User:Ascii/Theorems

The following are the $1000$ most linked to theorems (or axioms) as of 21st September 2019:

Pages in the Definition namespace and the Proven Results category were used.

They are also in descending order (that is, Principle of Mathematical Induction is more linked to than Proof by Contradiction):


 * 1) Principle of Mathematical Induction
 * 2) Proof by Contradiction
 * 3) Definition:Basis for the Induction
 * 4) Definition:Induction Hypothesis
 * 5) Linear Combination of Integrals
 * 6) Integration by Substitution
 * 7) Definition:Induction Step
 * 8) Primitive of Power
 * 9) Divisor Sum of Integer
 * 10) Integration by Parts
 * 11) Primitive of Constant Multiple of Function
 * 12) Power Rule for Derivatives
 * 13) Difference of Two Squares
 * 14) Method of Truth Tables
 * 15) Sum of Squares of Sine and Cosine
 * 16) Divisor Count Function from Prime Decomposition
 * 17) Chain Rule for Derivatives
 * 18) Derivative of Power
 * 19) Rule of Transposition
 * 20) Subset Relation is Transitive
 * 21) Binomial Theorem
 * 22) Proof by Counterexample
 * 23) Sum of Geometric Sequence
 * 24) Sum of Infinite Geometric Sequence
 * 25) Primitive of Reciprocal
 * 26) Intersection is Subset
 * 27) Primitive of Constant
 * 28) Set is Subset of Union
 * 29) Secant is Reciprocal of Cosine
 * 30) Axiom:Axiom of Choice
 * 31) Second Principle of Mathematical Induction
 * 32) Euler's Formula
 * 33) Tangent is Sine divided by Cosine
 * 34) Closed Form for Triangular Numbers
 * 35) Solution to Quadratic Equation
 * 36) Derivative of Exponential Function
 * 37) Exponential of Zero
 * 38) Primitive of Function of Constant Multiple
 * 39) Pythagoras's Theorem
 * 40) Difference of Logarithms
 * 41) Proof by Cases
 * 42) Well-Ordering Principle
 * 43) Cosine of Zero is One
 * 44) Exponential of Sum
 * 45) Intersection Distributes over Union
 * 46) Intersection with Subset is Subset
 * 47) Division Theorem
 * 48) Lagrange's Theorem (Group Theory)
 * 49) Euclidean Algorithm
 * 50) Equality of Mappings
 * 51) Two-Step Subgroup Test
 * 52) Axiom:Euclid's Second Postulate/Production
 * 53) Derivative of Sine Function
 * 54) Cosine Exponential Formulation
 * 55) Set is Closed iff Equals Topological Closure
 * 56) Triangle Inequality
 * 57) Cosecant is Reciprocal of Sine
 * 58) Law of Excluded Middle
 * 59) Derivative of Identity Function
 * 60) Relative Complement of Relative Complement
 * 61) Euler Phi Function of Integer
 * 62) Sine of Integer Multiple of Pi
 * 63) Derivative of Cosine Function
 * 64) Real Number Line is Metric Space
 * 65) Pascal's Rule
 * 66) Translation of Index Variable of Summation
 * 67) Sine Function is Odd
 * 68) Fundamental Theorem of Calculus
 * 69) Derivative of Constant
 * 70) De Moivre's Formula
 * 71) Sine Exponential Formulation
 * 72) Logarithm of Power
 * 73) Gamma Difference Equation
 * 74) Divisor Sum of Non-Square Semiprime
 * 75) Primitive of Reciprocal of x squared plus a squared/Arctangent Form
 * 76) Cosine Function is Even
 * 77) Sine of Zero is Zero
 * 78) Cosine of Integer Multiple of Pi
 * 79) Derivative of Natural Logarithm Function
 * 80) Separation of Variables
 * 81) Sum of Squares of Sine and Cosine/Corollary 1
 * 82) Intersection is Largest Subset
 * 83) Solution to Quadratic Equation/Real Coefficients
 * 84) Union is Smallest Superset
 * 85) Product Rule for Derivatives
 * 86) Principle of Recursive Definition
 * 87) Double Angle Formulas/Sine
 * 88) Countable Union of Countable Sets is Countable
 * 89) Squeeze Theorem
 * 90) Axiom:Axiomatization of 1-Based Natural Numbers
 * 91) Integers form Integral Domain
 * 92) Divisor Sum of Square-Free Integer
 * 93) Triangle Inequality/Real Numbers
 * 94) Primitive of Function under its Derivative
 * 95) Derivative of Monotone Function
 * 96) Primitive of Exponential of a x
 * 97) Set is Subset of Itself
 * 98) One-Step Subgroup Test
 * 99) Archimedean Principle
 * 100) Triangle Side-Angle-Side Equality
 * 101) Difference of Squares of Hyperbolic Cosine and Sine
 * 102) Exponential Tends to Zero and Infinity
 * 103) Primitive of Cosine Function/Corollary
 * 104) Subgroup of Abelian Group is Normal
 * 105) Composition of Mappings is Associative
 * 106) Square of Sum
 * 107) Empty Set is Subset of All Sets
 * 108) Cotangent is Cosine divided by Sine
 * 109) Solution of Constant Coefficient Homogeneous LSOODE
 * 110) Fubini's Theorem
 * 111) Euclid's Lemma
 * 112) Sum of Angles of Triangle equals Two Right Angles
 * 113) Gamma Function Extends Factorial
 * 114) Image of Subset under Mapping is Subset of Image
 * 115) Supremum of Subset
 * 116) Complex Modulus of Product of Complex Numbers
 * 117) Continuum Property
 * 118) Real Multiplication Distributes over Addition
 * 119) Meet Precedes Operands
 * 120) Sine of Sum
 * 121) Sine of Right Angle
 * 122) Union with Empty Set
 * 123) Subring Test
 * 124) Opposite Sides and Angles of Parallelogram are Equal
 * 125) Subset Relation is Ordering
 * 126) Basis Representation Theorem
 * 127) General Solution of Linear 2nd Order ODE from Homogeneous 2nd Order ODE and Particular Solution
 * 128) Transfinite Induction/Schema 2
 * 129) Euler Phi Function of Square-Free Integer
 * 130) Construction of Parallel Line
 * 131) Cancellation Laws
 * 132) Cosine of Sum
