") [2600]Prop. 10.020: Quotient of Rational Numbers is RationalProof (related to "Prop. 10.020: Quotient of Rational Numbers is Rational") [2601]Prop. 10.021: Medial is IrrationalProof (related to "Prop. 10.021: Medial is Irrational") [2602]Prop. 10.022: Square on Medial Straight LineProof (related to "Prop. 10.022: Square on Medial Straight Line") [2603]Prop. 10.023: Segment Commensurable with Medial Segment is MedialProof (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 MedialProof (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 SquareProof (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 AreaProof (related to "Prop. 10.026: Medial Area not greater than Medial Area by Rational Area") [2607]Prop. 10.027: Construction of Components of First BimedialProof (related to "Prop. 10.027: Construction of Components of First Bimedial") [2608]Prop. 10.028: Construction of Components of Second BimedialProof (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 GreProof (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 GProof (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 IProof (related to "Prop. 10.031: Constructing Medial Commensurable in Square I") [2612]Prop. 10.032: Constructing Medial Commensurable in Square IIProof (related to "Prop. 10.032: Constructing Medial Commensurable in Square II") [2613]Prop. 10.033: Construction of Components of MajorProof (related to "Prop. 10.033: Construction of Components of Major") [2614]Prop. 10.034: Construction of Components of Side of Rational plus Medial AreaProof (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 AreasProof (related to "Prop. 10.035: Construction of Components of Side of Sum of Medial Areas") [2616]Prop. 10.036: Binomial is IrrationalProof (related to "Prop. 10.036: Binomial is Irrational") [2617]Prop. 10.037: First Bimedial is IrrationalProof (related to "Prop. 10.037: First Bimedial is Irrational") [2618]Prop. 10.038: Second Bimedial is IrrationalProof (related to "Prop. 10.038: Second Bimedial is Irrational") [2619]Prop. 10.039: Major is IrrationalProof (related to "Prop. 10.039: Major is Irrational") [2620]Prop. 10.040: Side of Rational plus Medial Area is IrrationalProof (related to "Prop. 10.040: Side of Rational plus Medial Area is Irrational") [2621]Prop. 10.041: Side of Sum of Medial Areas is IrrationalProof (related to "Prop. 10.041: Side of Sum of Medial Areas is Irrational") [2622]Prop. 10.042: Binomial Straight Line is Divisible into Terms UniquelyProof (related to "Prop. 10.042: Binomial Straight Line is Divisible into Terms Uniquely") [2623]Prop. 10.043: First Bimedial Straight Line is Divisible UniquelyProof (related to "Prop. 10.043: First Bimedial Straight Line is Divisible Uniquely") [2624]Prop. 10.044: Second Bimedial Straight Line is Divisible UniquelyProof (related to "Prop. 10.044: Second Bimedial Straight Line is Divisible Uniquely") [2625]Prop. 10.045: Major Straight Line is Divisible UniquelyProof (related to "Prop. 10.045: Major Straight Line is Divisible Uniquely") [2626]Prop. 10.046: Side of Rational Plus Medial Area is Divisible UniquelyProof (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 UniquelyProof (related to "Prop. 10.047: Side of Sum of Two Medial Areas is Divisible Uniquely") [2628]Prop. 10.048: Construction of First Binomial Straight LineProof (related to "Prop. 10.048: Construction of First Binomial Straight Line") [2629]Prop. 10.049: Construction of Second Binomial Straight LineProof (related to "Prop. 10.049: Construction of Second Binomial Straight Line") [2630]Prop. 10.050: Construction of Third Binomial Straight LineProof (related to "Prop. 10.050: Construction of Third Binomial Straight Line") [2631]Prop. 10.051: Construction of Fourth Binomial Straight LineProof (related to "Prop. 10.051: Construction of Fourth Binomial Straight Line") [2632]Prop. 10.052: Construction of Fifth Binomial Straight LineProof (related to "Prop. 10.052: Construction of Fifth Binomial Straight Line") [2633]Prop. 10.053: Construction of Sixth Binomial Straight LineProof (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 BinomialProof (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 BinomialProof (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 BinomialProof (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 BinomialProof (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 BinomialProof (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 BinomialProof (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 LineProof (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 LineProof (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 LineProof (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 LineProof (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 LineProof (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 LineProof (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 OrderProof (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 OrderProof (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 MajorProof (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 AreaProof (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 AreasProof (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 LinesProof (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 LinesProof (related to "Prop. 10.072: Sum of two Incommensurable Medial Areas give rise to two Irrational Straight Lines") [2653]Prop. 10.073: Apotome is IrrationalProof (related to "Prop. 10.073: Apotome is Irrational") [2654]Prop. 10.074: First Apotome of Medial is IrrationalProof (related to "Prop. 10.074: First Apotome of Medial is Irrational") [2655]Prop. 10.075: Second Apotome of Medial is IrrationalProof (related to "Prop. 10.075: Second Apotome of Medial is Irrational") [2656]Prop. 10.076: Minor is IrrationalProof (related to "Prop. 10.076: Minor is Irrational") [2657]Prop. 10.077: That which produces Medial Whole with Rational Area is IrrationalProof (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 IrrationalProof (related to "Prop. 10.078: That which produces Medial Whole with Medial Area is Irrational") [2659]Prop. 10.079: