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Table of Contents Summary
PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      1.1  Pre-logic
      1.2  Propositional calculus
      1.3  Other axiomatizations related to classical propositional calculus
      1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
      1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
      1.6  Uniqueness and unique existence
      1.7  Other axiomatizations related to classical predicate calculus
PART 2  ZF (ZERMELO-FRAENKEL) SET THEORY
      2.1  ZF Set Theory - start with the Axiom of Extensionality
      2.2  ZF Set Theory - add the Axiom of Replacement
      2.3  ZF Set Theory - add the Axiom of Power Sets
      2.4  ZF Set Theory - add the Axiom of Union
      2.5  ZF Set Theory - add the Axiom of Regularity
      2.6  ZF Set Theory - add the Axiom of Infinity
PART 3  ZFC (ZERMELO-FRAENKEL WITH CHOICE) SET THEORY
      3.1  ZFC Set Theory - add Countable Choice and Dependent Choice
      3.2  ZFC Set Theory - add the Axiom of Choice
      3.3  ZFC Axioms with no distinct variable requirements
      3.4  The Generalized Continuum Hypothesis
PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      4.1  Inaccessibles
      4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
PART 5  REAL AND COMPLEX NUMBERS
      5.1  Construction and axiomatization of real and complex numbers
      5.2  Derive the basic properties from the field axioms
      5.3  Real and complex numbers - basic operations
      5.4  Integer sets
      5.5  Order sets
      5.6  Elementary integer functions
      5.7  Words over a set
      5.8  Reflexive and transitive closures of relations
      5.9  Elementary real and complex functions
      5.10  Elementary limits and convergence
      5.11  Elementary trigonometry
      5.12  Cardinality of real and complex number subsets
PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
      6.2  Elementary prime number theory
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
      7.2  Moore spaces
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
      8.2  Arrows (disjointified hom-sets)
      8.3  Examples of categories
      8.4  Categorical constructions
PART 9  BASIC ORDER THEORY
      9.1  Preordered sets and directed sets using extensible structures
      9.2  Posets and lattices using extensible structures
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
      10.2  Groups
      10.3  Rings
      10.4  Division rings and fields
      10.5  Left modules
      10.6  Vector spaces
      10.7  Ideals
      10.8  Associative algebras
      10.9  Abstract multivariate polynomials
      10.10  The complex numbers as an algebraic extensible structure
      10.11  Generalized pre-Hilbert and Hilbert spaces
PART 11  BASIC LINEAR ALGEBRA
      11.1  Vectors and free modules
      11.2  Matrices
      11.3  The determinant
      11.4  Polynomial matrices
      11.5  The characteristic polynomial
PART 12  BASIC TOPOLOGY
      12.1  Topology
      12.2  Filters and filter bases
      12.3  Uniform Structures and Spaces
      12.4  Metric spaces
      12.5  Metric subcomplex vector spaces
PART 13  BASIC REAL AND COMPLEX ANALYSIS
      13.1  Continuity
      13.2  Integrals
      13.3  Derivatives
PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      14.1  Polynomials
      14.2  Sequences and series
      14.3  Basic trigonometry
      14.4  Basic number theory
PART 15  ELEMENTARY GEOMETRY
      15.1  Definition and Tarski's Axioms of Geometry
      15.2  Tarskian Geometry
      15.3  Properties of geometries
      15.4  Geometry in Hilbert spaces
PART 16  GRAPH THEORY
      16.1  Vertices and edges
      16.2  Undirected graphs
      16.3  Walks, paths and cycles
      16.4  Eulerian paths and the Konigsberg Bridge problem
      16.5  The Friendship Theorem
PART 17  GUIDES AND MISCELLANEA
      17.1  Guides (conventions, explanations, and examples)
      17.2  Humor
      17.3  (Future - to be reviewed and classified)
PART 18  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      18.1  Additional material on group theory (deprecated)
      18.2  Complex vector spaces
      18.3  Normed complex vector spaces
      18.4  Operators on complex vector spaces
      18.5  Inner product (pre-Hilbert) spaces
      18.6  Complex Banach spaces
      18.7  Complex Hilbert spaces
PART 19  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      19.1  Axiomatization of complex pre-Hilbert spaces
      19.2  Inner product and norms
      19.3  Cauchy sequences and completeness axiom
      19.4  Subspaces and projections
      19.5  Properties of Hilbert subspaces
      19.6  Operators on Hilbert spaces
      19.7  States on a Hilbert lattice and Godowski's equation
      19.8  Cover relation, atoms, exchange axiom, and modular symmetry
PART 20  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      20.1  Mathboxes for user contributions
      20.2  Mathbox for Stefan Allan
      20.3  Mathbox for Thierry Arnoux
      20.4  Mathbox for Jonathan Ben-Naim
      20.5  Mathbox for BTernaryTau
      20.6  Mathbox for Mario Carneiro
      20.7  Mathbox for Filip Cernatescu
      20.8  Mathbox for Paul Chapman
      20.9  Mathbox for Scott Fenton
      20.10  Mathbox for Jeff Hankins
      20.11  Mathbox for Anthony Hart
      20.12  Mathbox for Chen-Pang He
      20.13  Mathbox for Jeff Hoffman
      20.14  Mathbox for Asger C. Ipsen
      20.15  Mathbox for BJ
      20.16  Mathbox for Jim Kingdon
      20.17  Mathbox for ML
      20.18  Mathbox for Wolf Lammen
      20.19  Mathbox for Brendan Leahy
      20.20  Mathbox for Jeff Madsen
      20.21  Mathbox for Giovanni Mascellani
      20.22  Mathbox for Peter Mazsa
      20.23  Mathbox for Rodolfo Medina
      20.24  Mathbox for Norm Megill
      20.25  Mathbox for Steven Nguyen
      20.26  Mathbox for Igor Ieskov
      20.27  Mathbox for OpenAI
      20.28  Mathbox for Stefan O'Rear
      20.29  Mathbox for Jon Pennant
      20.30  Mathbox for Richard Penner
      20.31  Mathbox for Stanislas Polu
      20.32  Mathbox for Rohan Ridenour
      20.33  Mathbox for Steve Rodriguez
      20.34  Mathbox for Andrew Salmon
      20.35  Mathbox for Alan Sare
      20.36  Mathbox for Glauco Siliprandi
      20.37  Mathbox for Saveliy Skresanov
      20.38  Mathbox for Jarvin Udandy
      20.39  Mathbox for Adhemar
      20.40  Mathbox for Alexander van der Vekens
      20.41  Mathbox for Emmett Weisz
      20.42  Mathbox for David A. Wheeler
      20.43  Mathbox for Kunhao Zheng

Detailed Table of Contents
(* means the section header has a description)
*PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      *1.1  Pre-logic
            *1.1.1  Inferences for assisting proof development   idi 1
      *1.2  Propositional calculus
            1.2.1  Recursively define primitive wffs for propositional calculus   wn 3
            *1.2.2  The axioms of propositional calculus   ax-mp 5
            *1.2.3  Logical implication   mp2 9
            *1.2.4  Logical negation   con4 113
            *1.2.5  Logical equivalence   wb 207
            *1.2.6  Logical conjunction   wa 396
            *1.2.7  Logical disjunction   wo 841
            *1.2.8  Mixed connectives   jaao 948
            *1.2.9  The conditional operator for propositions   wif 1054
            *1.2.10  The weak deduction theorem for propositional calculus   elimh 1072
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1078
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1475
            1.2.13  Logical "xor"   wxo 1495
            1.2.14  Logical "nor"   wnor 1512
            1.2.15  True and false constants   wal 1526
                  *1.2.15.1  Universal quantifier for use by df-tru   wal 1526
                  *1.2.15.2  Equality predicate for use by df-tru   cv 1527
                  1.2.15.3  The true constant   wtru 1529
                  1.2.15.4  The false constant   wfal 1540
            *1.2.16  Truth tables   truimtru 1551
                  1.2.16.1  Implication   truimtru 1551
                  1.2.16.2  Negation   nottru 1555
                  1.2.16.3  Equivalence   trubitru 1557
                  1.2.16.4  Conjunction   truantru 1561
                  1.2.16.5  Disjunction   truortru 1565
                  1.2.16.6  Alternative denial   trunantru 1569
                  1.2.16.7  Exclusive disjunction   truxortru 1573
                  1.2.16.8  Joint denial   trunortru 1577
            *1.2.17  Half adder and full adder in propositional calculus   whad 1584
                  1.2.17.1  Full adder: sum   whad 1584
                  1.2.17.2  Full adder: carry   wcad 1598
      1.3  Other axiomatizations related to classical propositional calculus
            *1.3.1  Minimal implicational calculus   minimp 1613
            *1.3.2  Implicational Calculus   impsingle 1619
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1633
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1650
            *1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1661
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1667
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1686
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1690
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1705
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1728
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1741
            *1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1760
      *1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
            *1.4.1  Universal quantifier (continued); define "exists" and "not free"   wex 1771
                  1.4.1.1  Existential quantifier   wex 1771
                  1.4.1.2  Non-freeness predicate   wnf 1775
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1787
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1801
                  *1.4.3.1  The empty domain of discourse   empty 1898
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1902
            *1.4.5  Equality predicate (continued)   weq 1955
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 1961
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2006
            1.4.8  Define proper substitution   sbjust 2059
            1.4.9  Membership predicate   wcel 2105
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2107
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2115
            *1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2123
      *1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2136
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2151
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2167
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2383
            *1.5.5  Alternate definition of substitution   sbimiALT 2573
      1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2616
            1.6.2  Unique existence: the unique existential quantifier   weu 2649
      1.7  Other axiomatizations related to classical predicate calculus
            *1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2746
            *1.7.2  Intuitionistic logic   axia1 2776
*PART 2  ZF (ZERMELO-FRAENKEL) SET THEORY
      2.1  ZF Set Theory - start with the Axiom of Extensionality
            2.1.1  Introduce the Axiom of Extensionality   ax-ext 2793
            2.1.2  Classes   cab 2799
                  2.1.2.1  Class abstractions   cab 2799
                  *2.1.2.2  Class equality   df-cleq 2814
                  2.1.2.3  Class membership   df-clel 2893
                  2.1.2.4  Elementary properties of class abstractions   abeq2 2945
            2.1.3  Class form not-free predicate   wnfc 2961
            2.1.4  Negated equality and membership   wne 3016
                  2.1.4.1  Negated equality   wne 3016
                  2.1.4.2  Negated membership   wnel 3123
            2.1.5  Restricted quantification   wral 3138
            2.1.6  The universal class   cvv 3495
            *2.1.7  Conditional equality (experimental)   wcdeq 3753
            2.1.8  Russell's Paradox   rru 3769
            2.1.9  Proper substitution of classes for sets   wsbc 3771
            2.1.10  Proper substitution of classes for sets into classes   csb 3882
            2.1.11  Define basic set operations and relations   cdif 3932
            2.1.12  Subclasses and subsets   df-ss 3951
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4090
                  2.1.13.1  The difference of two classes   dfdif3 4090
                  2.1.13.2  The union of two classes   elun 4124
                  2.1.13.3  The intersection of two classes   elin 4168
                  2.1.13.4  The symmetric difference of two classes   csymdif 4217
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4230
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unab 4269
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuss2 4282
            2.1.14  The empty set   c0 4290
            *2.1.15  The conditional operator for classes   cif 4465
            *2.1.16  The weak deduction theorem for set theory   dedth 4521