 * 133) Set is Subset of its Topological Closure
 * 134) Derivative of Inverse Function
 * 135) Fundamental Theorem of Arithmetic
 * 136) Roots of Complex Number
 * 137) Axiom:Euclid's Common Notions
 * 138) Numbers whose Divisor Sum is Square
 * 139) Primitive of Reciprocal of a x + b
 * 140) Derivative of Function of Constant Multiple
 * 141) Set Union Preserves Subsets
 * 142) Subset of Finite Set is Finite
 * 143) Square of Difference
 * 144) Composite of Injections is Injection
 * 145) Inverse of Group Product
 * 146) Sum of Logarithms
 * 147) Sine and Cosine are Periodic on Reals
 * 148) Primitive of Sine Function/Corollary
 * 149) Quotient Rule for Derivatives
 * 150) Natural Logarithm of 1 is 0
 * 151) Intersection is Commutative
 * 152) Area of Triangle in Terms of Side and Altitude
 * 153) Composite of Continuous Mappings is Continuous
 * 154) Prime Group is Cyclic
 * 155) Power Series Expansion for Exponential Function
 * 156) Solution to Exact Differential Equation
 * 157) Cosine of Complement equals Sine
 * 158) Sequence of Powers of Number less than One
 * 159) Sequence of Powers of Reciprocals is Null Sequence
 * 160) Fundamental Theorem on Equivalence Relations
 * 161) Derivative of Sine Function/Corollary
 * 162) Fourth Sylow Theorem
 * 163) Square of Real Number is Non-Negative
 * 164) Set Difference is Subset
 * 165) Exponent Combination Laws
 * 166) Derivative of Constant Multiple
 * 167) Composite of Bijections is Bijection
 * 168) Symmetry Rule for Binomial Coefficients
 * 169) Equality of Ordered Pairs
 * 170) Fermat's Little Theorem
 * 171) Linear Combination of Laplace Transforms
 * 172) Real Number Ordering is Compatible with Multiplication
 * 173) Logarithm of Reciprocal
 * 174) Cosine of Right Angle
 * 175) Union is Commutative
 * 176) Cyclic Group is Abelian
 * 177) Congruence of Powers
 * 178) Ordering of Reciprocals
 * 179) Two Linearly Independent Solutions of Homogeneous Linear Second Order ODE generate General Solution
 * 180) Combination Theorem for Limits of Functions/Real/Product Rule
 * 181) Mean Value Theorem
 * 182) Zorn's Lemma
 * 183) Intersection with Subset is Subset
 * 184) Axiom:Euclid's First Postulate
 * 185) Rational Numbers are Countably Infinite
 * 186) Dual Pairs (Order Theory)
 * 187) Axiom:Axiom of Countable Choice
 * 188) Cardinality of Set of Subsets
 * 189) Numbers such that Divisor Count divides Phi divides Divisor Sum
 * 190) Fifth Sylow Theorem
 * 191) Transfinite Induction/Schema 2
 * 192) Transfinite Induction/Schema 2
 * 193) Transfinite Induction/Schema 2
 * 194) Squeeze Theorem/Sequences/Real Numbers
 * 195) Image of Subset under Relation is Subset of Image
 * 196) Topological Closure is Closed
 * 197) Set is Subset of Union/General Result
 * 198) Sine of Complement equals Cosine
 * 199) L'Hôpital's Rule
 * 200) Isosceles Triangle has Two Equal Angles
 * 201) Empty Intersection iff Subset of Complement
 * 202) Restriction of Mapping to Image is Surjection
 * 203) Axiom:Axiom of Extension
 * 204) Intersection of Subgroups is Subgroup
 * 205) Change of Base of Logarithm
 * 206) Basis for Partition Topology
 * 207) Complex Plane is Metric Space
 * 208) First Sylow Theorem
 * 209) Derivative of Cosine Function/Corollary
 * 210) Triangle Inequality/Complex Numbers
 * 211) Ordinal Membership is Trichotomy
 * 212) Transitive Set is Proper Subset of Ordinal iff Element of Ordinal
 * 213) Injection to Image is Bijection
 * 214) Monotone Convergence Theorem (Real Analysis)
 * 215) Between two Real Numbers exists Rational Number
 * 216) Equality of Ratios is Transitive
 * 217) Intersection with Empty Set
 * 218) Order of Element Divides Order of Finite Group
 * 219) Intersection with Complement is Empty iff Subset
 * 220) Ring Product with Zero
 * 221) Set Difference as Intersection with Complement
 * 222) Principle of Finite Induction
 * 223) Metric Induces Topology
 * 224) Modulus in Terms of Conjugate
 * 225) Set Union is Idempotent
 * 226) Solution of Linear Diophantine Equation
 * 227) Bijection iff Inverse is Bijection
 * 228) Real Addition is Commutative
 * 229) Semiperfect Number is not Deficient
 * 230) Laplace Transform of Exponential
 * 231) Double Angle Formulas/Cosine
 * 232) Sine of Complex Number
 * 233) Sine of 45 Degrees
 * 234) Cosine of 45 Degrees
 * 235) Set Intersection Preserves Subsets
 * 236) Orbit-Stabilizer Theorem
 * 237) Gamma Function of One Half
 * 238) Union with Relative Complement
 * 239) Inverse of Group Inverse
 * 240) Sylow p-Subgroup is Unique iff Normal
 * 241) Axiom:Axiom of Foundation
 * 242) Identity Mapping is Bijection
 * 243) Logarithm is Strictly Increasing
 * 244) Modus Ponendo Ponens
 * 245) Triangle Side-Side-Side Equality
 * 246) Laplace Transform of Derivative
 * 247) De Polignac's Formula
 * 248) Solution to Linear First Order Ordinary Differential Equation/Solution by Integrating Factor
 * 249) Hyperbolic Cosine in terms of Cosine
 * 250) T2 Space is T1 Space
 * 251) Linear Combination of Derivatives
 * 252) Union with Superset is Superset
 * 253) Construction of Equal Straight Lines from Unequal
 * 254) Euclid's Lemma for Prime Divisors
 * 255) Preceding and Way Below implies Way Below
 * 256) Identity is Only Group Element of Order 1
 * 257) Variance as Expectation of Square minus Square of Expectation