Construction of Apotome is UniqueProof (related to "Prop. 10.079: Construction of Apotome is Unique") [2660]Prop. 10.080: Construction of First Apotome of Medial is UniqueProof (related to "Prop. 10.080: Construction of First Apotome of Medial is Unique") [2661]Prop. 10.081: Construction of Second Apotome of Medial is UniqueProof (related to "Prop. 10.081: Construction of Second Apotome of Medial is Unique") [2662]Prop. 10.082: Construction of Minor is UniqueProof (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 UniqueProof (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 UniqueProof (related to "Prop. 10.084: Construction of that which produces Medial Whole with Medial Area is Unique") [2665]Prop. 10.085: Construction of First ApotomeProof (related to "Prop. 10.085: Construction of First Apotome") [2666]Prop. 10.086: Construction of Second ApotomeProof (related to "Prop. 10.086: Construction of Second Apotome") [2667]Prop. 10.087: Construction of Third ApotomeProof (related to "Prop. 10.087: Construction of Third Apotome") [2668]Prop. 10.088: Construction of Fourth ApotomeProof (related to "Prop. 10.088: Construction of Fourth Apotome") [2669]Prop. 10.089: Construction of Fifth ApotomeProof (related to "Prop. 10.089: Construction of Fifth Apotome") [2670]Prop. 10.090: Construction of Sixth ApotomeProof (related to "Prop. 10.090: Construction of Sixth Apotome") [2671]Prop. 10.091: Side of Area Contained by Rational Straight Line and First ApotomeProof (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 ApotomeProof (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 ApotomeProof (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 ApotomeProof (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 ApotomeProof (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 ApotomeProof (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 LineProof (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 LineProof (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 LineProof (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 LineProof (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 LineProof (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 LineProof (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 ApotomeProof (related to "Prop. 10.103: Straight Line Commensurable with Apotome") [2684]Prop. 10.104: Straight Line Commensurable with Apotome of Medial Straight LineProof (related to "Prop. 10.104: Straight Line Commensurable with Apotome of Medial Straight Line") [2685]Prop. 10.105: Straight Line Commensurable with Minor Straight LineProof (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 AreaProof (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 AreaProof (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 SubtractedProof (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 SubtractedProof (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 SubtractedProof (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 LineProof (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 LineProof (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 ApotomeProof (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 RatioProof (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 LinesProof (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 PlanesProof (related to "Prop. 11.01: Straight Line cannot be in Two Planes") [2697]Prop. 11.02: Two Intersecting Straight Lines are in One PlaneProof (related to "Prop. 11.02: Two Intersecting Straight Lines are in One Plane") [2698]Prop. 11.03: Common Section of Two Planes is Straight LineProof (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 PlaneProof (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 PlaneProof (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 ParallelProof (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 PlaneProof (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 PlaneProof (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 otherProof (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 AnglesProof (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 PlaneProof (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 PlaneProof (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 UniqueProof (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 ParallelProof (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 ParallelProof (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 ParallelProof (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 PlanesProof (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 PlaneProof (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 PlaneProof (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 AngleProof (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 AnglesProof (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 TriangleProof (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 AnglesProof (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 ParallelogramsProof (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 PlanesProof (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 AngleProof (related to "Prop. 11.26: Construction of Solid Angle equal to Given Solid Angle") [2723]Prop. 11.27: Construction of Parallelepiped Similar to Given ParallelepipedProof (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 BisectedProof (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 VolumeProof (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 VolumeProof (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 VolumeProof (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 BasesProof (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 SidesProof (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 HeightsProof (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 AnglesProof (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 formeProof (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 ProportionalProof (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 CubeProof (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 