            2.1.17  Power classes   cpw 4537
            2.1.18  Unordered and ordered pairs   snjust 4558
            2.1.19  The union of a class   cuni 4832
            2.1.20  The intersection of a class   cint 4869
            2.1.21  Indexed union and intersection   ciun 4912
            2.1.22  Disjointness   wdisj 5023
            2.1.23  Binary relations   wbr 5058
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5120
            2.1.25  Functions in maps-to notation   cmpt 5138
            2.1.26  Transitive classes   wtr 5164
      2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5182
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5193
            2.2.3  Derive the Null Set Axiom   axnulALT 5200
            2.2.4  Theorems requiring subset and intersection existence   nalset 5209
            2.2.5  Theorems requiring empty set existence   class2set 5246
      2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5258
            2.3.2  Derive the Axiom of Pairing   axprlem1 5315
            2.3.3  Ordered pair theorem   opnz 5357
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5404
            2.3.5  Power class of union and intersection   pwin 5446
            2.3.6  The identity relation   cid 5453
            2.3.7  The membership relation (or epsilon relation)   cep 5458
            *2.3.8  Partial and total orderings   wpo 5466
            2.3.9  Founded and well-ordering relations   wfr 5505
            2.3.10  Relations   cxp 5547
            2.3.11  The Predecessor Class   cpred 6141
            2.3.12  Well-founded induction   tz6.26 6173
            2.3.13  Ordinals   word 6184
            2.3.14  Definite description binder (inverted iota)   cio 6306
            2.3.15  Functions   wfun 6343
            2.3.16  Cantor's Theorem   canth 7100
            2.3.17  Restricted iota (description binder)   crio 7102
            2.3.18  Operations   co 7145
                  2.3.18.1  Variable-to-class conversion for operations   caovclg 7329
            2.3.19  Maps-to notation   mpondm0 7375
            2.3.20  Function operation   cof 7396
            2.3.21  Proper subset relation   crpss 7437
      2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7450
            2.4.2  Ordinals (continued)   epweon 7485
            2.4.3  Transfinite induction   tfi 7556
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7568
            2.4.5  Peano's postulates   peano1 7589
            2.4.6  Finite induction (for finite ordinals)   find 7595
            2.4.7  Relations and functions (cont.)   dmexg 7601
            2.4.8  First and second members of an ordered pair   c1st 7678
            *2.4.9  The support of functions   csupp 7821
            *2.4.10  Special maps-to operations   opeliunxp2f 7867
            2.4.11  Function transposition   ctpos 7882
            2.4.12  Curry and uncurry   ccur 7922
            2.4.13  Undefined values   cund 7929
            2.4.14  Well-founded recursion   cwrecs 7937
            2.4.15  Functions on ordinals; strictly monotone ordinal functions   iunon 7967
            2.4.16  "Strong" transfinite recursion   crecs 7998
            2.4.17  Recursive definition generator   crdg 8036
            2.4.18  Finite recursion   frfnom 8061
            2.4.19  Ordinal arithmetic   c1o 8086
            2.4.20  Natural number arithmetic   nna0 8220
            2.4.21  Equivalence relations and classes   wer 8276
            2.4.22  The mapping operation   cmap 8396
            2.4.23  Infinite Cartesian products   cixp 8450
            2.4.24  Equinumerosity   cen 8495
            2.4.25  Schroeder-Bernstein Theorem   sbthlem1 8616
            2.4.26  Equinumerosity (cont.)   xpf1o 8668
            2.4.27  Pigeonhole Principle   phplem1 8685
            2.4.28  Finite sets   onomeneq 8697
            2.4.29  Finitely supported functions   cfsupp 8822
            2.4.30  Finite intersections   cfi 8863
            2.4.31  Hall's marriage theorem   marypha1lem 8886
            2.4.32  Supremum and infimum   csup 8893
            2.4.33  Ordinal isomorphism, Hartogs's theorem   coi 8962
            2.4.34  Hartogs function, order types, weak dominance   char 9009
      2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9045
            2.5.2  Axiom of Infinity equivalents   inf0 9073
      2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9090
            2.6.2  Existence of omega (the set of natural numbers)   omex 9095
            2.6.3  Cantor normal form   ccnf 9113
            2.6.4  Transitive closure   trcl 9159
            2.6.5  Rank   cr1 9180
            2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 9303
            2.6.7  Disjoint union   cdju 9316
            2.6.8  Cardinal numbers   ccrd 9353
            2.6.9  Axiom of Choice equivalents   wac 9530
            *2.6.10  Cardinal number arithmetic   undjudom 9582
            2.6.11  The Ackermann bijection   ackbij2lem1 9630
            2.6.12  Cofinality (without Axiom of Choice)   cflem 9657
            2.6.13  Eight inequivalent definitions of finite set   sornom 9688
            2.6.14  Hereditarily size-limited sets without Choice   itunifval 9827
*PART 3  ZFC (ZERMELO-FRAENKEL WITH CHOICE) SET THEORY
      3.1  ZFC Set Theory - add Countable Choice and Dependent Choice
            3.1.1  Introduce the Axiom of Countable Choice   ax-cc 9846
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 9857
      3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 9870
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 9905
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 9957
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 9985
            3.2.5  Cofinality using the Axiom of Choice   alephreg 9993
      3.3  ZFC Axioms with no distinct variable requirements
      3.4  The Generalized Continuum Hypothesis
            3.4.1  Sets satisfying the Generalized Continuum Hypothesis   cgch 10031
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10089
*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10093
            4.1.2  Weak universes   cwun 10111
            4.1.3  Tarski classes   ctsk 10159
            4.1.4  Grothendieck universes   cgru 10201
      4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10234
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10237
            4.2.3  Tarski map function   ctskm 10248
*PART 5  REAL AND COMPLEX NUMBERS
      5.1  Construction and axiomatization of real and complex numbers
            5.1.1  Dedekind-cut construction of real and complex numbers   cnpi 10255
            5.1.2  Final derivation of real and complex number postulates   axaddf 10556
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 10582
      5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 10607
            5.2.2  Infinity and the extended real number system   cpnf 10661
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 10701
            5.2.4  Ordering on reals   lttr 10706
            5.2.5  Initial properties of the complex numbers   mul12 10794
      5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 10846
            5.3.2  Subtraction   cmin 10859
            5.3.3  Multiplication   kcnktkm1cn 11060
            5.3.4  Ordering on reals (cont.)   gt0ne0 11094
            5.3.5  Reciprocals   ixi 11258
            5.3.6  Division   cdiv 11286
            5.3.7  Ordering on reals (cont.)   elimgt0 11467
            5.3.8  Completeness Axiom and Suprema   fimaxre 11573
            5.3.9  Imaginary and complex number properties   inelr 11617
            5.3.10  Function operation analogue theorems   ofsubeq0 11624
      5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 11627
            5.4.2  Principle of mathematical induction   nnind 11645
            *5.4.3  Decimal representation of numbers   c2 11681
            *5.4.4  Some properties of specific numbers   neg1cn 11740
            5.4.5  Simple number properties   halfcl 11851
            5.4.6  The Archimedean property   nnunb 11882
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 11886
            *5.4.8  Extended nonnegative integers   cxnn0 11956
            5.4.9  Integers (as a subset of complex numbers)   cz 11970
            5.4.10  Decimal arithmetic   cdc 12087
            5.4.11  Upper sets of integers   cuz 12232
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12332
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12337
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 12366
      5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 12379
            5.5.2  Infinity and the extended real number system (cont.)   cxne 12494
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 12688
            5.5.4  Real number intervals   cioo 12728
            5.5.5  Finite intervals of integers   cfz 12882
            *5.5.6  Finite intervals of nonnegative integers   elfz2nn0 12988
            5.5.7  Half-open integer ranges   cfzo 13023
      5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13150
            5.6.2  The modulo (remainder) operation   cmo 13227
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13305
            5.6.4  Strong induction over upper sets of integers   uzsinds 13345
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 13348
            5.6.6  The infinite sequence builder "seq" - extension   cseq 13359
            5.6.7  Integer powers   cexp 13419
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 13617
            5.6.9  Factorial function   cfa 13623
            5.6.10  The binomial coefficient operation   cbc 13652
            5.6.11  The ` # ` (set size) function   chash 13680
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 13816
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 13840
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 13844
      *5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 13851
            5.7.2  Last symbol of a word   clsw 13904
            5.7.3  Concatenations of words   cconcat 13912
            5.7.4  Singleton words   cs1 13939
            5.7.5  Concatenations with singleton words   ccatws1cl 13960
            5.7.6  Subwords/substrings   csubstr 13992
            5.7.7  Prefixes of a word   cpfx 14022
            5.7.8  Subwords of subwords   swrdswrdlem 14056
            5.7.9  Subwords and concatenations   pfxcctswrd 14062
            5.7.10  Subwords of concatenations   swrdccatfn 14076
            5.7.11  Splicing words (substring replacement)   csplice 14101
            5.7.12  Reversing words   creverse 14110
            5.7.13  Repeated symbol words   creps 14120
            *5.7.14  Cyclical shifts of words   ccsh 14140
            5.7.15  Mapping words by a function   wrdco 14183
            5.7.16  Longer string literals   cs2 14193
      *5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14322
            5.8.2  Basic properties of closures   cleq1lem 14332
            5.8.3  Definitions and basic properties of transitive closures   ctcl 14335
            5.8.4  Exponentiation of relations   crelexp 14369
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 14404
            *5.8.6  Principle of transitive induction.   relexpindlem 14412
      5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 14415
            5.9.2  Signum (sgn or sign) function   csgn 14435
            5.9.3  Real and imaginary parts; conjugate   ccj 14445
            5.9.4  Square root; absolute value   csqrt 14582
      5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 14817
            5.10.2  Limits   cli 14831
            5.10.3  Finite and infinite sums   csu 15032
            5.10.4  The binomial theorem   binomlem 15174
            5.10.5  The inclusion/exclusion principle   incexclem 15181
            5.10.6  Infinite sums (cont.)   isumshft 15184
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15197
            5.10.8  Arithmetic series   arisum 15205
            5.10.9  Geometric series   expcnv 15209
            5.10.10  Ratio test for infinite series convergence   cvgrat 15229
            5.10.11  Mertens' theorem   mertenslem1 15230
            5.10.12  Finite and infinite products   prodf 15233
                  5.10.12.1  Product sequences   prodf 15233
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15243
                  5.10.12.3  Complex products   cprod 15249
                  5.10.12.4  Finite products   fprod 15285
                  5.10.12.5  Infinite products   iprodclim 15342
            5.10.13  Falling and Rising Factorial   cfallfac 15348
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 15390
      5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 15405
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 15545
            5.11.2  _e is irrational   eirrlem 15547
      5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 15554
            5.12.2  The reals are uncountable   rpnnen2lem1 15557
*PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 15591
            6.1.2  Some Number sets are chains of proper subsets   nthruc 15595