 * 258) Powers of Group Elements
 * 259) Intersection is Associative
 * 260) Union is Associative
 * 261) Complex Multiplication is Commutative
 * 262) Group has Latin Square Property
 * 263) Identity is Unique
 * 264) Set Intersection is Idempotent
 * 265) Group Homomorphism Preserves Identity
 * 266) Binomial Coefficient with Zero
 * 267) Inverse of Strictly Monotone Function
 * 268) Bézout's Identity
 * 269) Identity of Subgroup
 * 270) Double Angle Formulas/Hyperbolic Sine
 * 271) Completing the Square
 * 272) Derivative of Tangent Function
 * 273) Power Series is Termwise Integrable within Radius of Convergence
 * 274) Secant of Conjugate Angle
 * 275) Tangent of Conjugate Angle
 * 276) Complement of Complement
 * 277) Bisection of Straight Line
 * 278) Particular Solution to Homogeneous Linear Second Order ODE gives rise to Another
 * 279) Definite Integral of Even Function
 * 280) Set Difference with Self is Empty Set
 * 281) Join Succeeds Operands
 * 282) Cartesian Product of Countable Sets is Countable
 * 283) Natural Number Addition is Commutative
 * 284) Comparison Test
 * 285) Cardinality of Empty Set
 * 286) Surjection iff Right Inverse
 * 287) Real Multiplication is Commutative
 * 288) Axiom:Peano's Axioms
 * 289) Solutions of Pythagorean Equation
 * 290) Cartesian Product is Empty iff Factor is Empty
 * 291) Convergent Sequence in Metric Space is Cauchy Sequence
 * 292) Hyperbolic Cosecant of Complex Number
 * 293) Hyperbolic Secant of Complex Number
 * 294) Laplace Transform of Sine
 * 295) Primitive of Exponential Function
 * 296) Sum of Squares of Hyperbolic Secant and Tangent
 * 297) Difference of Two Powers
 * 298) Product with Ring Negative
 * 299) Cosecant of Complex Number
 * 300) Cotangent of Conjugate Angle
 * 301) Sine of Conjugate Angle
 * 302) Cosine of Conjugate Angle
 * 303) Secant of Complex Number
 * 304) Cosine of Complex Number
 * 305) Cosecant of Conjugate Angle
 * 306) Cauchy's Residue Theorem
 * 307) Exponent Combination Laws/Product of Powers
 * 308) Continuous Image of Compact Space is Compact
 * 309) Union is Smallest Superset/General Result
 * 310) Sum of Sequence of Squares
 * 311) Supremum of Singleton
 * 312) Combination Theorem for Sequences
 * 313) Set Difference Intersection with Second Set is Empty Set
 * 314) Intersection is Subset/General Result
 * 315) Primitive of Reciprocal of Root of a squared minus x squared/Arcsine Form
 * 316) De Morgan's Laws (Set Theory)/Relative Complement/Complement of Intersection
 * 317) Infimum of Subset
 * 318) Parallelism implies Equal Alternate Angles
 * 319) Structure Induced by Ring Operations is Ring
 * 320) Exponential of Natural Logarithm
 * 321) Sine of Half-Integer Multiple of Pi
 * 322) Euler's Identity
 * 323) Complex Roots of Unity/Examples/Cube Roots
 * 324) Newton's Laws of Motion/Second Law
 * 325) Diagonal Relation is Equivalence
 * 326) Real Number Line is Complete Metric Space
 * 327) Definition:Zermelo-Fraenkel Axioms
 * 328) Primitive of Reciprocal of x squared minus a squared/Logarithm Form
 * 329) Primitive of Hyperbolic Sine of a x
 * 330) Hyperbolic Cosine of Complex Number
 * 331) Hyperbolic Sine of Complex Number
 * 332) Laplace Transform of Cosine
 * 333) Derivative of Hyperbolic Sine Function
 * 334) Way Below implies Preceding
 * 335) Symmetric Group on 3 Letters
 * 336) Euler's Formula/Corollary
 * 337) Cosine of 30 Degrees
 * 338) De Morgan's Laws (Set Theory)/Set Difference/General Case/Difference with Union
 * 339) Prime not Divisor implies Coprime
 * 340) Common Divisor Divides Integer Combination
 * 341) Constant Function is Primitive Recursive
 * 342) Set in Discrete Topology is Clopen
 * 343) Compact Subspace of Hausdorff Space is Closed
 * 344) Binomial Theorem/General Binomial Theorem
 * 345) Expectation is Linear
 * 346) Summation is Linear
 * 347) T1 Space is T0 Space
 * 348) Integral of Constant
 * 349) Construction of Equal Angle
 * 350) Divisor Sum of Prime Number
 * 351) Two Angles on Straight Line make Two Right Angles
 * 352) Combination Theorem for Limits of Functions/Sum Rule
 * 353) Subgroup of Index 2 is Normal
 * 354) Negative of Absolute Value
 * 355) Taylor's Theorem
 * 356) Shape of Sine Function
 * 357) Complex Numbers form Field
 * 358) Set Equivalence behaves like Equivalence Relation
 * 359) Principle of Non-Contradiction
 * 360) Primitive of Hyperbolic Cosine of a x
 * 361) Test for Ideal
 * 362) Moment in terms of Moment Generating Function
 * 363) Left Cosets are Equal iff Product with Inverse in Subgroup
 * 364) Distributive Laws/Arithmetic
 * 365) Membership is Left Compatible with Ordinal Multiplication
 * 366) Derivative of Hyperbolic Cosine Function
 * 367) Exponent Combination Laws/Negative Power
 * 368) Derivative of Composite Function
 * 369) Primitive of Exponential of a x by Sine of b x
 * 370) Hyperbolic Sine in terms of Sine
 * 371) Exponential of Sum/Complex Numbers
 * 372) Limit of Subsequence equals Limit of Sequence/Real Numbers
 * 373) Preimage of Subset is Subset of Preimage
 * 374) Form of Geometric Sequence of Integers
 * 375) Sine of 30 Degrees
 * 376) T5 Space is T4 Space
 * 377) Subset of Countably Infinite Set is Countable
 * 378) Second Principle of Mathematical Induction
 * 379) Second Principle of Mathematical Induction