BaseProof (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 DiametersProof (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 DiametersProof (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 PrismsProof (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 PrismsProof (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 BasesProof (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 BasesProof (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 TetrahedraProof (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 SidesProof (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 HeightsProof (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 HeightProof (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 BasesProof (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 BasesProof (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 AxisProof (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 HeightsProof (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 HeightsProof (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 CirclesProof (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 SpheresProof (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 DiametersProof (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 RatioProof (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 RatioProof (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 RatioProof (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 RatioProof (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 SegmentProof (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 ApotomeProof (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 EqualProof (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 RatioProof (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 RatioProof (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 saProof (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 MinorProof (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 CircleProof (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 SphereProof (related to "Prop. 13.13: Construction of Regular Tetrahedron within Given Sphere") [2767]Prop. 13.14: Construction of Regular Octahedron within Given SphereProof (related to "Prop. 13.14: Construction of Regular Octahedron within Given Sphere") [2768]Prop. 13.15: Construction of Cube within Given SphereProof (related to "Prop. 13.15: Construction of Cube within Given Sphere") [2769]Prop. 13.16: Construction of Regular Icosahedron within Given SphereProof (related to "Prop. 13.16: Construction of Regular Icosahedron within Given Sphere") [2770]Prop. 13.17: Construction of Regular Dodecahedron within Given SphereProof (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 SolidsProof (related to "Prop. 13.18: Comparison of Sides of Platonic Figures - There are only Five Platonic Solids") [2772]Prop. 2.01: Summing Areas or RectanglesProof (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 RectanglesProof (related to "Prop. 2.02: Square is Sum of Two Rectangles") [2511]Prop. 2.03: Rectangle is Sum of Square and RectangleProof (related to "Prop. 2.03: Rectangle is Sum of Square and Rectangle") [2560]Prop. 2.04: Square of SumProof (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 SquaresProof (related to "Prop. 2.05: Rectangle is Difference of Two Squares") [6534]Prop. 2.06: Square of Sum with One Halved SummandProof (related to "Prop. 2.06: Square of Sum with One Halved Summand") [6535]Prop. 2.07: Sum of SquaresProof (related to "Prop. 2.07: Sum of Squares") [6536]Prop. 2.08: Square of Sum with One Doubled SummandProof (related to "Prop. 2.08: Square of Sum with One Doubled Summand") [6537]Prop. 2.09: Sum of Squares of Sum and DifferenceProof (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 SegmentProof (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 FigureProof (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 CircleProof (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 CircleProof (related to "Prop. 3.02: Chord Lies Inside its Circle") [6545]Prop. 3.03: Conditions for Diameter to be Perpendicular BisectorProof (related to "Prop. 3.03: Conditions for Diameter to be Perpendicular Bisector") [2373]Prop. 3.04: Chords do not Bisect Each OtherProof (related to "Prop. 3.04: Chords do not Bisect Each Other") [2374]Prop. 3.05: Intersecting Circles have Different CentersProof (related to "Prop. 3.05: Intersecting Circles have Different Centers") [2375]Prop. 3.06: Touching Circles have Different CentersProof (related to "Prop. 3.06: Touching Circles have Different Centers") [2376]Prop. 3.07: Relative Lengths of Lines Inside CircleProof (related to "Prop. 3.07: Relative Lengths of Lines Inside Circle") [2377]Prop. 3.08: Relative Lengths of Lines Outside CircleProof (related to "Prop. 3.08: Relative Lengths of Lines Outside Circle") [2378]Prop. 3.09: Condition for Point to be Center of CircleProof (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 IntersectionProof (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 InternallyProof (related to "Prop. 3.11: Line Joining Centers of Two Circles Touching Internally") [2381]Prop. 3.12: Line Joining Centers of Two Circles Touching ExternallyProof (related to "Prop. 3.12: Line Joining Centers of Two Circles Touching Externally") [2382]Prop. 3.13: Circles Touch at One Point at MostProof (related to "Prop. 3.13: Circles Touch at One Point at Most") [2383]Prop. 3.14: Equal Chords in CircleProof (related to "Prop. 3.14: Equal Chords in Circle") [2384]Prop. 3.15: Relative Lengths of Chords of Circles‎Proof (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 CircleProof (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 CircleProof (related to "Prop. 3.17: Construction of Tangent from Point to Circle") [2387]Prop. 3.18: Radius at Right Angle to TangentProof (related to "Prop. 3.18: Radius at Right Angle to Tangent") [2388]Prop. 3.19: Right Angle to Tangent of Circle goes through CenterProof (related to "Prop. 3.19: Right Angle to Tangent of Circle goes through Center") [2389]Prop. 3.20: Inscribed Angle TheoremProof (related to "Prop. 3.20: Inscribed Angle Theorem") [2390]Prop. 3.21: Angles in Same Segment of Circle are EqualProof (related to "Prop. 3.21: Angles in Same Segment of Circle are Equal") [2391]Prop. 3.22: Opposite Angles of Cyclic QuadrilateralProof (related to "Prop. 3.22: Opposite Angles of Cyclic Quadrilateral") [2392]Prop. 3.23: Segment on Given Base UniqueProof (related to "Prop. 3.23: Segment on Given Base Unique") [2393]Prop. 3.24: Similar Segments on Equal Bases are EqualProof (related to "Prop. 3.24: Similar Segments on Equal Bases are Equal") [2394]Prop. 3.25: Construction of Circle from SegmentProof (related to "Prop. 3.25: Construction of Circle from Segment") [2395]Prop. 3.26: Equal Angles in Equal CirclesProof (related to "Prop. 3.26: Equal Angles in Equal Circles") [2396]Prop. 3.27: Angles on Equal Arcs are EqualProof (related to "Prop. 3.27: Angles on Equal Arcs are Equal") [2397]Prop. 3.28: Straight Lines Cut Off Equal Arcs in Equal CirclesProof (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 LinesProof (related to "Prop. 3.29: Equal Arcs of Circles Subtended by Equal Straight Lines") [2399]Prop. 3.30: Bisection of ArcProof (related to "Prop. 3.30: Bisection of Arc") [2400]Prop. 3.31: Relative Sizes of Angles in SegmentsProof (related to "Prop. 3.31: Relative Sizes of Angles in Segments") [2401]Prop. 3.32: Angles made by Chord with Tangent‎Proof (related to "Prop. 3.32: Angles made by Chord with Tangent‎") [2402]Prop. 3.33: Construction of Segment on Given Line Admitting Given AngleProof (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 AngleProof (related to "Prop. 3.34: Construction of Segment on Given Circle Admitting Given Angle") [2404]Prop. 3.35: Intersecting Chord TheoremProof (related to "Prop. 3.35: Intersecting Chord Theorem") [2405]Prop. 3.36: Tangent Secant TheoremProof (related to "Prop. 3.36: Tangent Secant Theorem") [2406]Prop. 3.37: Converse of Tangent Secant TheoremProof (related to "Prop. 3.37: Converse of Tangent Secant Theorem") [2407]Prop. 4.01: Fitting Chord Into CircleProof (related to "Prop. 4.01: Fitting Chord Into Circle") [2408]Prop. 4.02: Inscribing in Circle Triangle Equiangular with GivenProof (related to "Prop. 4.02: Inscribing in Circle Triangle Equiangular with Given") [2409]Prop. 4.03: Circumscribing about Circle Triangle Equiangular with GivenProof (related to "Prop. 4.03: Circumscribing about Circle Triangle Equiangular with Given") [2410]Prop. 4.04: Inscribing Circle in TriangleProof (related to "Prop. 4.04: Inscribing Circle in Triangle") [2411]Prop. 4.05: Circumscribing Circle about TriangleProof (related to "Prop. 4.05: Circumscribing Circle about Triangle") [2412]Prop. 4.06: Inscribing Square in CircleProof (related to "Prop. 4.06: Inscribing Square in Circle") [2413]Prop. 4.07: Circumscribing Square about CircleProof (related to "Prop. 4.07: Circumscribing Square about Circle") [2414]Prop. 4.08: Inscribing Circle in SquareProof (related to "Prop. 4.08: Inscribing Circle in Square") [2415]Prop. 4.09: Circumscribing Circle about SquareProof (related to "Prop. 4.09: Circumscribing Circle about Square") [2416]Prop. 4.10: Construction of Isosceles Triangle whose Base Angle is Twice ApexProof (related to "Prop. 4.10: Construction of Isosceles Triangle whose Base Angle is Twice Apex") [2417]Prop. 4.11: Inscribing Regular Pentagon in CircleProof (related to "Prop. 4.11: Inscribing Regular Pentagon in Circle") [2418]Prop. 4.12: Circumscribing Regular Pentagon about CircleProof (related to "Prop. 4.12: Circumscribing Regular Pentagon about Circle") [2419]Prop. 4.13: Inscribing Circle in Regular PentagonProof (related to "Prop. 4.13: Inscribing Circle in Regular Pentagon") [2420]Prop. 4.14: Circumscribing Circle about Regular PentagonProof (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 CircleProof (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 CircleProof (related to "Prop. 4.16: Inscribing Regular 15-gon in Circle") [2423]Prop. 5.01: Multiplication of Numbers is Left Distributive over AdditionProof (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 AdditionProof (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 AssociativeProof (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 RatiosProof (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 SubtractionProof (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 Subtraction‎Proof (related to "Prop. 5.06: Multiplication of Real Numbers is Right Distributive over Subtraction‎") [2429]Prop. 5.07: Ratios of Equal MagnitudesProof (related to "Prop. 5.07: Ratios of Equal Magnitudes") [2430]Prop. 5.08: Relative Sizes of Ratios on Unequal MagnitudesProof (related to "Prop. 5.08: Relative Sizes of Ratios on Unequal Magnitudes") [2431]Prop. 5.09: Magnitudes with Same Ratios are EqualProof (related to "Prop. 5.09: Magnitudes with Same Ratios are Equal") [2432]Prop. 5.10: Relative Sizes of Magnitudes on Unequal RatiosProof (related to "Prop. 5.10: Relative Sizes of Magnitudes on Unequal Ratios") [2433]Prop. 5.11: Equality of Ratios is TransitiveProof (related to "Prop. 5.11: Equality of Ratios is Transitive") [2434]Prop. 5.12: Sum of Components of Equal RatiosProof (related to "Prop. 5.12: Sum of Components of Equal Ratios") [2435]Prop. 5.13: Relative Sizes of Proportional MagnitudesProof (related to "Prop. 5.13: Relative Sizes of Proportional Magnitudes") [2436]Prop. 5.14: Relative Sizes of Components of RatiosProof (related to "Prop. 5.14: Relative Sizes of Components of Ratios") [2437]Prop. 5.15: Ratio Equals its MultiplesProof (related to "Prop. 5.15: Ratio Equals its Multiples") [2438]Prop. 5.16: Proportional Magnitudes are Proportional AlternatelyProof (related to "Prop. 5.16: Proportional Magnitudes are Proportional Alternately") [2439]Prop. 5.17: Magnitudes Proportional Compounded are Proportional SeparatedProof (related to "Prop. 5.17: Magnitudes Proportional Compounded are Proportional Separated") [2440]Prop. 5.18: Magnitudes Proportional Separated are Proportional CompoundedProof (related to "Prop. 5.18: Magnitudes Proportional Separated are Proportional Compounded") [2441]Prop. 5.19: Proportional Magnitudes have Proportional