            6.1.3  The divides relation   cdvds 15597
            *6.1.4  Even and odd numbers   evenelz 15675
            6.1.5  The division algorithm   divalglem0 15734
            6.1.6  Bit sequences   cbits 15758
            6.1.7  The greatest common divisor operator   cgcd 15833
            6.1.8  Bézout's identity   bezoutlem1 15877
            6.1.9  Algorithms   nn0seqcvgd 15904
            6.1.10  Euclid's Algorithm   eucalgval2 15915
            *6.1.11  The least common multiple   clcm 15922
            *6.1.12  Coprimality and Euclid's lemma   coprmgcdb 15983
            6.1.13  Cancellability of congruences   congr 15998
      6.2  Elementary prime number theory
            *6.2.1  Elementary properties   cprime 16005
            *6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16045
            6.2.3  Properties of the canonical representation of a rational   cnumer 16063
            6.2.4  Euler's theorem   codz 16090
            6.2.5  Arithmetic modulo a prime number   modprm1div 16124
            6.2.6  Pythagorean Triples   coprimeprodsq 16135
            6.2.7  The prime count function   cpc 16163
            6.2.8  Pocklington's theorem   prmpwdvds 16230
            6.2.9  Infinite primes theorem   unbenlem 16234
            6.2.10  Sum of prime reciprocals   prmreclem1 16242
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16249
            6.2.12  Lagrange's four-square theorem   cgz 16255
            6.2.13  Van der Waerden's theorem   cvdwa 16291
            6.2.14  Ramsey's theorem   cram 16325
            *6.2.15  Primorial function   cprmo 16357
            *6.2.16  Prime gaps   prmgaplem1 16375
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 16389
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 16417
            6.2.19  Specific prime numbers   prmlem0 16429
            6.2.20  Very large primes   1259lem1 16454
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
            *7.1.1  Basic definitions   cstr 16469
            7.1.2  Slot definitions   cplusg 16555
            7.1.3  Definition of the structure product   crest 16684
            7.1.4  Definition of the structure quotient   cordt 16762
      7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 16867
            7.2.2  Independent sets in a Moore system   mrisval 16891
            7.2.3  Algebraic closure systems   isacs 16912
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
            8.1.1  Categories   ccat 16925
            8.1.2  Opposite category   coppc 16971
            8.1.3  Monomorphisms and epimorphisms   cmon 16988
            8.1.4  Sections, inverses, isomorphisms   csect 17004
            *8.1.5  Isomorphic objects   ccic 17055
            8.1.6  Subcategories   cssc 17067
            8.1.7  Functors   cfunc 17114
            8.1.8  Full & faithful functors   cful 17162
            8.1.9  Natural transformations and the functor category   cnat 17201
            8.1.10  Initial, terminal and zero objects of a category   cinito 17238
      8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 17303
      8.3  Examples of categories
            8.3.1  The category of sets   csetc 17325
            8.3.2  The category of categories   ccatc 17344
            *8.3.3  The category of extensible structures   fncnvimaeqv 17360
      8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 17408
            8.4.2  Functor evaluation   cevlf 17449
            8.4.3  Hom functor   chof 17488
PART 9  BASIC ORDER THEORY
      9.1  Preordered sets and directed sets using extensible structures
      9.2  Posets and lattices using extensible structures
            9.2.1  Posets   cpo 17540
            9.2.2  Lattices   clat 17645
            9.2.3  The dual of an ordered set   codu 17728
            9.2.4  Subset order structures   cipo 17751
            9.2.5  Distributive lattices   latmass 17788
            9.2.6  Posets and lattices as relations   cps 17798
            9.2.7  Directed sets, nets   cdir 17828
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
            *10.1.1  Magmas   cplusf 17839
            *10.1.2  Identity elements   mgmidmo 17860
            *10.1.3  Iterated sums in a magma   gsumvalx 17876
            *10.1.4  Semigroups   csgrp 17890
            *10.1.5  Definition and basic properties of monoids   cmnd 17901
            10.1.6  Monoid homomorphisms and submonoids   cmhm 17944
            *10.1.7  Iterated sums in a monoid   gsumvallem2 17988
            10.1.8  Free monoids   cfrmd 18002
            10.1.9  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18023
      10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18043
            *10.2.2  Group multiple operation   cmg 18164
            10.2.3  Subgroups and Quotient groups   csubg 18213
            *10.2.4  Cyclic monoids and groups   cycsubmel 18283
            10.2.5  Elementary theory of group homomorphisms   cghm 18295
            10.2.6  Isomorphisms of groups   cgim 18337
            10.2.7  Group actions   cga 18359
            10.2.8  Centralizers and centers   ccntz 18385
            10.2.9  The opposite group   coppg 18413
            10.2.10  Symmetric groups   csymg 18435
                  *10.2.10.1  Definition and basic properties   csymg 18435
                  10.2.10.2  Cayley's theorem   cayleylem1 18471
                  10.2.10.3  Permutations fixing one element   symgfix2 18475
                  *10.2.10.4  Transpositions in the symmetric group   cpmtr 18500
                  10.2.10.5  The sign of a permutation   cpsgn 18548
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 18583
            10.2.12  Direct products   clsm 18690
                  10.2.12.1  Direct products (extension)   smndlsmidm 18712
            10.2.13  Free groups   cefg 18763
            10.2.14  Abelian groups   ccmn 18837
                  10.2.14.1  Definition and basic properties   ccmn 18837
                  10.2.14.2  Cyclic groups   ccyg 18927
                  10.2.14.3  Group sum operation   gsumval3a 18954
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 19034
                  10.2.14.5  Internal direct products   cdprd 19046
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 19118
            10.2.15  Simple groups   csimpg 19143
                  10.2.15.1  Definition and basic properties   csimpg 19143
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 19157
      10.3  Rings
            10.3.1  Multiplicative Group   cmgp 19170
            10.3.2  Ring unit   cur 19182
                  10.3.2.1  Semirings   csrg 19186
                  *10.3.2.2  The binomial theorem for semirings   srgbinomlem1 19221
            10.3.3  Definition and basic properties of unital rings   crg 19228
            10.3.4  Opposite ring   coppr 19303
            10.3.5  Divisibility   cdsr 19319
            10.3.6  Ring primes   crpm 19393
            10.3.7  Ring homomorphisms   crh 19395
      10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 19433
            10.4.2  Subrings of a ring   csubrg 19462
                  10.4.2.1  Sub-division rings   csdrg 19503
            10.4.3  Absolute value (abstract algebra)   cabv 19518
            10.4.4  Star rings   cstf 19545
      10.5  Left modules
            10.5.1  Definition and basic properties   clmod 19565
            10.5.2  Subspaces and spans in a left module   clss 19634
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 19722
            10.5.4  Subspace sum; bases for a left module   clbs 19777
      10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 19805
      10.7  Ideals
            10.7.1  The subring algebra; ideals   csra 19871
            10.7.2  Two-sided ideals and quotient rings   c2idl 19934
            10.7.3  Principal ideal rings. Divisibility in the integers   clpidl 19944
            10.7.4  Nonzero rings and zero rings   cnzr 19960
            10.7.5  Left regular elements. More kinds of rings   crlreg 19982
      10.8  Associative algebras
            10.8.1  Definition and basic properties   casa 20012
      10.9  Abstract multivariate polynomials
            10.9.1  Definition and basic properties   cmps 20061
            10.9.2  Polynomial evaluation   ces 20214
            10.9.3  Additional definitions for (multivariate) polynomials   cslv 20251
            *10.9.4  Univariate polynomials   cps1 20273
            10.9.5  Univariate polynomial evaluation   ces1 20406
      10.10  The complex numbers as an algebraic extensible structure
            10.10.1  Definition and basic properties   cpsmet 20459
            *10.10.2  Ring of integers   zring 20547
            10.10.3  Algebraic constructions based on the complex numbers   czrh 20577
            10.10.4  Signs as subgroup of the complex numbers   cnmsgnsubg 20651
            10.10.5  Embedding of permutation signs into a ring   zrhpsgnmhm 20658
            10.10.6  The ordered field of real numbers   crefld 20678
      10.11  Generalized pre-Hilbert and Hilbert spaces
            10.11.1  Definition and basic properties   cphl 20698
            10.11.2  Orthocomplements and closed subspaces   cocv 20734
            10.11.3  Orthogonal projection and orthonormal bases   cpj 20774
*PART 11  BASIC LINEAR ALGEBRA
      11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 20805
            *11.1.2  Free modules   cfrlm 20820
            *11.1.3  Standard basis (unit vectors)   cuvc 20856
            *11.1.4  Independent sets and families   clindf 20878
            11.1.5  Characterization of free modules   lmimlbs 20910
      *11.2  Matrices
            *11.2.1  The matrix multiplication   cmmul 20924
            *11.2.2  Square matrices   cmat 20946
            *11.2.3  The matrix algebra   matmulr 20977
            *11.2.4  Matrices of dimension 0 and 1   mat0dimbas0 21005
            *11.2.5  The subalgebras of diagonal and scalar matrices   cdmat 21027
            *11.2.6  Multiplication of a matrix with a "column vector"   cmvmul 21079
            11.2.7  Replacement functions for a square matrix   cmarrep 21095
            11.2.8  Submatrices   csubma 21115
      11.3  The determinant
            11.3.1  Definition and basic properties   cmdat 21123
            11.3.2  Determinants of 2 x 2 -matrices   m2detleiblem1 21163
            11.3.3  The matrix adjugate/adjunct   cmadu 21171
            *11.3.4  Laplace expansion of determinants (special case)   symgmatr01lem 21192
            11.3.5  Inverse matrix   invrvald 21215
            *11.3.6  Cramer's rule   slesolvec 21218
      *11.4  Polynomial matrices
            11.4.1  Basic properties   pmatring 21231
            *11.4.2  Constant polynomial matrices   ccpmat 21241
            *11.4.3  Collecting coefficients of polynomial matrices   cdecpmat 21300
            *11.4.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 21330
      *11.5  The characteristic polynomial
            *11.5.1  Definition and basic properties   cchpmat 21364
            *11.5.2  The characteristic factor function G   fvmptnn04if 21387
            *11.5.3  The Cayley-Hamilton theorem   cpmadurid 21405
PART 12  BASIC TOPOLOGY
      12.1  Topology
            *12.1.1  Topological spaces   ctop 21431
                  12.1.1.1  Topologies   ctop 21431
                  12.1.1.2  Topologies on sets   ctopon 21448
                  12.1.1.3  Topological spaces   ctps 21470
            12.1.2  Topological bases   ctb 21483
            12.1.3  Examples of topologies   distop 21533
            12.1.4  Closure and interior   ccld 21554
            12.1.5  Neighborhoods   cnei 21635
            12.1.6  Limit points and perfect sets   clp 21672
            12.1.7  Subspace topologies   restrcl 21695
            12.1.8  Order topology   ordtbaslem 21726
            12.1.9  Limits and continuity in topological spaces   ccn 21762
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 21844
            12.1.11  Compactness   ccmp 21924
            12.1.12  Bolzano-Weierstrass theorem   bwth 21948
            12.1.13  Connectedness   cconn 21949
            12.1.14  First- and second-countability   c1stc 21975
            12.1.15  Local topological properties   clly 22002
            12.1.16  Refinements   cref 22040
            12.1.17  Compactly generated spaces   ckgen 22071
            12.1.18  Product topologies   ctx 22098
            12.1.19  Continuous function-builders   cnmptid 22199
            12.1.20  Quotient maps and quotient topology   ckq 22231
            12.1.21  Homeomorphisms   chmeo 22291
      12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 22365
            12.2.2  Filters   cfil 22383
            12.2.3  Ultrafilters   cufil 22437
            12.2.4  Filter limits   cfm 22471
            12.2.5  Extension by continuity   ccnext 22597
            12.2.6  Topological groups   ctmd 22608
            12.2.7  Infinite group sum on topological groups   ctsu 22663
            12.2.8  Topological rings, fields, vector spaces   ctrg 22693
      12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 22737