 * 380) Condition for Point being in Closure
 * 381) Set Intersection Preserves Subsets/Corollary
 * 382) Properties of Norm on Division Ring/Norm of Negative
 * 383) Tangent Function is Odd
 * 384) Integral of Constant/Definite
 * 385) Negated Upper Index of Binomial Coefficient
 * 386) De Morgan's Laws (Set Theory)/Set Difference/Difference with Intersection
 * 387) Absolute Value of Integer is not less than Divisors
 * 388) Axiom:Axiom of Replacement
 * 389) Sine of Angle plus Right Angle
 * 390) Integer Multiplication is Commutative
 * 391) Derivative at Maximum or Minimum
 * 392) Kernel is Normal Subgroup of Domain
 * 393) Field is Integral Domain
 * 394) Square Root of 2 is Irrational
 * 395) Shape of Cosine Function
 * 396) Solution to Homogeneous Differential Equation
 * 397) Harmonic Series is Divergent
 * 398) Duality Principle (Order Theory)
 * 399) Surjection from Natural Numbers iff Countable
 * 400) Zermelo's Well-Ordering Theorem
 * 401) Smallest Element is Unique
 * 402) Primitive of Square of Secant of a x
 * 403) Primitive of Reciprocal of x by Root of x squared minus a squared
 * 404) Primitive of Hyperbolic Tangent of a x
 * 405) Hyperbolic Cotangent of Complex Number
 * 406) Primitive of x over x squared plus a squared
 * 407) Factors of Binomial Coefficient
 * 408) Euler Phi Function of 2 times Odd Prime
 * 409) Period of Complex Exponential Function
 * 410) Preceding iff Meet equals Less Operand
 * 411) Derivative of Arcsecant Function/Corollary 1
 * 412) Derivative of Arctangent Function
 * 413) Primitive of Square of Sine of a x
 * 414) Excluded Point Topology is Open Extension Topology of Discrete Topology
 * 415) Cotangent of Complex Number
 * 416) Cosine of Angle plus Right Angle
 * 417) Derivative of Arctangent Function/Corollary
 * 418) Derivatives of Function of a x + b
 * 419) Euler Phi Function of Prime
 * 420) Integer Divisor Results/One Divides all Integers
 * 421) Cosine of Three Right Angles
 * 422) Sine of Straight Angle
 * 423) Preimage of Intersection under Mapping
 * 424) De Morgan's Laws (Set Theory)/Set Difference/General Case/Difference with Intersection
 * 425) Metric Space is Hausdorff
 * 426) Combination Theorem for Limits of Functions/Real/Multiple Rule
 * 427) Restriction of Commutative Operation is Commutative
 * 428) Composite of Surjections is Surjection
 * 429) Cardinality of Set Union
 * 430) Cardinality of Cartesian Product of Finite Sets
 * 431) Subset implies Cardinal Inequality
 * 432) Set Complement inverts Subsets
 * 433) Set is Subset of Union/Family of Sets
 * 434) Brahmagupta-Fibonacci Identity
 * 435) Exponential is Strictly Increasing
 * 436) Construction of Perpendicular Line
 * 437) Translation of Index Variable of Product
 * 438) Combination Theorem for Limits of Functions
 * 439) Sum of Squares of Sine and Cosine/Corollary 2
 * 440) Inverse of Algebraic Structure Isomorphism is Isomorphism
 * 441) Tangent Exponential Formulation
 * 442) Divisor Sum of Power of Prime
 * 443) Absolute Value Function is Completely Multiplicative
 * 444) Sequence of Implications of Separation Axioms
 * 445) Stirling's Formula
 * 446) Triangles with Two Equal Angles are Similar
 * 447) Set Difference as Intersection with Relative Complement
 * 448) Ordering on Ordinal is Subset Relation
 * 449) Trichotomy Law (Ordering)
 * 450) Empty Set is Element of Topology
 * 451) Open Sets in Real Number Line
 * 452) Set is Open iff Neighborhood of all its Points
 * 453) Rule of Commutation
 * 454) Real Numbers form Field
 * 455) Complex Multiplication is Associative
 * 456) Integer Multiplication is Associative
 * 457) Natural Number Multiplication is Commutative
 * 458) Reverse Triangle Inequality
 * 459) Meet Semilattice is Ordered Structure
 * 460) Identity Mapping is Right Identity
 * 461) Divisor Relation on Positive Integers is Partial Ordering
 * 462) Rule of Material Implication
 * 463) Complex Number equals Conjugate iff Wholly Real
 * 464) Boundary is Intersection of Closure with Closure of Complement
 * 465) Inverse of Bijection is Bijection
 * 466) Composite of Bijection with Inverse is Identity Mapping
 * 467) Roots of Complex Number/Corollary
 * 468) Primitive of Reciprocal of a x + b squared
 * 469) Primitive of Square of Hyperbolic Cosecant of a x
 * 470) Primitive of Square of Cosecant of a x
 * 471) Restriction of Associative Operation is Associative
 * 472) Hyperbolic Tangent of Complex Number
 * 473) Extended Transitivity
 * 474) Euler-Binet Formula
 * 475) Dual of Dual Statement (Order Theory)
 * 476) Product of Ring Negatives
 * 477) Union of Subsets is Subset
 * 478) Ratio Test
 * 479) Derivative of Arccotangent Function/Corollary
 * 480) Derivative of Arccosine Function/Corollary
 * 481) Derivative of Arcsine Function/Corollary
 * 482) Derivative of Arccosecant Function/Corollary
 * 483) Basis for Discrete Topology
 * 484) Symmetric Group on 3 Letters/Cayley Table
 * 485) Sine of Three Right Angles
 * 486) Tangent of Complex Number
 * 487) Secant of Angle plus Right Angle
 * 488) Cotangent Exponential Formulation
 * 489) Reciprocal Function is Strictly Decreasing
 * 490) Projection from Product Topology is Continuous
 * 491) Areas of Triangles and Parallelograms Proportional to Base
 * 492)
 * 493) Sine of 60 Degrees
 * 494) Cosine of 60 Degrees