RemaindersProof (related to "Prop. 5.19: Proportional Magnitudes have Proportional Remainders") [2442]Prop. 5.20: Relative Sizes of Successive RatiosProof (related to "Prop. 5.20: Relative Sizes of Successive Ratios") [2443]Prop. 5.21: Relative Sizes of Elements in Perturbed ProportionProof (related to "Prop. 5.21: Relative Sizes of Elements in Perturbed Proportion") [2444]Prop. 5.22: Equality of Ratios Ex AequaliProof (related to "Prop. 5.22: Equality of Ratios Ex Aequali") [2445]Prop. 5.23: Equality of Ratios in Perturbed ProportionProof (related to "Prop. 5.23: Equality of Ratios in Perturbed Proportion") [2446]Prop. 5.24: Sum of Antecedents of ProportionProof (related to "Prop. 5.24: Sum of Antecedents of Proportion") [2447]Prop. 5.25: Sum of Antecedent and Consequent of ProportionProof (related to "Prop. 5.25: Sum of Antecedent and Consequent of Proportion") [2448]Prop. 6.01: Areas of Triangles and Parallelograms Proportional to BaseProof (related to "Prop. 6.01: Areas of Triangles and Parallelograms Proportional to Base") [2449]Prop. 6.02: Parallel Line in Triangle Cuts Sides ProportionallyProof (related to "Prop. 6.02: Parallel Line in Triangle Cuts Sides Proportionally") [2450]Prop. 6.03: Angle Bisector TheoremProof (related to "Prop. 6.03: Angle Bisector Theorem") [2451]Prop. 6.04: Equiangular Triangles are SimilarProof (related to "Prop. 6.04: Equiangular Triangles are Similar") [2452]Prop. 6.05: Triangles with Proportional Sides are SimilarProof (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 SimilarProof (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 SimilarProof (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 TrianglesProof (related to "Prop. 6.08: Perpendicular in Right-Angled Triangle makes two Similar Triangles") [2456]Prop. 6.09: Construction of Part of LineProof (related to "Prop. 6.09: Construction of Part of Line") [2457]Prop. 6.10: Construction of Similarly Cut Straight LineProof (related to "Prop. 6.10: Construction of Similarly Cut Straight Line") [2458]Prop. 6.11: Construction of Third Proportional Straight Line‎Proof (related to "Prop. 6.11: Construction of Third Proportional Straight Line‎") [2459]Prop. 6.12: Construction of Fourth Proportional Straight LineProof (related to "Prop. 6.12: Construction of Fourth Proportional Straight Line") [2460]Prop. 6.13: Construction of Mean Proportional‎Proof (related to "Prop. 6.13: Construction of Mean Proportional‎") [2461]Prop. 6.14: Sides of Equal and Equiangular Parallelograms are Reciprocally Proportional‎Proof (related to "Prop. 6.14: Sides of Equal and Equiangular Parallelograms are Reciprocally Proportional‎") [2462]Prop. 6.15: Sides of Equiangular Triangles are Reciprocally ProportionalProof (related to "Prop. 6.15: Sides of Equiangular Triangles are Reciprocally Proportional") [2463]Prop. 6.16: Rectangles Contained by Proportional Straight LinesProof (related to "Prop. 6.16: Rectangles Contained by Proportional Straight Lines") [2464]Prop. 6.17: Rectangles Contained by Three Proportional Straight LinesProof (related to "Prop. 6.17: Rectangles Contained by Three Proportional Straight Lines") [2465]Prop. 6.18: Construction of Similar PolygonProof (related to "Prop. 6.18: Construction of Similar Polygon") [2466]Prop. 6.19: Ratio of Areas of Similar TrianglesProof (related to "Prop. 6.19: Ratio of Areas of Similar Triangles") [2467]Prop. 6.20: Similar Polygons are Composed of Similar TrianglesProof (related to "Prop. 6.20: Similar Polygons are Composed of Similar Triangles") [2468]Prop. 6.21: Similarity of Polygons is Equivalence‎ RelationProof (related to "Prop. 6.21: Similarity of Polygons is Equivalence‎ Relation") [2469]Prop. 6.22: Similar Figures on Proportional Straight LinesProof (related to "Prop. 6.22: Similar Figures on Proportional Straight Lines") [2470]Prop. 6.23: Ratio of Areas of Equiangular ParallelogramsProof (related to "Prop. 6.23: Ratio of Areas of Equiangular Parallelograms") [2471]Prop. 6.24: Parallelograms About Diameter are SimilarProof (related to "Prop. 6.24: Parallelograms About Diameter are Similar") [2472]Prop. 6.25: Construction of Figure Similar to One and Equal to AnotherProof (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 DiameterProof (related to "Prop. 6.26: Parallelogram Similar and in Same Angle has Same Diameter") [2474]Prop. 6.27: Similar Parallelogram on Half a Straight LineProof (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 ParallelogramProof (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 ParallelogramProof (related to "Prop. 6.29: Construction of Parallelogram Equal to Given Figure Exceeding a Parallelogram") [2477]Prop. 6.30: Construction of Golden SectionProof (related to "Prop. 6.30: Construction of Golden Section") [2478]Prop. 6.31: Similar Figures on Sides of Right-Angled TriangleProof (related to "Prop. 6.31: Similar Figures on Sides of Right-Angled Triangle") [2479]Prop. 6.32: Triangles with Two Sides Parallel and EqualProof (related to "Prop. 6.32: Triangles with Two Sides Parallel and Equal") [2480]Prop. 6.33: Angles in Circles have Same Ratio as ArcsProof (related to "Prop. 6.33: Angles in Circles have Same Ratio as Arcs") [2481]Prop. 7.01: Sufficient Condition for CoprimalityProof (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 AlgorithmProof (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 NumbersProof (related to "Prop. 7.03: Greatest Common Divisor of Three Numbers") [2484]Prop. 7.04: Natural Number Divisor or Multiple of Divisor of AnotherProof (related to "Prop. 7.04: Natural Number Divisor or Multiple of Divisor of Another") [2485]Prop. 7.05: Divisors obey Distributive LawProof (related to "Prop. 7.05: Divisors obey Distributive Law") [2486]Prop. 7.06: Multiples of Divisors Obey Distributive LawProof (related to "Prop. 7.06: Multiples of Divisors Obey Distributive Law") [2487]Prop. 7.07: Subtraction of Divisors Obeys Distributive LawProof (related to "Prop. 7.07: Subtraction of Divisors Obeys Distributive Law") [2488]Prop. 7.08: Subtraction of Multiples of Divisors Obeys Distributive LawProof (related to "Prop. 7.08: Subtraction of Multiples of Divisors Obeys Distributive