            12.3.2  The topology induced by an uniform structure   cutop 22768
            12.3.3  Uniform Spaces   cuss 22791
            12.3.4  Uniform continuity   cucn 22813
            12.3.5  Cauchy filters in uniform spaces   ccfilu 22824
            12.3.6  Complete uniform spaces   ccusp 22835
      12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 22843
            12.4.2  Basic metric space properties   cxms 22856
            12.4.3  Metric space balls   blfvalps 22922
            12.4.4  Open sets of a metric space   mopnval 22977
            12.4.5  Continuity in metric spaces   metcnp3 23079
            12.4.6  The uniform structure generated by a metric   metuval 23088
            12.4.7  Examples of metric spaces   dscmet 23111
            *12.4.8  Normed algebraic structures   cnm 23115
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 23243
            12.4.10  Topology on the reals   qtopbaslem 23296
            12.4.11  Topological definitions using the reals   cii 23412
            12.4.12  Path homotopy   chtpy 23500
            12.4.13  The fundamental group   cpco 23533
      12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 23595
            *12.5.2  Subcomplex vector spaces   ccvs 23656
            *12.5.3  Normed subcomplex vector spaces   isncvsngp 23682
            12.5.4  Subcomplex pre-Hilbert space   ccph 23699
            12.5.5  Convergence and completeness   ccfil 23784
            12.5.6  Baire's Category Theorem   bcthlem1 23856
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 23864
                  12.5.7.1  The complete ordered field of the real numbers   retopn 23911
            12.5.8  Euclidean spaces   crrx 23915
            12.5.9  Minimizing Vector Theorem   minveclem1 23956
            12.5.10  Projection Theorem   pjthlem1 23969
PART 13  BASIC REAL AND COMPLEX ANALYSIS
      13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 23978
      13.2  Integrals
            13.2.1  Lebesgue measure   covol 23992
            13.2.2  Lebesgue integration   cmbf 24144
                  13.2.2.1  Lesbesgue integral   cmbf 24144
                  13.2.2.2  Lesbesgue directed integral   cdit 24373
      13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 24389
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 24389
                  13.3.1.2  Results on real differentiation   dvferm1lem 24510
PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 24576
            14.1.2  The division algorithm for univariate polynomials   cmn1 24648
            14.1.3  Elementary properties of complex polynomials   cply 24703
            14.1.4  The division algorithm for polynomials   cquot 24808
            14.1.5  Algebraic numbers   caa 24832
            14.1.6  Liouville's approximation theorem   aalioulem1 24850
      14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 24870
            14.2.2  Uniform convergence   culm 24893
            14.2.3  Power series   pserval 24927
      14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 24960
            14.3.2  Properties of pi = 3.14159...   pilem1 24968
            14.3.3  Mapping of the exponential function   efgh 25052
            14.3.4  The natural logarithm on complex numbers   clog 25065
            *14.3.5  Logarithms to an arbitrary base   clogb 25269
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 25306
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 25344
            14.3.8  Inverse trigonometric functions   casin 25367
            14.3.9  The Birthday Problem   log2ublem1 25452
            14.3.10  Areas in R^2   carea 25461
            14.3.11  More miscellaneous converging sequences   rlimcnp 25471
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 25490
            14.3.13  Euler-Mascheroni constant   cem 25497
            14.3.14  Zeta function   czeta 25518
            14.3.15  Gamma function   clgam 25521
      14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 25573
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 25578
            14.4.3  The Basel problem (ζ(2) = π2/6)   basellem1 25586
            14.4.4  Number-theoretical functions   ccht 25596
            14.4.5  Perfect Number Theorem   mersenne 25731
            14.4.6  Characters of Z/nZ   cdchr 25736
            14.4.7  Bertrand's postulate   bcctr 25779
            *14.4.8  Quadratic residues and the Legendre symbol   clgs 25798
            *14.4.9  Gauss' Lemma   gausslemma2dlem0a 25860
            14.4.10  Quadratic reciprocity   lgseisenlem1 25879
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 25921
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 25973
            14.4.13  The Prime Number Theorem   mudivsum 26034
            14.4.14  Ostrowski's theorem   abvcxp 26119
*PART 15  ELEMENTARY GEOMETRY
      15.1  Definition and Tarski's Axioms of Geometry
            15.1.1  Justification for the congruence notation   tgjustf 26187
      15.2  Tarskian Geometry
            15.2.1  Congruence   tgcgrcomimp 26191
            15.2.2  Betweenness   tgbtwntriv2 26201
            15.2.3  Dimension   tglowdim1 26214
            15.2.4  Betweenness and Congruence   tgifscgr 26222
            15.2.5  Congruence of a series of points   ccgrg 26224
            15.2.6  Motions   cismt 26246
            15.2.7  Colinearity   tglng 26260
            15.2.8  Connectivity of betweenness   tgbtwnconn1lem1 26286
            15.2.9  Less-than relation in geometric congruences   cleg 26296
            15.2.10  Rays   chlg 26314
            15.2.11  Lines   btwnlng1 26333
            15.2.12  Point inversions   cmir 26366
            15.2.13  Right angles   crag 26407
            15.2.14  Half-planes   islnopp 26453
            15.2.15  Midpoints and Line Mirroring   cmid 26486
            15.2.16  Congruence of angles   ccgra 26521
            15.2.17  Angle Comparisons   cinag 26549
            15.2.18  Congruence Theorems   tgsas1 26568
            15.2.19  Equilateral triangles   ceqlg 26579
      15.3  Properties of geometries
            15.3.1  Isomorphisms between geometries   f1otrgds 26583
      15.4  Geometry in Hilbert spaces
            15.4.1  Geometry in the complex plane   cchhllem 26601
            15.4.2  Geometry in Euclidean spaces   cee 26602
                  15.4.2.1  Definition of the Euclidean space   cee 26602
                  15.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 26627
                  15.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 26691
*PART 16  GRAPH THEORY
      *16.1  Vertices and edges
            16.1.1  The edge function extractor for extensible structures   cedgf 26702
            *16.1.2  Vertices and indexed edges   cvtx 26709
                  16.1.2.1  Definitions and basic properties   cvtx 26709
                  16.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 26716
                  16.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 26724
                  16.1.2.4  Representations of graphs without edges   snstrvtxval 26750
                  16.1.2.5  Degenerated cases of representations of graphs   vtxval0 26752
            16.1.3  Edges as range of the edge function   cedg 26760
      *16.2  Undirected graphs
            16.2.1  Undirected hypergraphs   cuhgr 26769
            16.2.2  Undirected pseudographs and multigraphs   cupgr 26793
            *16.2.3  Loop-free graphs   umgrislfupgrlem 26835
            16.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 26839
            *16.2.5  Undirected simple graphs   cuspgr 26861
            16.2.6  Examples for graphs   usgr0e 26946
            16.2.7  Subgraphs   csubgr 26977
            16.2.8  Finite undirected simple graphs   cfusgr 27026
            16.2.9  Neighbors, complete graphs and universal vertices   cnbgr 27042
                  16.2.9.1  Neighbors   cnbgr 27042
                  16.2.9.2  Universal vertices   cuvtx 27095
                  16.2.9.3  Complete graphs   ccplgr 27119
            16.2.10  Vertex degree   cvtxdg 27175
            *16.2.11  Regular graphs   crgr 27265
      *16.3  Walks, paths and cycles
            *16.3.1  Walks   cewlks 27305
            16.3.2  Walks for loop-free graphs   lfgrwlkprop 27397
            16.3.3  Trails   ctrls 27400
            16.3.4  Paths and simple paths   cpths 27421
            16.3.5  Closed walks   cclwlks 27479
            16.3.6  Circuits and cycles   ccrcts 27493
            *16.3.7  Walks as words   cwwlks 27531
            16.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 27632
            16.3.9  Walks in regular graphs   rusgrnumwwlkl1 27675
            *16.3.10  Closed walks as words   cclwwlk 27687
                  16.3.10.1  Closed walks as words   cclwwlk 27687
                  16.3.10.2  Closed walks of a fixed length as words   cclwwlkn 27730
                  16.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 27794
            16.3.11  Examples for walks, trails and paths   0ewlk 27821
            16.3.12  Connected graphs   cconngr 27893
      16.4  Eulerian paths and the Konigsberg Bridge problem
            *16.4.1  Eulerian paths   ceupth 27904
            *16.4.2  The Königsberg Bridge problem   konigsbergvtx 27953
      16.5  The Friendship Theorem
            16.5.1  Friendship graphs - basics   cfrgr 27965
            16.5.2  The friendship theorem for small graphs   frgr1v 27978
            16.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 27989
            *16.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 28006
PART 17  GUIDES AND MISCELLANEA
      17.1  Guides (conventions, explanations, and examples)
            *17.1.1  Conventions   conventions 28107
            17.1.2  Natural deduction   natded 28110
            *17.1.3  Natural deduction examples   ex-natded5.2 28111
            17.1.4  Definitional examples   ex-or 28128
            17.1.5  Other examples   aevdemo 28167
      17.2  Humor
            17.2.1  April Fool's theorem   avril1 28170
      17.3  (Future - to be reviewed and classified)
            17.3.1  Planar incidence geometry   cplig 28179
            *17.3.2  Aliases kept to prevent broken links   dummylink 28192
*PART 18  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      *18.1  Additional material on group theory (deprecated)
            18.1.1  Definitions and basic properties for groups   cgr 28194
            18.1.2  Abelian groups   cablo 28249
      18.2  Complex vector spaces
            18.2.1  Definition and basic properties   cvc 28263
            18.2.2  Examples of complex vector spaces   cnaddabloOLD 28286
      18.3  Normed complex vector spaces
            18.3.1  Definition and basic properties   cnv 28289
            18.3.2  Examples of normed complex vector spaces   cnnv 28382
            18.3.3  Induced metric of a normed complex vector space   imsval 28390
            18.3.4  Inner product   cdip 28405
            18.3.5  Subspaces   css 28426
      18.4  Operators on complex vector spaces
            18.4.1  Definitions and basic properties   clno 28445
      18.5  Inner product (pre-Hilbert) spaces
            18.5.1  Definition and basic properties   ccphlo 28517
            18.5.2  Examples of pre-Hilbert spaces   cncph 28524
            18.5.3  Properties of pre-Hilbert spaces   isph 28527
      18.6  Complex Banach spaces
            18.6.1  Definition and basic properties   ccbn 28567
            18.6.2  Examples of complex Banach spaces   cnbn 28574
            18.6.3  Uniform Boundedness Theorem   ubthlem1 28575
            18.6.4  Minimizing Vector Theorem   minvecolem1 28579
      18.7  Complex Hilbert spaces
            18.7.1  Definition and basic properties   chlo 28590
            18.7.2  Standard axioms for a complex Hilbert space   hlex 28603
            18.7.3  Examples of complex Hilbert spaces   cnchl 28621
            18.7.4  Hellinger-Toeplitz Theorem   htthlem 28622
*PART 19  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      19.1  Axiomatization of complex pre-Hilbert spaces
            19.1.1  Basic Hilbert space definitions   chba 28624
            19.1.2  Preliminary ZFC lemmas   df-hnorm 28673
            *19.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 28686
            *19.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 28704
            19.1.5  Vector operations   hvmulex 28716
            19.1.6  Inner product postulates for a Hilbert space   ax-hfi 28784
      19.2  Inner product and norms
            19.2.1  Inner product   his5 28791
            19.2.2  Norms   dfhnorm2 28827
            19.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 28865
            19.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 28884
      19.3  Cauchy sequences and completeness axiom
            19.3.1  Cauchy sequences and limits   hcau 28889
            19.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 28899
            19.3.3  Completeness postulate for a Hilbert space   ax-hcompl 28907
            19.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 28908