 * 495) Cosine of Straight Angle
 * 496) Cosine of Sum/Corollary
 * 497) Combination Theorem for Sequences/Real/Sum Rule
 * 498) Integral of Power
 * 499) Second Principle of Mathematical Induction
 * 500) Infimum of Singleton
 * 501) Sum of Integrals on Adjacent Intervals for Integrable Functions
 * 502) Combination Theorem for Sequences/Real/Multiple Rule
 * 503) Determinant of Transpose
 * 504) Set Difference Union Intersection
 * 505) Pigeonhole Principle
 * 506) Relation between Two Ordinals
 * 507) Restriction of Injection is Injection
 * 508) Subsemigroup Closure Test
 * 509) Integers Divided by GCD are Coprime
 * 510) Closure of Finite Union equals Union of Closures
 * 511) Topological Closure of Subset is Subset of Topological Closure
 * 512) Closed Subspace of Compact Space is Compact
 * 513) Complement of Interior equals Closure of Complement
 * 514) Heine-Borel Theorem
 * 515) Intermediate Value Theorem
 * 516) Proof by Contraposition
 * 517) Cosine of Half-Integer Multiple of Pi
 * 518) Singleton of Element is Subset
 * 519) Triangle with Two Equal Angles is Isosceles
 * 520) First Isomorphism Theorem/Groups
 * 521) Integer Combination of Coprime Integers
 * 522) Metric Space fulfils all Separation Axioms
 * 523) Real Multiplication Identity is One
 * 524) Epimorphism Preserves Identity
 * 525) Lower and Upper Bounds for Sequences
 * 526) Union Distributes over Intersection
 * 527) Parallelism implies Equal Corresponding Angles
 * 528) Limit of Sine of X over X at Zero
 * 529) Riemann Zeta Function at Even Integers
 * 530) Two Straight Lines make Equal Opposite Angles
 * 531) Equivalence of Definitions of Normal Subgroup
 * 532) Real Multiplication is Associative
 * 533) Second Principle of Finite Induction
 * 534) Rationals are Everywhere Dense in Topological Space of Reals
 * 535) Identity Mapping is Left Identity
 * 536) Equation of Straight Line in Plane
 * 537) Axiom:Axiom of Powers
 * 538) Euclid's Theorem
 * 539) Axiom:Axiom of Pairing
 * 540) Convergent Sequence in Metric Space is Bounded
 * 541) Equivalence of Definitions of Reflexive Relation
 * 542) Continuous Real Function is Darboux Integrable
 * 543) Bijection iff Left and Right Inverse
 * 544) Primitive of Square of Hyperbolic Secant of a x
 * 545) Primitive of Function of Root of a x + b
 * 546) Primitive of Composite Function
 * 547) Primitive of Hyperbolic Cotangent of a x
 * 548) Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form
 * 549) Sine of Sum/Corollary
 * 550) Motion of Cart attached to Wall by Spring under Damping/Problem Definition
 * 551) Reciprocal of Strictly Positive Real Number is Strictly Positive
 * 552) Weierstrass Substitution
 * 553) Hyperbolic Sine Function is Odd
 * 554) GCD from Prime Decomposition
 * 555) Primitive of x by Exponential of a x
 * 556) Isomorphism Theorems
 * 557) Sylow Theorems
 * 558) Real Polynomial Function is Continuous
 * 559) Path-Connected Space is Connected
 * 560) Double Angle Formulas/Hyperbolic Cosine
 * 561) Membership is Left Compatible with Ordinal Addition
 * 562) Group Homomorphism Preserves Inverses
 * 563) Dihedral Group D4/Matrix Representation/Formulation 2/Cayley Table
 * 564) Derivative of Laplace Transform
 * 565) Derivative of Hyperbolic Tangent Function
 * 566) Series of Power over Factorial Converges
 * 567) Derivative of Arcsine Function
 * 568) Difference of Squares of Hyperbolic Cotangent and Cosecant
 * 569) Derivative of Hyperbolic Cotangent Function
 * 570) Residue at Simple Pole
 * 571) De Morgan's Laws (Predicate Logic)
 * 572) Powers of Group Elements/Sum of Indices
 * 573) Cotangent of Angle plus Right Angle
 * 574) Sum of Two Sides of Triangle Greater than Third Side
 * 575) Real Number is Integer iff equals Floor
 * 576) Geometrical Interpretation of Complex Subtraction
 * 577) Surjection if Composite is Surjection
 * 578) Product of Complex Numbers in Polar Form
 * 579) Exponent Combination Laws/Power of Product
 * 580) Geometrical Interpretation of Complex Addition
 * 581) True Statement is implied by Every Statement
 * 582) Compact Space is Countably Compact
 * 583) Integer Multiplication Distributes over Addition
 * 584) Sum Rule for Derivatives
 * 585) De Morgan's Laws (Logic)/Disjunction of Negations
 * 586) Sum of Sequence of Cubes
 * 587) Equation of Circle in Complex Plane
 * 588) Supremum of Suprema
 * 589) Derivative of Exponential Function/Corollary 1
 * 590) Determinant of Matrix Product
 * 591) Set of Integers Bounded Below by Integer has Smallest Element
 * 592) Signum Function is Primitive Recursive
 * 593) Sigma-Algebra Closed under Finite Intersection
 * 594) Principle of Structural Induction
 * 595) Finite Union of Finite Sets is Finite
 * 596) Cantor-Bernstein-Schröder Theorem
 * 597) Axiom:Axiom of Dependent Choice
 * 598) Natural Number Addition is Associative
 * 599) Degree of Product of Polynomials over Ring/Corollary 2
 * 600) Cosine of Angle plus Full Angle
 * 601) Power Reduction Formulas/Sine Squared
 * 602) Cosecant of Supplementary Angle
 * 603) Product of GCD and LCM
 * 604) Set Difference and Intersection form Partition
 * 605) Relative Sizes of Definite Integrals
 * 606) Axiom of Subsets Equivalents
 * 607) Union of Singleton
 * 608) Basel Problem
 * 609) Properties of Restriction of Relation?