Law") [2489]Prop. 7.09: Alternate Ratios of Equal FractionsProof (related to "Prop. 7.09: Alternate Ratios of Equal Fractions") [2490]Prop. 7.10: Multiples of Alternate Ratios of Equal FractionsProof (related to "Prop. 7.10: Multiples of Alternate Ratios of Equal Fractions") [2491]Prop. 7.11: Proportional Numbers have Proportional DifferencesProof (related to "Prop. 7.11: Proportional Numbers have Proportional Differences") [2492]Prop. 7.12: Ratios of Numbers is Distributive over AdditionProof (related to "Prop. 7.12: Ratios of Numbers is Distributive over Addition") [2493]Prop. 7.13: Proportional Numbers are Proportional AlternatelyProof (related to "Prop. 7.13: Proportional Numbers are Proportional Alternately") [2494]Prop. 7.14: Proportion of Numbers is TransitiveProof (related to "Prop. 7.14: Proportion of Numbers is Transitive") [2495]Prop. 7.15: Alternate Ratios of MultiplesProof (related to "Prop. 7.15: Alternate Ratios of Multiples") [2496]Prop. 7.16: Natural Number Multiplication is CommutativeProof (related to "Prop. 7.16: Natural Number Multiplication is Commutative") [2497]Prop. 7.17: Multiples of Ratios of NumbersProof (related to "Prop. 7.17: Multiples of Ratios of Numbers") [2498]Prop. 7.18: Ratios of Multiples of NumbersProof (related to "Prop. 7.18: Ratios of Multiples of Numbers") [2499]Prop. 7.19: Relation of Ratios to Products‎Proof (related to "Prop. 7.19: Relation of Ratios to Products‎") [2500]Prop. 7.20: Ratios of Fractions in Lowest TermsProof (related to "Prop. 7.20: Ratios of Fractions in Lowest Terms") [2501]Prop. 7.21: Coprime Numbers form Fraction in Lowest TermsProof (related to "Prop. 7.21: Coprime Numbers form Fraction in Lowest Terms") [2502]Prop. 7.22: Numbers forming Fraction in Lowest Terms are CoprimeProof (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 OtherProof (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 WholeProof (related to "Prop. 7.24: Integer Coprime to all Factors is Coprime to Whole") [2505]Prop. 7.25: Square of Coprime Number is CoprimeProof (related to "Prop. 7.25: Square of Coprime Number is Coprime") [2506]Prop. 7.26: Product of Coprime Pairs is CoprimeProof (related to "Prop. 7.26: Product of Coprime Pairs is Coprime") [2507]Prop. 7.27: Powers of Coprime Numbers are CoprimeProof (related to "Prop. 7.27: Powers of Coprime Numbers are Coprime") [2774]Prop. 7.28: Numbers are Coprime iff Sum is Coprime to BothProof (related to "Prop. 7.28: Numbers are Coprime iff Sum is Coprime to Both") [2508]Prop. 7.29: Prime not Divisor implies CoprimeProof (related to "Prop. 7.29: Prime not Divisor implies Coprime") [2509]Prop. 7.30: Euclidean LemmaProof (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 DivisorsProof (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 FactorProof (related to "Prop. 7.32: Natural Number is Prime or has Prime Factor") [2512]Prop. 7.33: Least Ratio of NumbersProof (related to "Prop. 7.33: Least Ratio of Numbers") [2513]Prop. 7.34: Existence of Lowest Common MultipleProof (related to "Prop. 7.34: Existence of Lowest Common Multiple") [2514]Prop. 7.35: Least Common Multiple Divides Common MultipleProof (related to "Prop. 7.35: Least Common Multiple Divides Common Multiple") [2515]Prop. 7.36: Least Common Multiple of Three NumbersProof (related to "Prop. 7.36: Least Common Multiple of Three Numbers") [2516]Prop. 7.37: Integer Divided by Divisor is IntegerProof (related to "Prop. 7.37: Integer Divided by Divisor is Integer") [2517]Prop. 7.38: Divisor is Reciprocal of Divisor of IntegerProof (related to "Prop. 7.38: Divisor is Reciprocal of Divisor of Integer") [2518]Prop. 7.39: Least Number with Three Given FractionsProof (related to "Prop. 7.39: Least Number with Three Given Fractions") [2519]Prop. 8.01: Geometric Progression with Coprime Extremes is in Lowest TermsProof (related to "Prop. 8.01: Geometric Progression with Coprime Extremes is in Lowest Terms") [2520]Prop. 8.02: Construction of Geometric Progression in Lowest TermsProof (related to "Prop. 8.02: Construction of Geometric Progression in Lowest Terms") [2521]Prop. 8.03: Geometric Progression in Lowest Terms has Coprime ExtremesProof (related to "Prop. 8.03: Geometric Progression in Lowest Terms has Coprime Extremes") [2522]Prop. 8.04: Construction of Sequence of Numbers with Given RatiosProof (related to "Prop. 8.04: Construction of Sequence of Numbers with Given Ratios") [2523]Prop. 8.05: Ratio of Products of Sides of Plane NumbersProof (related to "Prop. 8.05: Ratio of Products of Sides of Plane Numbers") [2524]Prop. 8.06: First Element of Geometric Progression not dividing SecondProof (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 SecondProof (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 ElementsProof (related to "Prop. 8.08: Geometric Progressions in Proportion have Same Number of Elements") [2527]Prop. 8.09: Elements of Geometric Progression between Coprime NumbersProof (related to "Prop. 8.09: Elements of Geometric Progression between Coprime Numbers") [2528]Prop. 8.10: Product of Geometric Progressions from OneProof (related to "Prop. 8.10: Product of Geometric Progressions from One") [2529]Prop. 8.11: Between two Squares exists one Mean ProportionalProof (related to "Prop. 8.11: Between two Squares exists one Mean Proportional") [2530]Prop. 8.12: Between two Cubes exist two Mean ProportionalsProof (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 ProgressionProof (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 SquareProof (related to "Prop. 8.14: Number divides Number iff Square divides Square") [2533]Prop. 8.15: Number divides Number iff Cube divides CubeProof (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 SquareProof (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 CubeProof (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 ProportionalProof (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 ProportionalsProof (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 PlaneProof (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 SolidProof (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 SquareProof (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 CubeProof (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 SquareProof (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 CubeProof (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 SquaresProof (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 CubesProof (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 SquareProof (related to "Prop. 9.01: Product of Similar Plane Numbers is Square") [2547]Prop. 9.02: Numbers whose Product is Square are Similar Plane NumbersProof (related to "Prop. 9.02: Numbers whose Product is Square are Similar Plane Numbers") [2548]Prop. 9.03: Square of Cube Number is CubeProof (related to "Prop. 9.03: Square of Cube Number is Cube") [2549]Prop. 9.04: Cube Number multiplied by Cube Number is CubeProof (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 CubeProof (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 CubeProof (related to "Prop. 9.06: Number Squared making Cube is itself Cube") [2552]Prop. 9.07: Product of Composite Number with Number is Solid NumberProof (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 NumberProof (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 NumberProof (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 NumberProof (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 ElementsProof (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 PrimeProof (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 PrimeProof (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 ArithmeticProof (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 ElementProof (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 ProportionalProof (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 SecondProof (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 NumbersProof (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 NumbersProof (related to "Prop. 9.19: Condition for Existence of Fourth Number Proportional to Three Numbers") [2565]Prop. 9.20: Infinite Number of PrimesProof (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 EvenProof (related to "Prop. 9.21: Sum of Even Numbers is Even") [2566]Prop. 9.22: Sum of Even Number of Odd Numbers is EvenProof (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 OddProof (related to "Prop. 9.23: Sum of Odd Number of Odd Numbers is Odd") [2568]Prop. 9.24: Even Number minus Even Number is EvenProof (related to "Prop. 9.24: Even Number minus Even Number is Even") [2569]Prop. 9.25: Even Number minus Odd Number is OddProof (related to "Prop. 9.25: Even Number minus Odd Number is Odd") [2570]Prop. 9.26: Odd Number minus Odd Number is EvenProof (related to "Prop. 9.26: Odd Number minus Odd Number is Even") [2571]Prop. 9.27: Odd Number minus Even Number is OddProof (related to "Prop. 9.27: Odd Number minus Even Number is Odd") [2572]Prop. 9.28: Odd Number multiplied by Even Number is EvenProof (related to "Prop. 9.28: Odd Number multiplied by Even Number is Even") [2573]Prop. 9.29: Odd Number multiplied by Odd Number is OddProof (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 HalfProof (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 DoubleProof (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 OnlyProof (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 OddProof (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 OddProof (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 ProgressionProof (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 HomomorphismDirect Proof (related to "Properties of a Group Homomorphism") [681]Properties of CosetsProof (related to "Properties of Cosets") [830]Properties of Ordinal NumbersProof (related to "Properties of Ordinal Numbers") [725]Properties of the Absolute ValueProof (related to "Properties of the Absolute Value") [1089]Properties of Transitive SetsProof (related to "Properties of Transitive Sets") [722]Pythagorean IdentityProof (related to "Pythagorean Identity") [1795]Quadratic FormulaProof (related to "Quadratic Formula") [6826]Quotient of Convergent Complex SequencesProof (related to "Quotient of Convergent Complex Sequences") [1723]Quotient of Convergent Real SequencesProof (related to "Quotient of Convergent Real Sequences") [1143]Quotient SpaceProof (related to "Quotient Space") [6331]Ratio Test For Absolutely Convergent Complex SeriesProof (related to "Ratio Test For Absolutely Convergent Complex Series") [1730]Ratio Test For Absolutely Convergent SeriesProof (related to "Ratio Test For Absolutely Convergent Series") [1355]Rational Cauchy Sequence Members Are BoundedProof (related to "Rational Cauchy Sequence Members Are Bounded") [1490]Rational Cauchy Sequences Build a Commutative Group With Respect To AdditionProof (related to "Rational Cauchy Sequences Build a Commutative Group With Respect To Addition") [1519]Rational Cauchy Sequences Build a Commutative Monoid With Respect To MultiplicationProof (related to "Rational Cauchy Sequences Build a Commutative Monoid With Respect To Multiplication") [1521]Rational Powers of Positive NumbersProof (related to "Rational Powers of Positive Numbers") [1623]Real Cauchy Sequences are BoundedProof (related to "Real Cauchy Sequences are Bounded") [6873]Rearrangement of Absolutely Convergent SeriesProof (related to "Rearrangement of Absolutely Convergent Series") [1365]Reciprocity Law of Falling And Rising Factorial PowersProof (related to "Reciprocity Law of Falling And Rising Factorial Powers") [1413]Reciprocity of Complex Exponential Function, Non-Zero PropertyProof (related to "Reciprocity of Complex Exponential Function, Non-Zero Property") [1741]Reciprocity of Exponential Function of General Base, Non-Zero PropertyProof (related to "Reciprocity of Exponential Function of General Base, Non-Zero Property") [1615]Reciprocity of Exponential Function, Non-Zero PropertyProof (related to "Reciprocity of Exponential Function, Non-Zero Property") [1418]Recursive Formula for Binomial CoefficientsProof (related to "Recursive Formula for Binomial Coefficients") [995]Relationship Between the Greatest Common Divisor and the Least Common MultipleProof (related to "Relationship Between