      19.4  Subspaces and projections
            19.4.1  Subspaces   df-sh 28912
            19.4.2  Closed subspaces   df-ch 28926
            19.4.3  Orthocomplements   df-oc 28957
            19.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 29013
            19.4.5  Projection theorem   pjhthlem1 29096
            19.4.6  Projectors   df-pjh 29100
      19.5  Properties of Hilbert subspaces
            19.5.1  Orthomodular law   omlsilem 29107
            19.5.2  Projectors (cont.)   pjhtheu2 29121
            19.5.3  Hilbert lattice operations   sh0le 29145
            19.5.4  Span (cont.) and one-dimensional subspaces   spansn0 29246
            19.5.5  Commutes relation for Hilbert lattice elements   df-cm 29288
            19.5.6  Foulis-Holland theorem   fh1 29323
            19.5.7  Quantum Logic Explorer axioms   qlax1i 29332
            19.5.8  Orthogonal subspaces   chscllem1 29342
            19.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 29359
            19.5.10  Projectors (cont.)   pjorthi 29374
            19.5.11  Mayet's equation E_3   mayete3i 29433
      19.6  Operators on Hilbert spaces
            *19.6.1  Operator sum, difference, and scalar multiplication   df-hosum 29435
            19.6.2  Zero and identity operators   df-h0op 29453
            19.6.3  Operations on Hilbert space operators   hoaddcl 29463
            19.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 29544
            19.6.5  Linear and continuous functionals and norms   df-nmfn 29550
            19.6.6  Adjoint   df-adjh 29554
            19.6.7  Dirac bra-ket notation   df-bra 29555
            19.6.8  Positive operators   df-leop 29557
            19.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 29558
            19.6.10  Theorems about operators and functionals   nmopval 29561
            19.6.11  Riesz lemma   riesz3i 29767
            19.6.12  Adjoints (cont.)   cnlnadjlem1 29772
            19.6.13  Quantum computation error bound theorem   unierri 29809
            19.6.14  Dirac bra-ket notation (cont.)   branmfn 29810
            19.6.15  Positive operators (cont.)   leopg 29827
            19.6.16  Projectors as operators   pjhmopi 29851
      19.7  States on a Hilbert lattice and Godowski's equation
            19.7.1  States on a Hilbert lattice   df-st 29916
            19.7.2  Godowski's equation   golem1 29976
      19.8  Cover relation, atoms, exchange axiom, and modular symmetry
            19.8.1  Covers relation; modular pairs   df-cv 29984
            19.8.2  Atoms   df-at 30043
            19.8.3  Superposition principle   superpos 30059
            19.8.4  Atoms, exchange and covering properties, atomicity   chcv1 30060
            19.8.5  Irreducibility   chirredlem1 30095
            19.8.6  Atoms (cont.)   atcvat3i 30101
            19.8.7  Modular symmetry   mdsymlem1 30108
PART 20  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      20.1  Mathboxes for user contributions
            20.1.1  Mathbox guidelines   mathbox 30147
      20.2  Mathbox for Stefan Allan
      20.3  Mathbox for Thierry Arnoux
            20.3.1  Propositional Calculus - misc additions   bian1d 30152
            20.3.2  Predicate Calculus   sbc2iedf 30158
                  20.3.2.1  Predicate Calculus - misc additions   sbc2iedf 30158
                  20.3.2.2  Restricted quantification - misc additions   ralcom4f 30161
                  20.3.2.3  Equality   eqtrb 30166
                  20.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 30167
                  20.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 30169
                  20.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 30178
                  20.3.2.7  Existential "at most one" - misc additions   moel 30180
                  20.3.2.8  Existential uniqueness - misc additions   reuxfrdf 30183
                  20.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 30185
            20.3.3  General Set Theory   dmrab 30188
                  20.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 30188
                  20.3.3.2  Image Sets   abrexdomjm 30195
                  20.3.3.3  Set relations and operations - misc additions   elunsn 30201
                  20.3.3.4  Unordered pairs   eqsnd 30217
                  20.3.3.5  Conditional operator - misc additions   ifeqeqx 30225
                  20.3.3.6  Set union   uniinn0 30230
                  20.3.3.7  Indexed union - misc additions   cbviunf 30236
                  20.3.3.8  Disjointness - misc additions   disjnf 30249
            20.3.4  Relations and Functions   xpdisjres 30277
                  20.3.4.1  Relations - misc additions   xpdisjres 30277
                  20.3.4.2  Functions - misc additions   ac6sf2 30299
                  20.3.4.3  Operations - misc additions   mpomptxf 30354
                  20.3.4.4  Explicit Functions with one or two points as a domain   brsnop 30356
                  20.3.4.5  Isomorphisms - misc. add.   gtiso 30363
                  20.3.4.6  Disjointness (additional proof requiring functions)   disjdsct 30365
                  20.3.4.7  First and second members of an ordered pair - misc additions   df1stres 30366
                  20.3.4.8  Supremum - misc additions   supssd 30372
                  20.3.4.9  Finite Sets   imafi2 30374
                  20.3.4.10  Countable Sets   snct 30376
            20.3.5  Real and Complex Numbers   creq0 30398
                  20.3.5.1  Complex operations - misc. additions   creq0 30398
                  20.3.5.2  Ordering on reals - misc additions   lt2addrd 30402
                  20.3.5.3  Extended reals - misc additions   xrlelttric 30403
                  20.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 30421
                  20.3.5.5  Real number intervals - misc additions   joiniooico 30424
                  20.3.5.6  Finite intervals of integers - misc additions   uzssico 30434
                  20.3.5.7  Half-open integer ranges - misc additions   iundisjfi 30446
                  20.3.5.8  The ` # ` (set size) function - misc additions   hashunif 30455
                  20.3.5.9  The greatest common divisor operator - misc. add   dvdszzq 30458
                  20.3.5.10  Integers   nnindf 30462
                  20.3.5.11  Decimal numbers   dfdec100 30474
            *20.3.6  Decimal expansion   cdp2 30475
                  *20.3.6.1  Decimal point   cdp 30492
                  20.3.6.2  Division in the extended real number system   cxdiv 30521
            20.3.7  Words over a set - misc additions   wrdfd 30540
                  20.3.7.1  Splicing words (substring replacement)   splfv3 30560
                  20.3.7.2  Cyclic shift of words   1cshid 30561
            20.3.8  Extensible Structures   ressplusf 30565
                  20.3.8.1  Structure restriction operator   ressplusf 30565
                  20.3.8.2  The opposite group   oppgle 30568
                  20.3.8.3  Posets   ressprs 30570
                  20.3.8.4  Complete lattices   clatp0cl 30586
                  20.3.8.5  Extended reals Structure - misc additions   ax-xrssca 30588
                  20.3.8.6  The extended nonnegative real numbers commutative monoid   xrge0base 30600
            20.3.9  Algebra   abliso 30611
                  20.3.9.1  Monoids Homomorphisms   abliso 30611
                  20.3.9.2  Finitely supported group sums - misc additions   gsumsubg 30612
                  20.3.9.3  Centralizers and centers - misc additions   cntzun 30623
                  20.3.9.4  Totally ordered monoids and groups   comnd 30626
                  20.3.9.5  The symmetric group   symgfcoeu 30654
                  20.3.9.6  Transpositions   pmtridf1o 30664
                  20.3.9.7  Permutation Signs   psgnid 30667
                  20.3.9.8  Permutation cycles   ctocyc 30676
                  20.3.9.9  The Alternating Group   evpmval 30715
                  20.3.9.10  Signum in an ordered monoid   csgns 30728
                  20.3.9.11  The Archimedean property for generic ordered algebraic structures   cinftm 30733
                  20.3.9.12  Semiring left modules   cslmd 30756
                  20.3.9.13  Simple groups   prmsimpcyc 30784
                  20.3.9.14  Rings - misc additions   rngurd 30785
                  20.3.9.15  Subfields   primefldchr 30795
                  20.3.9.16  Totally ordered rings and fields   corng 30796
                  20.3.9.17  Ring homomorphisms - misc additions   rhmdvdsr 30819
                  20.3.9.18  Scalar restriction operation   cresv 30825
                  20.3.9.19  The commutative ring of gaussian integers   gzcrng 30840
                  20.3.9.20  The archimedean ordered field of real numbers   reofld 30841
                  20.3.9.21  The quotient map and quotient modules   qusker 30846
                  20.3.9.22  Univariate Polynomials   fply1 30859
                  20.3.9.23  Independent sets and families   islinds5 30860
                  20.3.9.24  Prime Ideals   cprmidl 30872
                  20.3.9.25  The subring algebra   sra1r 30886
                  20.3.9.26  Division Ring Extensions   drgext0g 30892
                  20.3.9.27  Vector Spaces   lvecdimfi 30898
                  20.3.9.28  Vector Space Dimension   cldim 30899
            20.3.10  Field Extensions   cfldext 30928
            20.3.11  Matrices   csmat 30958
                  20.3.11.1  Submatrices   csmat 30958
                  20.3.11.2  Matrix literals   clmat 30976
                  20.3.11.3  Laplace expansion of determinants   mdetpmtr1 30988
            20.3.12  Topology   txomap 30998
                  20.3.12.1  Open maps   txomap 30998
                  20.3.12.2  Topology of the unit circle   qtopt1 30999
                  20.3.12.3  Refinements   reff 31003
                  20.3.12.4  Open cover refinement property   ccref 31006
                  20.3.12.5  Lindelöf spaces   cldlf 31016
                  20.3.12.6  Paracompact spaces   cpcmp 31019
                  20.3.12.7  Pseudometrics   cmetid 31026
                  20.3.12.8  Continuity - misc additions   hauseqcn 31038
                  20.3.12.9  Topology of the closed unit interval   unitsscn 31039
                  20.3.12.10  Topology of ` ( RR X. RR ) `   unicls 31046
                  20.3.12.11  Order topology - misc. additions   cnvordtrestixx 31056
                  20.3.12.12  Continuity in topological spaces - misc. additions   mndpluscn 31069
                  20.3.12.13  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 31075
                  20.3.12.14  Limits - misc additions   lmlim 31090
                  20.3.12.15  Univariate polynomials   pl1cn 31098
            20.3.13  Uniform Stuctures and Spaces   chcmp 31099
                  20.3.13.1  Hausdorff uniform completion   chcmp 31099
            20.3.14  Topology and algebraic structures   zringnm 31101
                  20.3.14.1  The norm on the ring of the integer numbers   zringnm 31101
                  20.3.14.2  Topological ` ZZ ` -modules   zlm0 31103
                  20.3.14.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 31113
                  20.3.14.4  Canonical embedding of the real numbers into a complete ordered field   crrh 31134
                  20.3.14.5  Embedding from the extended real numbers into a complete lattice   cxrh 31157
                  20.3.14.6  Canonical embeddings into the ordered field of the real numbers   zrhre 31160
                  *20.3.14.7  Topological Manifolds   cmntop 31163
            20.3.15  Real and complex functions   nexple 31168
                  20.3.15.1  Integer powers - misc. additions   nexple 31168
                  20.3.15.2  Indicator Functions   cind 31169
                  20.3.15.3  Extended sum   cesum 31186
            20.3.16  Mixed Function/Constant operation   cofc 31254
            20.3.17  Abstract measure   csiga 31267
                  20.3.17.1  Sigma-Algebra   csiga 31267
                  20.3.17.2  Generated sigma-Algebra   csigagen 31297
                  *20.3.17.3  lambda and pi-Systems, Rings of Sets   ispisys 31311
                  20.3.17.4  The Borel algebra on the real numbers   cbrsiga 31340
                  20.3.17.5  Product Sigma-Algebra   csx 31347
                  20.3.17.6  Measures   cmeas 31354
                  20.3.17.7  The counting measure   cntmeas 31385
                  20.3.17.8  The Lebesgue measure - misc additions   voliune 31388
                  20.3.17.9  The Dirac delta measure   cdde 31391
                  20.3.17.10  The 'almost everywhere' relation   cae 31396
                  20.3.17.11  Measurable functions   cmbfm 31408
                  20.3.17.12  Borel Algebra on ` ( RR X. RR ) `   br2base 31427
                  *20.3.17.13  Caratheodory's extension theorem   coms 31449
            20.3.18  Integration   itgeq12dv 31484
                  20.3.18.1  Lebesgue integral - misc additions   itgeq12dv 31484
                  20.3.18.2  Bochner integral   citgm 31485