 * 610) Quotient Group is Group
 * 611) Cardinality of Power Set of Finite Set
 * 612) Natural Numbers are Non-Negative Integers
 * 613) General Periodicity Property
 * 614) Rule of Idempotence
 * 615) Axiom:Euclid's Second Postulate
 * 616) Inverse in Monoid is Unique
 * 617) Subset Relation on Power Set is Partial Ordering
 * 618) Compactness Theorem
 * 619) Real Numbers form Ordered Field
 * 620) Ring of Integers Modulo Prime is Field
 * 621) Cardinality of Set of All Mappings
 * 622) Distance Formula
 * 623) Axiom:Leibniz's Law
 * 624) Rule of Idempotence/Disjunction/Formulation 2/Reverse Implication
 * 625) Rule of Commutation/Disjunction/Formulation 2/Forward Implication
 * 626) Rule of Addition/Sequent Form/Formulation 2/Form 2
 * 627) Equivalence of Definitions of Compact Topological Space
 * 628) Equation of Circle
 * 629) Cauchy's Mean Theorem
 * 630) Modulo Addition is Well-Defined
 * 631) Complex Addition is Associative
 * 632) Real Numbers form Ordered Integral Domain
 * 633) Primitive of Reciprocal of a x squared plus b x plus c
 * 634) Primitive of Function of a x + b
 * 635) Primitive of Power of Secant of a x
 * 636) Divisor Sum Function is Multiplicative
 * 637) Intersection with Set Difference is Set Difference with Intersection
 * 638) Laplace Transform of Exponential times Function
 * 639) Natural Number Multiplication Distributes over Addition
 * 640) Identity Element is Idempotent
 * 641) LCM from Prime Decomposition
 * 642) Primitive of x by Sine of a x
 * 643) Primitive of x squared by Cosine of a x
 * 644) Group Action on Sets with k Elements
 * 645) First Supplement to Law of Quadratic Reciprocity
 * 646) Exponential of Real Number is Strictly Positive
 * 647) Inverse of Order Isomorphism is Order Isomorphism
 * 648) Unsigned Stirling Number of the First Kind of 0
 * 649) Integer Divisor Results/Integer Divides Itself
 * 650) Polynomial Factor Theorem
 * 651) Hyperbolic Sine of Sum
 * 652) Hyperbolic Cosine of Sum
 * 653) Open Ball is Neighborhood of all Points Inside
 * 654) False Statement implies Every Statement
 * 655) Square Modulo 4
 * 656) Difference between Two Squares equal to Repunit
 * 657) Derivative of Real Area Hyperbolic Cotangent of x over a
 * 658) Derivative of Real Area Hyperbolic Tangent of x over a
 * 659) Derivative of Real Area Hyperbolic Sine of x over a
 * 660) Derivative of Hyperbolic Secant Function
 * 661) Fundamental Theorem of Calculus/First Part/Corollary
 * 662) Estimation Lemma for Contour Integrals
 * 663) Primitive of Exponential of a x by Cosine of b x
 * 664) Primitive of Square of Cosine of a x
 * 665) Tangent of Sum
 * 666) Square of Golden Mean equals One plus Golden Mean
 * 667) Trigonometric Functions in terms of each other
 * 668) Cosecant Exponential Formulation
 * 669) Tangent of Angle plus Right Angle
 * 670) Sine of 15 Degrees
 * 671) Contour Integral of Concatenation of Contours
 * 672) Strictly Monotone Real Function is Bijective
 * 673) Epimorphism Preserves Commutativity
 * 674) Codomain of Composite Relation
 * 675) Domain of Composite Relation
 * 676) Continuity Defined from Closed Sets
 * 677) Intersection Distributes over Union/General Result
 * 678) Finite Topological Space is Compact
 * 679) Rational Numbers form Field
 * 680) Substitutivity of Equality
 * 681) Interior of Closed Real Interval is Open Real Interval
 * 682) Solution of Linear Congruence
 * 683) Union is Smallest Superset/Family of Sets
 * 684) Properties of Ordered Ring
 * 685) Properties of Norm on Division Ring/Norm of Inverse
 * 686) Product of Complex Conjugates
 * 687) Honsberger's Identity
 * 688) Cardinality of Subset of Finite Set
 * 689) Identity Mapping is Injection
 * 690) Half Angle Formulas/Cosine
 * 691) De Morgan's Laws (Set Theory)/Relative Complement/Complement of Union
 * 692) Inverse of Identity Element is Itself
 * 693) Union of Finite Sets is Finite
 * 694) Zero Choose n
 * 695) Strictly Monotone Mapping with Totally Ordered Domain is Injective
 * 696) Inverse of Many-to-One Relation is One-to-Many
 * 697) Primitive of x by Cosine of a x
 * 698) Sine of Angle plus Full Angle
 * 699) Identity Mapping is Surjection
 * 700) Hyperbolic Tangent in terms of Tangent
 * 701) Cosine of Supplementary Angle
 * 702) Magnitudes with Same Ratios are Equal
 * 703) Power Series Expansion for Logarithm of 1 + x
 * 704) Infimum of Upper Closure of Element
 * 705) Relative Complement with Self is Empty Set
 * 706) Addition is Primitive Recursive
 * 707) Double Angle Formulas/Cosine/Corollary 1
 * 708) Binomial Theorem/Integral Index
 * 709) Equality is Transitive
 * 710) Axiom:Axiom of Specification
 * 711) Particular Values of Stirling Numbers of the Second Kind
 * 712) Particular Values of Unsigned Stirling Numbers of the First Kind
 * 713) Elementary Properties of Probability Measure
 * 714) Number of Permutations
 * 715) Open Ball of Metric Space is Open Set
 * 716) Theorem of Even Perfect Numbers
 * 717) Measure is Finitely Additive Function
 * 718) Injection iff Left Inverse
 * 719) Internal Direct Product Theorem
 * 720) Axiom:Axiom of Infinity
 * 721) Factorial/Examples
 * 722) Dihedral Group/Group Presentation
 * 723) Factor Principles/Disjunction on Left/Formulation 2
 * 724) Russell's Paradox
 * 725) Real Numbers are Uncountable
 * 726) Real Numbers under Addition form Infinite Abelian Group
 * 727) Group equals Center iff Abelian
 * 728) Acceleration Due to Gravity
 * 729) Unsigned Stirling Number of the First Kind of 1
 * 730) Set is Subset of Union/Set of Sets
 * 731) Method of Variation of Parameters
 * 732) Primitive of Hyperbolic Cosecant of a x
 * 733) Primitive of Cotangent of a x
 * 734) Primitive of Function of Arcsine
 * 735) Primitive of x over Root of a squared minus x squared
 * 736) Modus Tollendo Tollens
 * 737) Laplace Transform of Positive Integer Power
 * 738) Left Coset Space forms Partition
 * 739) Real Number Ordering is Compatible with Addition
 * 740) Zero Wronskian of Solutions of Homogeneous Linear Second Order ODE iff Linearly Dependent
 * 741) Unsigned Stirling Number of the First Kind of Number with Self
 * 742) Half Angle Formulas/Sine
 * 743) Matrix Multiplication is Associative