the Greatest Common Divisor and the Least Common Multiple") [1282]Replacing Mutually Independent Events by Their ComplementsProof (related to "Replacing Mutually Independent Events by Their Complements") [1811]Representing Real Cosine by Complex Exponential FunctionProof (related to "Representing Real Cosine by Complex Exponential Function") [1787]Representing Real Sine by Complex Exponential FunctionProof (related to "Representing Real Sine by Complex Exponential Function") [1789]Riemann Integral for Step FunctionsProof (related to "Riemann Integral for Step Functions") [1753]Riemann Upper and Riemann Lower Integrals for Bounded Real FunctionsProof (related to "Riemann Upper and Riemann Lower Integrals for Bounded Real Functions") [1762]Right-Distributivity Law For Natural NumbersProof (related to "Right-Distributivity Law For Natural Numbers") [1437]Rule of Combining Different Sets of IndicesProof (related to "Rule of Combining Different Sets of Indices") [1120]Rules of Calculation with InequalitiesElementary Proof (related to "Rules of Calculation with Inequalities") [595]Set Intersection is AssociativeProof of Equality of Sets (related to "Set Intersection is Associative") [6850]Set Intersection is CommutativeProof of Equality of Sets (related to "Set Intersection is Commutative") [6842]Set Union is AssociativeProof of Equality of Sets (related to "Set Union is Associative") [6851]Set Union is CommutativeProof of Equality of Sets (related to "Set Union is Commutative") [6843]Sets are Subsets of Their UnionProof (related to "Sets are Subsets of Their Union") [6833]Simple Binomial IdentitiesProof (related to "Simple Binomial Identities") [1840]Simulating LOOP Programs Using WHILE ProgramsProof (related to "Simulating LOOP Programs Using WHILE Programs") [1200]Size of an $$r$$-Regular Graph with $$n$$ VerticesProof (related to "Size of an $$r$$-Regular Graph with $$n$$ Vertices") [6356]Splitting a Graph with Even Degree Vertices into CyclesProof (related to "Splitting a Graph with Even Degree Vertices into Cycles") [6383]Square RootsProof (related to "Square Roots") [1162]Subgroups of Cyclic GroupsProof (related to "Subgroups of Cyclic Groups") [820]Subgroups of Finite Cyclic GroupsProof (related to "Subgroups of Finite Cyclic Groups") [826]Subset of Powers is SubmonoidProof (related to "Subset of Powers is Submonoid") [6818]Subsets of Finite SetsProof (related to "Subsets of Finite Sets") [987]Subsets of Natural Numbers Relatively Prime to a Natural Number are Divisor-ClosedProof (related to "Subsets of Natural Numbers Relatively Prime to a Natural Number are Divisor-Closed") [6408]Successor of OridinalProof (related to "Successor of Oridinal") [775]Sum of Arithmetic ProgressionProof (related to "Sum of Arithmetic Progression") [1118]Sum of Binomial CoefficientsProof (related to "Sum of Binomial Coefficients") [1406]Sum of Binomial Coefficients IProof (related to "Sum of Binomial Coefficients I") [1842]Sum of Binomial Coefficients IIProof (related to "Sum of Binomial Coefficients II") [1844]Sum of Convergent Complex SequencesProof (related to "Sum of Convergent Complex Sequences") [1712]Sum of Convergent Real SequencesProof (related to "Sum of Convergent Real Sequences") [1132]Sum of Convergent Real SeriesProof (related to "Sum of Convergent Real Series") [6644]Supremum Property, Infimum PropertyProof 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 PrincipleProof (related to "The Fundamental Counting Principle") [992]The General Perturbation MethodProof (related to "The General Perturbation Method") [1122]The Proving Principle by ContradictionProof (related to "The Proving Principle by Contradiction") [745]The Proving Principle By Contraposition, ContrapositiveProof (related to "The Proving Principle By Contraposition, Contrapositive") [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 functionsProof (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-WeierstrassProof (related to "Theorem of Bolzano-Weierstrass") [6609]Theorem of Large Numbers for Relative FrequenciesProof (related to "Theorem of Large Numbers for Relative Frequencies") [1848]Time Dilation, Lorentz FactorProof (related to "Time Dilation, Lorentz Factor") [6298]Transitivity of the Order Relation of Natural NumbersProof (related to "Transitivity of the Order Relation of Natural Numbers") [1550]Triangle InequalityProof (related to "Triangle Inequality") [1088]Trichotomy of OrdinalsProof (related to "Trichotomy of Ordinals") [731]Trichotomy of the Order Relation for Natural NumbersProof (related to "Trichotomy of the Order Relation for Natural Numbers") [1553]Union of Countable Many Countable SetsProof (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 BasisProof by Contradiction (related to "Uniqueness Lemma of a Finite Basis") [1040]Uniqueness of 1Elementary Proof of Uniqueness (related to "Uniqueness of 1") [49]Uniqueness of Complex ZeroProof (related to "Uniqueness of Complex Zero") [1687]Uniqueness of Integer ZeroProof (related to "Uniqueness of Integer Zero") [1683]Uniqueness of Natural ZeroProof (related to "Uniqueness of Natural Zero") [1681]Uniqueness of Negative NumbersElementary Proof of Uniqueness (related to "Uniqueness of Negative Numbers") [60]Uniqueness Of Predecessors Of Natural NumbersProof (related to "Uniqueness Of Predecessors Of Natural Numbers") [1543]Uniqueness of Rational ZeroProof (related to "Uniqueness of Rational Zero") [1685]Uniqueness of Real ZeroElementary Proof of Uniqueness (related to "Uniqueness of Real Zero") [44]Uniqueness of Reciprocal NumbersElementary Proof of Uniqueness (related to "Uniqueness of Reciprocal Numbers") [61]Uniqueness of the Limit of a SequenceProof (related to "Uniqueness of the Limit of a Sequence") [1130]Unit CircleProof (related to "Unit Circle") [1750]Unit Ring of All Rational Cauchy SequencesProof (related to "Unit Ring of All Rational Cauchy Sequences") [1104]Urn Model With ReplacementProof (related to "Urn Model With Replacement") [1800]Urn Model Without ReplacementProof (related to "Urn Model Without Replacement") [1798]Well-Ordering PrincipleProof (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]