            20.3.19  Euler's partition theorem   oddpwdc 31512
            20.3.20  Sequences defined by strong recursion   csseq 31541
            20.3.21  Fibonacci Numbers   cfib 31554
            20.3.22  Probability   cprb 31565
                  20.3.22.1  Probability Theory   cprb 31565
                  20.3.22.2  Conditional Probabilities   ccprob 31589
                  20.3.22.3  Real-valued Random Variables   crrv 31598
                  20.3.22.4  Preimage set mapping operator   corvc 31613
                  20.3.22.5  Distribution Functions   orvcelval 31626
                  20.3.22.6  Cumulative Distribution Functions   orvclteel 31630
                  20.3.22.7  Probabilities - example   coinfliplem 31636
                  20.3.22.8  Bertrand's Ballot Problem   ballotlemoex 31643
            20.3.23  Signum (sgn or sign) function - misc. additions   sgncl 31696
                  20.3.23.1  Operations on words   ccatmulgnn0dir 31712
            20.3.24  Polynomials with real coefficients - misc additions   plymul02 31716
            20.3.25  Descartes's rule of signs   signspval 31722
                  20.3.25.1  Sign changes in a word over real numbers   signspval 31722
                  20.3.25.2  Counting sign changes in a word over real numbers   signslema 31732
            20.3.26  Number Theory   efcld 31762
                  20.3.26.1  Representations of a number as sums of integers   crepr 31779
                  20.3.26.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 31806
                  20.3.26.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 31815
            20.3.27  Elementary Geometry   cstrkg2d 31835
                  *20.3.27.1  Two-dimensional geometry   cstrkg2d 31835
                  20.3.27.2  Outer Five Segment (not used, no need to move to main)   cafs 31840
            *20.3.28  LeftPad Project   clpad 31845
      *20.4  Mathbox for Jonathan Ben-Naim
            20.4.1  First-order logic and set theory   bnj170 31868
            20.4.2  Well founded induction and recursion   bnj110 32030
            20.4.3  The existence of a minimal element in certain classes   bnj69 32180
            20.4.4  Well-founded induction   bnj1204 32182
            20.4.5  Well-founded recursion, part 1 of 3   bnj60 32232
            20.4.6  Well-founded recursion, part 2 of 3   bnj1500 32238
            20.4.7  Well-founded recursion, part 3 of 3   bnj1522 32242
      20.5  Mathbox for BTernaryTau
            20.5.1  Acyclic graphs   cacycgr 32287
      20.6  Mathbox for Mario Carneiro
            20.6.1  Predicate calculus with all distinct variables   ax-7d 32304
            20.6.2  Miscellaneous stuff   quartfull 32310
            20.6.3  Derangements and the Subfactorial   deranglem 32311
            20.6.4  The Erdős-Szekeres theorem   erdszelem1 32336
            20.6.5  The Kuratowski closure-complement theorem   kur14lem1 32351
            20.6.6  Retracts and sections   cretr 32362
            20.6.7  Path-connected and simply connected spaces   cpconn 32364
            20.6.8  Covering maps   ccvm 32400
            20.6.9  Normal numbers   snmlff 32474
            20.6.10  Godel-sets of formulas - part 1   cgoe 32478
            20.6.11  Godel-sets of formulas - part 2   cgon 32577
            20.6.12  Models of ZF   cgze 32591
            *20.6.13  Metamath formal systems   cmcn 32605
            20.6.14  Grammatical formal systems   cm0s 32730
            20.6.15  Models of formal systems   cmuv 32750
            20.6.16  Splitting fields   citr 32772
            20.6.17  p-adic number fields   czr 32788
      *20.7  Mathbox for Filip Cernatescu
      20.8  Mathbox for Paul Chapman
            20.8.1  Real and complex numbers (cont.)   climuzcnv 32812
            20.8.2  Miscellaneous theorems   elfzm12 32816
      20.9  Mathbox for Scott Fenton
            20.9.1  ZFC Axioms in primitive form   axextprim 32825
            20.9.2  Untangled classes   untelirr 32832
            20.9.3  Extra propositional calculus theorems   3orel2 32839
            20.9.4  Misc. Useful Theorems   nepss 32846
            20.9.5  Properties of real and complex numbers   sqdivzi 32857
            20.9.6  Infinite products   iprodefisumlem 32870
            20.9.7  Factorial limits   faclimlem1 32873
            20.9.8  Greatest common divisor and divisibility   pdivsq 32879
            20.9.9  Properties of relationships   brtp 32883
            20.9.10  Properties of functions and mappings   funpsstri 32906
            20.9.11  Set induction (or epsilon induction)   setinds 32921
            20.9.12  Ordinal numbers   elpotr 32924
            20.9.13  Defined equality axioms   axextdfeq 32940
            20.9.14  Hypothesis builders   hbntg 32948
            20.9.15  (Trans)finite Recursion Theorems   tfisg 32953
            20.9.16  Transitive closure under a relationship   ctrpred 32954
            20.9.17  Founded Induction   frpomin 32976
            20.9.18  Ordering Ordinal Sequences   orderseqlem 32992
            20.9.19  Well-founded zero, successor, and limits   cwsuc 32995
            20.9.20  Founded Partial Recursion   cfrecs 33015
            20.9.21  Surreal Numbers   csur 33045
            20.9.22  Surreal Numbers: Ordering   sltsolem1 33078
            20.9.23  Surreal Numbers: Birthday Function   bdayfo 33080
            20.9.24  Surreal Numbers: Density   fvnobday 33081
            20.9.25  Surreal Numbers: Full-Eta Property   bdayimaon 33095
            20.9.26  Surreal numbers - ordering theorems   csle 33121
            20.9.27  Surreal numbers - birthday theorems   bdayfun 33140
            20.9.28  Surreal numbers: Conway cuts   csslt 33148
            20.9.29  Surreal numbers - cuts and options   cmade 33177
            20.9.30  Quantifier-free definitions   ctxp 33189
            20.9.31  Alternate ordered pairs   caltop 33315
            20.9.32  Geometry in the Euclidean space   cofs 33341
                  20.9.32.1  Congruence properties   cofs 33341
                  20.9.32.2  Betweenness properties   btwntriv2 33371
                  20.9.32.3  Segment Transportation   ctransport 33388
                  20.9.32.4  Properties relating betweenness and congruence   cifs 33394
                  20.9.32.5  Connectivity of betweenness   btwnconn1lem1 33446
                  20.9.32.6  Segment less than or equal to   csegle 33465
                  20.9.32.7  Outside-of relationship   coutsideof 33478
                  20.9.32.8  Lines and Rays   cline2 33493
            20.9.33  Forward difference   cfwddif 33517
            20.9.34  Rank theorems   rankung 33525
            20.9.35  Hereditarily Finite Sets   chf 33531
      20.10  Mathbox for Jeff Hankins
            20.10.1  Miscellany   a1i14 33546
            20.10.2  Basic topological facts   topbnd 33570
            20.10.3  Topology of the real numbers   ivthALT 33581
            20.10.4  Refinements   cfne 33582
            20.10.5  Neighborhood bases determine topologies   neibastop1 33605
            20.10.6  Lattice structure of topologies   topmtcl 33609
            20.10.7  Filter bases   fgmin 33616
            20.10.8  Directed sets, nets   tailfval 33618
      20.11  Mathbox for Anthony Hart
            20.11.1  Propositional Calculus   tb-ax1 33629
            20.11.2  Predicate Calculus   nalfal 33649
            20.11.3  Miscellaneous single axioms   meran1 33657
            20.11.4  Connective Symmetry   negsym1 33663
      20.12  Mathbox for Chen-Pang He
            20.12.1  Ordinal topology   ontopbas 33674
      20.13  Mathbox for Jeff Hoffman
            20.13.1  Inferences for finite induction on generic function values   fveleq 33697
            20.13.2  gdc.mm   nnssi2 33701
      20.14  Mathbox for Asger C. Ipsen
            20.14.1  Continuous nowhere differentiable functions   dnival 33708
      *20.15  Mathbox for BJ
            *20.15.1  Propositional calculus   bj-mp2c 33777
                  *20.15.1.1  Derived rules of inference   bj-mp2c 33777
                  *20.15.1.2  A syntactic theorem   bj-0 33779
                  20.15.1.3  Minimal implicational calculus   bj-a1k 33781
                  *20.15.1.4  Positive calculus   bj-syl66ib 33788
                  20.15.1.5  Implication and negation   bj-con2com 33794
                  *20.15.1.6  Disjunction   bj-jaoi1 33802
                  *20.15.1.7  Logical equivalence   bj-dfbi4 33804
                  20.15.1.8  The conditional operator for propositions   bj-consensus 33809
                  *20.15.1.9  Propositional calculus: miscellaneous   bj-imbi12 33814
            *20.15.2  Modal logic   bj-axdd2 33824
            *20.15.3  Provability logic   cprvb 33829
            *20.15.4  First-order logic   bj-genr 33838
                  20.15.4.1  Adding ax-gen   bj-genr 33838
                  20.15.4.2  Adding ax-4   bj-2alim 33842
                  20.15.4.3  Adding ax-5   bj-ax12wlem 33875
                  20.15.4.4  Equality and substitution   bj-ssbeq 33884
                  20.15.4.5  Adding ax-6   bj-spimvwt 33900
                  20.15.4.6  Adding ax-7   bj-cbvexw 33907
                  20.15.4.7  Membership predicate, ax-8 and ax-9   bj-ax89 33909
                  20.15.4.8  Adding ax-11   bj-alcomexcom 33912
                  20.15.4.9  Adding ax-12   axc11n11 33914
                  20.15.4.10  Nonfreeness   wnnf 33953
                  20.15.4.11  Adding ax-13   bj-axc10 34003
                  *20.15.4.12  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 34013
                  *20.15.4.13  Distinct var metavariables   bj-hbaeb2 34039
                  *20.15.4.14  Around ~ equsal   bj-equsal1t 34043
                  *20.15.4.15  Some Principia Mathematica proofs   stdpc5t 34048
                  20.15.4.16  Alternate definition of substitution   bj-sbsb 34058
                  20.15.4.17  Lemmas for substitution   bj-sbf3 34060
                  20.15.4.18  Existential uniqueness   bj-eu3f 34063
                  *20.15.4.19  First-order logic: miscellaneous   bj-sblem1 34064
            20.15.5  Set theory   eliminable1 34080
                  *20.15.5.1  Eliminability of class terms   eliminable1 34080
                  *20.15.5.2  Classes without the axiom of extensionality   bj-denotes 34086
                  20.15.5.3  Characterization among sets versus among classes   elelb 34111
                  *20.15.5.4  The nonfreeness quantifier for classes   bj-nfcsym 34113
                  *20.15.5.5  Proposal for the definitions of class membership and class equality   bj-ax9 34114
                  *20.15.5.6  Lemmas for class substitution   bj-sbeqALT 34115
                  20.15.5.7  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 34125
                  *20.15.5.8  Class abstractions   bj-unrab 34142
                  *20.15.5.9  Restricted nonfreeness   wrnf 34149
                  *20.15.5.10  Russell's paradox   bj-ru0 34151
                  20.15.5.11  Curry's paradox in set theory   currysetlem 34154
                  *20.15.5.12  Some disjointness results   bj-n0i 34160
                  *20.15.5.13  Complements on direct products   bj-xpimasn 34165
                  *20.15.5.14  "Singletonization" and tagging   bj-snsetex 34173
                  *20.15.5.15  Tuples of classes   bj-cproj 34200
                  *20.15.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 34235
                  *20.15.5.17  Set theory: miscellaneous   bj-elpwg 34240
                  *20.15.5.18  Evaluation   bj-evaleq 34258
                  20.15.5.19  Elementwise operations   celwise 34265
                  *20.15.5.20  Elementwise intersection (families of sets induced on a subset)   bj-rest00 34267
                  20.15.5.21  Moore collections (complements)   bj-intss 34286
                  20.15.5.22  Maps-to notation for functions with three arguments   bj-0nelmpt 34301
                  *20.15.5.23  Currying   csethom 34307
                  *20.15.5.24  Setting components of extensible structures   cstrset 34319
            *20.15.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 34322
                  20.15.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 34322
                  *20.15.6.2  Identity relation (complements)   bj-opabssvv 34335
                  *20.15.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 34357
                  *20.15.6.4  Direct image and inverse image   cimdir 34363
                  *20.15.6.5  Extended numbers and projective lines as sets   cfractemp 34371
                  *20.15.6.6  Addition and opposite   caddcc 34412
                  *20.15.6.7  Order relation on the extended reals   cltxr 34416
                  *20.15.6.8  Argument, multiplication and inverse   carg 34418
                  20.15.6.9  The canonical bijection from the finite ordinals   ciomnn 34424
                  20.15.6.10  Divisibility   cnnbar 34435
            *20.15.7  Monoids   bj-cmnssmnd 34443
                  *20.15.7.1  Finite sums in monoids   cfinsum 34454