 * 744) Ring of Integers is Principal Ideal Domain
 * 745) Primitive of Reciprocal of x by a x + b
 * 746) Primitive of x over x squared minus a squared
 * 747) Order of Real Numbers is Dual of Order of their Negatives
 * 748) Euler Phi Function of Non-Square Semiprime
 * 749) Division Theorem for Polynomial Forms over Field
 * 750) Triple Angle Formulas/Sine
 * 751) Set Difference with Union is Set Difference
 * 752) Epimorphism Preserves Semigroups
 * 753) Index Laws for Monoids/Sum of Indices
 * 754) Double Angle Formulas/Cosine/Corollary 2
 * 755) Powers of Group Elements/Product of Indices
 * 756) Differentiable Function is Continuous
 * 757) Derivative of Real Area Hyperbolic Cosine of x over a
 * 758) Derivative of Hyperbolic Cosecant Function
 * 759) Zero Derivative implies Constant Function
 * 760) Derivative of Secant Function
 * 761) De Morgan's Laws (Predicate Logic)/Denial of Existence
 * 762) Linear Combination of Integrals/Definite
 * 763) Reversal of Limits of Definite Integral
 * 764) Exponent Combination Laws/Power of Power
 * 765) Cosine of 15 Degrees
 * 766) Hyperbolic Cosine Function is Even
 * 767) Sine of Angle plus Straight Angle
 * 768) Cosine in terms of Hyperbolic Cosine
 * 769) Sine in terms of Hyperbolic Sine
 * 770) Secant Exponential Formulation
 * 771) Upper Closure is Upper Section
 * 772) Construction of Square on Given Straight Line
 * 773) Euler Phi Function of Prime Power/Corollary
 * 774) Power of Conjugate equals Conjugate of Power
 * 775) Equality of Ratios Ex Aequali
 * 776) Modulus of Limit
 * 777) Limit of Function by Convergent Sequences
 * 778) Combination Theorem for Limits of Functions/Quotient Rule
 * 779) De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection
 * 780) Lower Closure is Lower Section
 * 781) Indexed Union Subset
 * 782) Intersection with Relative Complement is Empty
 * 783) No Membership Loops
 * 784) Subgroup of Abelian Group is Abelian
 * 785) Finding Center of Circle
 * 786) Sequence of Powers of Reciprocals is Null Sequence/Corollary
 * 787) Properties of Norm on Division Ring/Norm of Unity
 * 788) Negative of Absolute Value/Corollary 1
 * 789) Bolzano-Weierstrass Theorem
 * 790) Composite of Homomorphisms is Homomorphism/Algebraic Structure
 * 791) Fibonacci Number with Negative Index
 * 792) Cassini's Identity
 * 793) Image of Union under Mapping
 * 794) Set Difference with Superset is Empty Set
 * 795) Injection if Composite is Injection
 * 796) Topology Defined by Closed Sets
 * 797) Closed Set in Topological Subspace
 * 798) Subset of Domain is Subset of Preimage of Image
 * 799) Dot Product Operator is Commutative
 * 800) Inclusion-Exclusion Principle
 * 801) Binomial Coefficient with One
 * 802) Tangent of Zero
 * 803) Parallel Transversal Theorem
 * 804) Modulus of Complex Integral
 * 805) Uniform Limit of Analytic Functions is Analytic
 * 806) First Isomorphism Theorem/Rings
 * 807) Integer Addition is Commutative
 * 808) Mediant is Between
 * 809) Expectation of Function of Discrete Random Variable
 * 810) Equal Alternate Angles implies Parallel Lines
 * 811) Equivalence of Definitions of Symmetric Relation
 * 812) Hypothetical Syllogism
 * 813) Equiangular Triangles are Similar
 * 814) Axiom:Axiom of Unions
 * 815) Maximal Ideal iff Quotient Ring is Field
 * 816) P-adic Norm on Rational Numbers is Non-Archimedean Norm
 * 817) Basic Results about Modules
 * 818) Image is Subset of Codomain
 * 819) Equivalence of Definitions of Isometry of Metric Spaces
 * 820) Shape of Tangent Function
 * 821) Equivalence of Definitions of T1 Space
 * 822) Greatest Element is Unique
 * 823) Rational Numbers form Metric Space
 * 824) Dihedral Group D4/Group Presentation
 * 825) Rule of Implication
 * 826) Cotangent of Complement equals Tangent
 * 827) Cotangent is Reciprocal of Tangent
 * 828) Shape of Cotangent Function
 * 829) Reparameterization of Directed Smooth Curve Maps Endpoints To Endpoints
 * 830) Equivalence of Definitions of Connected Topological Space
 * 831) Area of Circle
 * 832) Principle of Finite Choice
 * 833) Korselt's Theorem
 * 834) Cancellable Elements of Semigroup form Subsemigroup
 * 835) Bézout's Identity
 * 836) Equivalence of Definitions of Analytic Basis
 * 837) General Associativity Theorem
 * 838) Cosine Formula for Dot Product
 * 839) Integers under Addition form Abelian Group
 * 840) Identity Element of Addition on Numbers
 * 841) Rule of Association
 * 842) Square of Small-Digit Palindromic Number is Palindromic
 * 843) Stirling Number of the Second Kind of 1
 * 844) Solution of Second Order Differential Equation with Missing Independent Variable
 * 845) Solution of Second Order Differential Equation with Missing Dependent Variable
 * 846) Schanuel's Conjecture
 * 847) Primitive of x over Root of x squared minus a squared
 * 848) Primitive of Square of Hyperbolic Cotangent of a x
 * 849) Primitive of Square of Hyperbolic Tangent of a x
 * 850) Reduction Formula for Primitive of Power of x by Power of a x + b/Increment of Power of x
 * 851) Primitive of Power of a x + b over Power of p x + q/Formulation 3
 * 852) Primitive of x over a squared minus x squared
 * 853) Primitive of x over Root of x squared plus a squared
 * 854) Primitive of Power of Hyperbolic Secant of a x
 * 855) Primitive of Power of Hyperbolic Cosecant of a x
 * 856) Primitive of Power of Cosecant of a x
 * 857) Primitive of Tangent Function/Cosine Form
 * 858) Power Reduction Formulas/Hyperbolic Cosine Squared
 * 859) Orthogonal Trajectories of One-Parameter Family of Curves
 * 860) Harmonic Mean of Divisors in terms of Divisor Count and Divisor Sum
 * 861) Closure of Subset of Metric Space by Convergent Sequence
 * 862) Singleton is Directed and Filtered Subset
 * 863) Magic Square/Examples/Order 3
 * 864) Supremum of Lower Closure of Set
 * 865) Hyperbolic Tangent Function is Odd
 * 866) Principle of Mathematical Induction/Naturally Ordered Semigroup
 * 867) Homomorphism with Identity Preserves Inverses
 * 868) Power of Product of Commutative Elements in Group
 * 869) Reflexive Reduction of Relation Compatible with Group Operation is Compatible
 * 870) Inverse in Group is Unique
 * 871) Limit Ordinal Equals its Union