            *20.15.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 34457
                  *20.15.8.1  Real vector spaces   bj-fvimacnv0 34457
                  *20.15.8.2  Complex numbers (supplements)   bj-subcom 34478
                  *20.15.8.3  Barycentric coordinates   bj-bary1lem 34480
      20.16  Mathbox for Jim Kingdon
                  20.16.0.1  Circle constant   taupilem3 34483
                  20.16.0.2  Number theory   dfgcd3 34488
      20.17  Mathbox for ML
            20.17.1  Miscellaneous   csbdif 34489
            20.17.2  Cartesian exponentiation   cfinxp 34547
            20.17.3  Topology   iunctb2 34567
                  *20.17.3.1  Pi-base theorems   pibp16 34577
      20.18  Mathbox for Wolf Lammen
            20.18.1  1. Bootstrapping   wl-section-boot 34586
            20.18.2  Implication chains   wl-section-impchain 34610
            20.18.3  An alternative axiom ~ ax-13   ax-wl-13v 34628
            20.18.4  Other stuff   wl-mps 34630
            20.18.5  1. Restricted Quantifiers   wl-ral 34713
      20.19  Mathbox for Brendan Leahy
      20.20  Mathbox for Jeff Madsen
            20.20.1  Logic and set theory   anim12da 34870
            20.20.2  Real and complex numbers; integers   filbcmb 34898
            20.20.3  Sequences and sums   sdclem2 34900
            20.20.4  Topology   subspopn 34910
            20.20.5  Metric spaces   metf1o 34913
            20.20.6  Continuous maps and homeomorphisms   constcncf 34920
            20.20.7  Boundedness   ctotbnd 34927
            20.20.8  Isometries   cismty 34959
            20.20.9  Heine-Borel Theorem   heibor1lem 34970
            20.20.10  Banach Fixed Point Theorem   bfplem1 34983
            20.20.11  Euclidean space   crrn 34986
            20.20.12  Intervals (continued)   ismrer1 34999
            20.20.13  Operation properties   cass 35003
            20.20.14  Groups and related structures   cmagm 35009
            20.20.15  Group homomorphism and isomorphism   cghomOLD 35044
            20.20.16  Rings   crngo 35055
            20.20.17  Division Rings   cdrng 35109
            20.20.18  Ring homomorphisms   crnghom 35121
            20.20.19  Commutative rings   ccm2 35150
            20.20.20  Ideals   cidl 35168
            20.20.21  Prime rings and integral domains   cprrng 35207
            20.20.22  Ideal generators   cigen 35220
      20.21  Mathbox for Giovanni Mascellani
            *20.21.1  Tools for automatic proof building   efald2 35239
            *20.21.2  Tseitin axioms   fald 35290
            *20.21.3  Equality deductions   iuneq2f 35317
            *20.21.4  Miscellanea   orcomdd 35328
      20.22  Mathbox for Peter Mazsa
            20.22.1  Notations   cxrn 35335
            20.22.2  Preparatory theorems   el2v1 35373
            20.22.3  Range Cartesian product   df-xrn 35505
            20.22.4  Cosets by ` R `   df-coss 35541
            20.22.5  Relations   df-rels 35607
            20.22.6  Subset relations   df-ssr 35620
            20.22.7  Reflexivity   df-refs 35632
            20.22.8  Converse reflexivity   df-cnvrefs 35645
            20.22.9  Symmetry   df-syms 35660
            20.22.10  Reflexivity and symmetry   symrefref2 35681
            20.22.11  Transitivity   df-trs 35690
            20.22.12  Equivalence relations   df-eqvrels 35701
            20.22.13  Redundancy   df-redunds 35740
            20.22.14  Domain quotients   df-dmqss 35755
            20.22.15  Equivalence relations on domain quotients   df-ers 35779
            20.22.16  Functions   df-funss 35795
            20.22.17  Disjoints vs. converse functions   df-disjss 35818
      20.23  Mathbox for Rodolfo Medina
            20.23.1  Partitions   prtlem60 35871
      *20.24  Mathbox for Norm Megill
            *20.24.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 35901
            *20.24.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 35911
            *20.24.3  Legacy theorems using obsolete axioms   ax5ALT 35925
            20.24.4  Experiments with weak deduction theorem   elimhyps 35979
            20.24.5  Miscellanea   cnaddcom 35990
            20.24.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 35992
            20.24.7  Functionals and kernels of a left vector space (or module)   clfn 36075
            20.24.8  Opposite rings and dual vector spaces   cld 36141
            20.24.9  Ortholattices and orthomodular lattices   cops 36190
            20.24.10  Atomic lattices with covering property   ccvr 36280
            20.24.11  Hilbert lattices   chlt 36368
            20.24.12  Projective geometries based on Hilbert lattices   clln 36509
            20.24.13  Construction of a vector space from a Hilbert lattice   cdlema1N 36809
            20.24.14  Construction of involution and inner product from a Hilbert lattice   clpoN 38498
      20.25  Mathbox for Steven Nguyen
            20.25.1  Utility theorems   ioin9i8 38980
            *20.25.2  Arithmetic theorems   c0exALT 39032
            20.25.3  Exponents   oexpreposd 39059
            20.25.4  Real subtraction   cresub 39075
            *20.25.5  Projective spaces   cprjsp 39131
            20.25.6  Equivalent formulations of Fermat's Last Theorem   dffltz 39151
      20.26  Mathbox for Igor Ieskov
      20.27  Mathbox for OpenAI
      20.28  Mathbox for Stefan O'Rear
            20.28.1  Additional elementary logic and set theory   moxfr 39169
            20.28.2  Additional theory of functions   imaiinfv 39170
            20.28.3  Additional topology   elrfi 39171
            20.28.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 39175
            20.28.5  Algebraic closure systems   cnacs 39179
            20.28.6  Miscellanea 1. Map utilities   constmap 39190
            20.28.7  Miscellanea for polynomials   mptfcl 39197
            20.28.8  Multivariate polynomials over the integers   cmzpcl 39198
            20.28.9  Miscellanea for Diophantine sets 1   coeq0i 39230
            20.28.10  Diophantine sets 1: definitions   cdioph 39232
            20.28.11  Diophantine sets 2 miscellanea   ellz1 39244
            20.28.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 39249
            20.28.13  Diophantine sets 3: construction   diophrex 39252
            20.28.14  Diophantine sets 4 miscellanea   2sbcrex 39261
            20.28.15  Diophantine sets 4: Quantification   rexrabdioph 39271
            20.28.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 39278
            20.28.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 39288
            20.28.18  Pigeonhole Principle and cardinality helpers   fphpd 39293
            20.28.19  A non-closed set of reals is infinite   rencldnfilem 39297
            20.28.20  Lagrange's rational approximation theorem   irrapxlem1 39299
            20.28.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 39306
            20.28.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 39313
            20.28.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 39355
            *20.28.24  Logarithm laws generalized to an arbitrary base   reglogcl 39367
            20.28.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 39375
            20.28.26  X and Y sequences 1: Definition and recurrence laws   crmx 39377
            20.28.27  Ordering and induction lemmas for the integers   monotuz 39418
            20.28.28  X and Y sequences 2: Order properties   rmxypos 39424
            20.28.29  Congruential equations   congtr 39442
            20.28.30  Alternating congruential equations   acongid 39452
            20.28.31  Additional theorems on integer divisibility   coprmdvdsb 39462
            20.28.32  X and Y sequences 3: Divisibility properties   jm2.18 39465
            20.28.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 39482
            20.28.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 39492
            20.28.35  Uncategorized stuff not associated with a major project   setindtr 39501
            20.28.36  More equivalents of the Axiom of Choice   axac10 39510
            20.28.37  Finitely generated left modules   clfig 39547
            20.28.38  Noetherian left modules I   clnm 39555
            20.28.39  Addenda for structure powers   pwssplit4 39569
            20.28.40  Every set admits a group structure iff choice   unxpwdom3 39575
            20.28.41  Noetherian rings and left modules II   clnr 39589
            20.28.42  Hilbert's Basis Theorem   cldgis 39601
            20.28.43  Additional material on polynomials [DEPRECATED]   cmnc 39611
            20.28.44  Degree and minimal polynomial of algebraic numbers   cdgraa 39620
            20.28.45  Algebraic integers I   citgo 39637
            20.28.46  Endomorphism algebra   cmend 39655
            20.28.47  Cyclic groups and order   idomrootle 39675
            20.28.48  Cyclotomic polynomials   ccytp 39682
            20.28.49  Miscellaneous topology   fgraphopab 39690
      20.29  Mathbox for Jon Pennant
      20.30  Mathbox for Richard Penner
            20.30.1  Short Studies   ifpan123g 39704
                  20.30.1.1  Additional work on conditional logical operator   ifpan123g 39704
                  20.30.1.2  Sophisms   rp-fakeimass 39758
                  *20.30.1.3  Finite Sets   rp-isfinite5 39763
                  20.30.1.4  General Observations   intabssd 39765
                  20.30.1.5  Infinite Sets   pwelg 39799
                  *20.30.1.6  Finite intersection property   fipjust 39804
                  20.30.1.7  RP ADDTO: Subclasses and subsets   rababg 39813
                  20.30.1.8  RP ADDTO: The intersection of a class   elintabg 39814
                  20.30.1.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 39817
                  20.30.1.10  RP ADDTO: Relations   xpinintabd 39820
                  *20.30.1.11  RP ADDTO: Functions   elmapintab 39836
                  *20.30.1.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 39840
                  20.30.1.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 39841
                  20.30.1.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 39844
                  20.30.1.15  RP ADDTO: Basic properties of closures   cleq2lem 39848
                  20.30.1.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 39870
            20.30.2  Additional statements on relations and subclasses   al3im 39871
                  20.30.2.1  Transitive relations (not to be confused with transitive classes).   trrelind 39890
                  20.30.2.2  Reflexive closures   crcl 39897
                  *20.30.2.3  Finite relationship composition.   relexp2 39902
                  20.30.2.4  Transitive closure of a relation   dftrcl3 39945
                  *20.30.2.5  Adapted from Frege   frege77d 39971
            *20.30.3  Propositions from _Begriffsschrift_   dfxor4 39991
                  *20.30.3.1  _Begriffsschrift_ Chapter I   dfxor4 39991
                  *20.30.3.2  _Begriffsschrift_ Notation hints   rp-imass 39997
                  20.30.3.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 40016
                  20.30.3.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 40055
                  *20.30.3.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 40082
                  20.30.3.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 40113
                  *20.30.3.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 40140
                  *20.30.3.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 40158
                  *20.30.3.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 40165
                  *20.30.3.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 40188
                  *20.30.3.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 40204
            *20.30.4  Exploring Topology via Seifert and Threlfall   enrelmap 40223
                  *20.30.4.1  Equinumerosity of sets of relations and maps   enrelmap 40223
                  *20.30.4.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   sscon34b 40249
                  *20.30.4.3  Generic Neighborhood Spaces   gneispa 40360
            *20.30.5  Exploring Higher Homotopy via Kerodon   k0004lem1 40377
                  *20.30.5.1  Simplicial Sets   k0004lem1 40377
      20.31  Mathbox for Stanislas Polu
            20.31.1  IMO Problems   wwlemuld 40386
                  20.31.1.1  IMO 1972 B2   wwlemuld 40386
            *20.31.2  INT Inequalities Proof Generator   int-addcomd 40407
            *20.31.3  N-Digit Addition Proof Generator   unitadd 40429
            20.31.4  AM-GM (for k = 2,3,4)   gsumws3 40430
      20.32  Mathbox for Rohan Ridenour
            20.32.1  Misc   spALT 40435
            20.32.2  Shorter primitive equivalent of ax-groth   gru0eld 40445
                  20.32.2.1  Grothendieck universes are closed under collection   gru0eld 40445
                  20.32.2.2  Minimal universes   ismnu 40477
                  20.32.2.3  Primitive equivalent of ax-groth   expandan 40504
      20.33  Mathbox for Steve Rodriguez