 * 872) Inverse of Strictly Increasing Strictly Concave Real Function is Strictly Convex
 * 873) Indexed Union Equality
 * 874) Expectation of Discrete Random Variable from PGF
 * 875) Set Difference with Empty Set is Self
 * 876) Equivalence of Definitions of Well-Ordering/Definition 1 implies Definition 2
 * 877) Definite Integral on Zero Interval
 * 878) Subset Product within Semigroup is Associative/Corollary
 * 879) Rule of Conjunction
 * 880) Triple Angle Formulas/Cosine
 * 881) Empty Set is Element of Power Set
 * 882) Duality Principle (Order Theory)/Global Duality
 * 883) Power Reduction Formulas/Cosine Squared
 * 884) Praeclarum Theorema
 * 885) Inverse of Inverse of Bijection
 * 886) Derivative of Real Area Hyperbolic Cosecant of x over a
 * 887) Faà di Bruno's Formula
 * 888) Euler's Reflection Formula
 * 889) Definite Integral of Partial Derivative
 * 890) Set Difference is Right Distributive over Union
 * 891) Square Matrix with Duplicate Rows has Zero Determinant
 * 892) Hyperbolic Cotangent in terms of Cotangent
 * 893) Cosecant Function is Odd
 * 894) Cosine of i
 * 895) Sine of i
 * 896) Real Plus Epsilon
 * 897) Derivative of Complex Composite Function
 * 898) Intersection is Largest Subset/General Result
 * 899) Preimage of Union under Mapping/General Result
 * 900) Complements of Parallelograms are Equal
 * 901) Subset Product within Semigroup is Associative
 * 902) Divisor Relation is Transitive
 * 903) Real Function is Strictly Convex iff Derivative is Strictly Increasing
 * 904) Complex Roots of Unity occur in Conjugate Pairs
 * 905) Complex Multiplication Distributes over Addition
 * 906) Equation of Straight Line in Plane/General Equation
 * 907) Singleton is Finite
 * 908) Ordinal is Less than Successor
 * 909) Union of Ordinals is Least Upper Bound
 * 910) Set Difference with Set Difference
 * 911) Closed Set in Topological Subspace/Corollary
 * 912) Telescoping Series/Example 1
 * 913) Union of Open Sets of Metric Space is Open
 * 914) Universal Generalisation
 * 915) Radius at Right Angle to Tangent
 * 916) Vector Space has Basis Between Linearly Independent Set and Finite Spanning Set
 * 917) Group Homomorphism Preserves Subgroups
 * 918) Real Number is Integer iff equals Ceiling
 * 919) Cauchy-Goursat Theorem
 * 920) Combination Theorem for Sequences/Real/Combined Sum Rule
 * 921) Element in Left Coset iff Product with Inverse in Subgroup
 * 922) De Morgan's Laws (Logic)/Conjunction of Negations
 * 923) Product Rule for Counting
 * 924) Equivalence of Mappings between Finite Sets of Same Cardinality
 * 925) Surjection iff Cardinal Inequality
 * 926) Invertible Element of Monoid is Cancellable
 * 927) Cancellability of Congruences
 * 928) Ordinal is not Element of Itself
 * 929) Law of Cosines
 * 930) Bounds of Natural Logarithm
 * 931) Characterization of Measurable Functions
 * 932) Tail of Convergent Series tends to Zero
 * 933) Measure is Monotone
 * 934) Power Series is Differentiable on Interval of Convergence
 * 935) Viète's Formulas
 * 936) Way Below iff Second Operand Preceding Supremum of Ideal implies First Operand is Element of Ideal
 * 937) Identity Mapping is Automorphism
 * 938) Euler Phi Function of Prime Power
 * 939) Division of Complex Numbers in Polar Form
 * 940) De Morgan's Laws (Set Theory)/Relative Complement/General Case/Complement of Union
 * 941) Thales' Theorem
 * 942) Angles on Equal Arcs are Equal
 * 943) Supremum of Lower Closure of Element
 * 944) Directed iff Finite Subsets have Upper Bounds
 * 945) Intersection of Closed Sets is Closed in Topological Space
 * 946) Difference between Adjacent Convergents of Simple Continued Fraction
 * 947) Terms in Convergent Series Converge to Zero
 * 948) Element to Power of Multiple of Order is Identity
 * 949) Abel's Lemma/Formulation 2
 * 950) Integer Addition is Associative
 * 951) Complex Addition Identity is Zero
 * 952) Vector Scaled by Zero is Zero Vector
 * 953) Ring is Ideal of Itself
 * 954) Successor Set of Ordinal is Ordinal
 * 955) Particular Values of Signed Stirling Numbers of the First Kind
 * 956) Odd Number Theorem
 * 957) Finite Connected Graph is Tree iff Size is One Less than Order
 * 958) Union of Empty Set
 * 959) Complex Roots of Unity in Exponential Form
 * 960) Regular Representations in Group are Permutations
 * 961) Equivalence of Definitions of Prime Number
 * 962) Equality of Ordered Tuples
 * 963) Equivalence of Definitions of Normal Subset
 * 964) Characterization of Measures
 * 965) Equivalence of Definitions of Locally Connected Space
 * 966) Prime Number Theorem
 * 967) Equivalence of Definitions of Irreducible Space
 * 968) Shape of Secant Function
 * 969) Existence of Interval of Convergence of Power Series
 * 970) Equivalence of Definitions of Interior (Topology)
 * 971) Equivalence of Definitions of Injection
 * 972) Riemann Hypothesis
 * 973) Newton's Law of Universal Gravitation
 * 974) Equivalence of Definitions of Complex Exponential Function
 * 975) Elementary Properties of Event Space
 * 976) Equivalence of Definitions of Euler's Number
 * 977) Correspondence between Linear Group Actions and Linear Representations
 * 978) Relation Partitions Set iff Equivalence
 * 979) Supremum of Empty Set is Smallest Element
 * 980) Duality Principle (Category Theory)
 * 981) Size of Linearly Independent Subset is at Most Size of Finite Generator
 * 982) Cyclic Groups of Same Order are Isomorphic
 * 983) Cosine Function is Absolutely Convergent
 * 984) Matrix Form of Quaternion
 * 985) Equivalence of Definitions of Congruence
 * 986) Triangle Angle-Side-Angle Equality
 * 987) Sum of Complex Number with Conjugate
 * 988) Euclidean Space is Complete Metric Space
 * 989) Axiom:Euclid's Third Postulate
 * 990) Equivalence of Definitions of Closed Set
 * 991) Open and Closed Sets in Topological Space
 * 992) Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE
 * 993) Transfinite Recursion Theorem
 * 994) Metric Induced by Norm is Metric
 * 995) Solution to Bernoulli's Equation
 * 996) Bases of Vector Space have Equal Cardinality
 * 997) Sum of Arithmetic Sequence
 * 998) Full Angle measures 2 Pi Radians
 * 999) Transfinite Recursion Theorem/Theorem 2