            20.33.1  Miscellanea   nanorxor 40517
            20.33.2  Ratio test for infinite series convergence and divergence   dvgrat 40524
            20.33.3  Multiples   reldvds 40527
            20.33.4  Function operations   caofcan 40535
            20.33.5  Calculus   lhe4.4ex1a 40541
            20.33.6  The generalized binomial coefficient operation   cbcc 40548
            20.33.7  Binomial series   uzmptshftfval 40558
      20.34  Mathbox for Andrew Salmon
            20.34.1  Principia Mathematica * 10   pm10.12 40570
            20.34.2  Principia Mathematica * 11   2alanimi 40584
            20.34.3  Predicate Calculus   sbeqal1 40610
            20.34.4  Principia Mathematica * 13 and * 14   pm13.13a 40619
            20.34.5  Set Theory   elnev 40650
            20.34.6  Arithmetic   addcomgi 40668
            20.34.7  Geometry   cplusr 40669
      *20.35  Mathbox for Alan Sare
            20.35.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 40691
            20.35.2  Supplementary unification deductions   bi1imp 40695
            20.35.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 40715
            20.35.4  What is Virtual Deduction?   wvd1 40783
            20.35.5  Virtual Deduction Theorems   df-vd1 40784
            20.35.6  Theorems proved using Virtual Deduction   trsspwALT 41032
            20.35.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 41060
            20.35.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 41127
            20.35.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 41131
            20.35.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 41138
            *20.35.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 41141
      20.36  Mathbox for Glauco Siliprandi
            20.36.1  Miscellanea   evth2f 41152
            20.36.2  Functions   feq1dd 41303
            20.36.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 41420
            20.36.4  Real intervals   gtnelioc 41645
            20.36.5  Finite sums   fsumclf 41730
            20.36.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 41741
            20.36.7  Limits   clim1fr1 41762
                  20.36.7.1  Inferior limit (lim inf)   clsi 41912
                  *20.36.7.2  Limits for sequences of extended real numbers   clsxlim 41979
            20.36.8  Trigonometry   coseq0 42025
            20.36.9  Continuous Functions   mulcncff 42031
            20.36.10  Derivatives   dvsinexp 42075
            20.36.11  Integrals   itgsin0pilem1 42115
            20.36.12  Stone Weierstrass theorem - real version   stoweidlem1 42167
            20.36.13  Wallis' product for π   wallispilem1 42231
            20.36.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 42240
            20.36.15  Dirichlet kernel   dirkerval 42257
            20.36.16  Fourier Series   fourierdlem1 42274
            20.36.17  e is transcendental   elaa2lem 42399
            20.36.18  n-dimensional Euclidean space   rrxtopn 42450
            20.36.19  Basic measure theory   csalg 42474
                  *20.36.19.1  σ-Algebras   csalg 42474
                  20.36.19.2  Sum of nonnegative extended reals   csumge0 42525
                  *20.36.19.3  Measures   cmea 42612
                  *20.36.19.4  Outer measures and Caratheodory's construction   come 42652
                  *20.36.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 42699
                  *20.36.19.6  Measurable functions   csmblfn 42858
      20.37  Mathbox for Saveliy Skresanov
            20.37.1  Ceva's theorem   sigarval 42988
            20.37.2  Simple groups   simpcntrab 43008
      20.38  Mathbox for Jarvin Udandy
      20.39  Mathbox for Adhemar
            *20.39.1  Minimal implicational calculus   adh-minim 43118
      20.40  Mathbox for Alexander van der Vekens
            20.40.1  General auxiliary theorems (1)   eusnsn 43142
                  20.40.1.1  Unordered and ordered pairs - extension for singletons   eusnsn 43142
                  20.40.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 43145
                  20.40.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 43146
                  20.40.1.4  Relations - extension   eubrv 43151
                  20.40.1.5  Definite description binder (inverted iota) - extension   iota0def 43154
                  20.40.1.6  Functions - extension   fveqvfvv 43156
            20.40.2  Alternative for Russell's definition of a description binder   caiota 43164
            20.40.3  Double restricted existential uniqueness   r19.32 43177
                  20.40.3.1  Restricted quantification (extension)   r19.32 43177
                  20.40.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 43187
                  20.40.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 43190
                  20.40.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 43193
            *20.40.4  Alternative definitions of function and operation values   wdfat 43196
                  20.40.4.1  Restricted quantification (extension)   ralbinrald 43202
                  20.40.4.2  The universal class (extension)   nvelim 43203
                  20.40.4.3  Introduce the Axiom of Power Sets (extension)   alneu 43204
                  20.40.4.4  Predicate "defined at"   dfateq12d 43206
                  20.40.4.5  Alternative definition of the value of a function   dfafv2 43212
                  20.40.4.6  Alternative definition of the value of an operation   aoveq123d 43258
            *20.40.5  Alternative definitions of function values (2)   cafv2 43288
            20.40.6  General auxiliary theorems (2)   an4com24 43348
                  20.40.6.1  Logical conjunction - extension   an4com24 43348
                  20.40.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 43349
                  20.40.6.3  Negated membership (alternative)   cnelbr 43351
                  20.40.6.4  The empty set - extension   ralralimp 43358
                  20.40.6.5  Indexed union and intersection - extension   otiunsndisjX 43359
                  20.40.6.6  Functions - extension   fvifeq 43360
                  20.40.6.7  Maps-to notation - extension   fvmptrab 43372
                  20.40.6.8  Ordering on reals - extension   leltletr 43374
                  20.40.6.9  Subtraction - extension   cnambpcma 43375
                  20.40.6.10  Ordering on reals (cont.) - extension   leaddsuble 43378
                  20.40.6.11  Imaginary and complex number properties - extension   readdcnnred 43384
                  20.40.6.12  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 43389
                  20.40.6.13  Integers (as a subset of complex numbers) - extension   zgeltp1eq 43390
                  20.40.6.14  Decimal arithmetic - extension   1t10e1p1e11 43391
                  20.40.6.15  Upper sets of integers - extension   eluzge0nn0 43393
                  20.40.6.16  Infinity and the extended real number system (cont.) - extension   nltle2tri 43394
                  20.40.6.17  Finite intervals of integers - extension   ssfz12 43395
                  20.40.6.18  Half-open integer ranges - extension   fzopred 43403
                  20.40.6.19  The modulo (remainder) operation - extension   m1mod0mod1 43410
                  20.40.6.20  The infinite sequence builder "seq"   smonoord 43412
                  20.40.6.21  Finite and infinite sums - extension   fsummsndifre 43413
                  20.40.6.22  Extensible structures - extension   setsidel 43417
            *20.40.7  Partitions of real intervals   ciccp 43420
            20.40.8  Shifting functions with an integer range domain   fargshiftfv 43446
            20.40.9  Words over a set (extension)   lswn0 43451
                  20.40.9.1  Last symbol of a word - extension   lswn0 43451
            20.40.10  Unordered pairs   wich 43452
                  20.40.10.1  Interchangeable setvar variables   wich 43452
                  20.40.10.2  Set of unordered pairs   sprid 43483
                  *20.40.10.3  Proper (unordered) pairs   prpair 43510
                  20.40.10.4  Set of proper unordered pairs   cprpr 43521
            20.40.11  Number theory (extension)   cfmtno 43536
                  *20.40.11.1  Fermat numbers   cfmtno 43536
                  *20.40.11.2  Mersenne primes   m2prm 43600
                  20.40.11.3  Proth's theorem   modexp2m1d 43624
                  20.40.11.4  Solutions of quadratic equations   quad1 43632
            *20.40.12  Even and odd numbers   ceven 43636
                  20.40.12.1  Definitions and basic properties   ceven 43636
                  20.40.12.2  Alternate definitions using the "divides" relation   dfeven2 43661
                  20.40.12.3  Alternate definitions using the "modulo" operation   dfeven3 43670
                  20.40.12.4  Alternate definitions using the "gcd" operation   iseven5 43676
                  20.40.12.5  Theorems of part 5 revised   zneoALTV 43681
                  20.40.12.6  Theorems of part 6 revised   odd2np1ALTV 43686
                  20.40.12.7  Theorems of AV's mathbox revised   0evenALTV 43700
                  20.40.12.8  Additional theorems   epoo 43715
                  20.40.12.9  Perfect Number Theorem (revised)   perfectALTVlem1 43733
            20.40.13  Number theory (extension 2)   cfppr 43736
                  *20.40.13.1  Fermat pseudoprimes   cfppr 43736
                  *20.40.13.2  Goldbach's conjectures   cgbe 43757
            20.40.14  Graph theory (extension)   cgrisom 43830
                  *20.40.14.1  Isomorphic graphs   cgrisom 43830
                  20.40.14.2  Loop-free graphs - extension   1hegrlfgr 43854
                  20.40.14.3  Walks - extension   cupwlks 43855
                  20.40.14.4  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 43865
            20.40.15  Monoids (extension)   ovn0dmfun 43878
                  20.40.15.1  Auxiliary theorems   ovn0dmfun 43878
                  20.40.15.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 43886
                  20.40.15.3  Magma homomorphisms and submagmas   cmgmhm 43891
                  20.40.15.4  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 43921
                  20.40.15.5  Group sum operation (extension 1)   gsumsplit2f 43934
                  *20.40.15.6  Monoid of endofunctions   cefmnd 43937
            *20.40.16  Magmas and internal binary operations (alternate approach)   ccllaw 43988
                  *20.40.16.1  Laws for internal binary operations   ccllaw 43988
                  *20.40.16.2  Internal binary operations   cintop 44001
                  20.40.16.3  Alternative definitions for magmas and semigroups   cmgm2 44020
            20.40.17  Categories (extension)   idfusubc0 44034
                  20.40.17.1  Subcategories (extension)   idfusubc0 44034
            20.40.18  Rings (extension)   lmod0rng 44037
                  20.40.18.1  Nonzero rings (extension)   lmod0rng 44037
                  *20.40.18.2  Non-unital rings ("rngs")   crng 44043
                  20.40.18.3  Rng homomorphisms   crngh 44054
                  20.40.18.4  Ring homomorphisms (extension)   rhmfn 44087
                  20.40.18.5  Ideals as non-unital rings   lidldomn1 44090
                  20.40.18.6  The non-unital ring of even integers   0even 44100
                  20.40.18.7  A constructed not unital ring   cznrnglem 44122
                  *20.40.18.8  The category of non-unital rings   crngc 44126
                  *20.40.18.9  The category of (unital) rings   cringc 44172
                  20.40.18.10  Subcategories of the category of rings   srhmsubclem1 44242
            20.40.19  Basic algebraic structures (extension)   opeliun2xp 44279
                  20.40.19.1  Auxiliary theorems   opeliun2xp 44279
                  20.40.19.2  The binomial coefficient operation (extension)   bcpascm1 44297
                  20.40.19.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 44300
                  20.40.19.4  Group sum operation (extension 2)   mgpsumunsn 44307
                  20.40.19.5  Symmetric groups (extension)   exple2lt6 44310
                  20.40.19.6  Divisibility (extension)   invginvrid 44313
                  20.40.19.7  The support of functions (extension)   rmsupp0 44314
                  20.40.19.8  Finitely supported functions (extension)   rmsuppfi 44319
                  20.40.19.9  Left modules (extension)   lmodvsmdi 44328
                  20.40.19.10  Associative algebras (extension)   ascl1 44330
                  20.40.19.11  Univariate polynomials (extension)   ply1vr1smo 44333
                  20.40.19.12  Univariate polynomials (examples)   linply1 44345
            20.40.20  Linear algebra (extension)   cdmatalt 44349
                  *20.40.20.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 44349
                  *20.40.20.2  Linear combinations   clinc 44357
                  *20.40.20.3  Linear independence   clininds 44393
                  20.40.20.4  Simple left modules and the ` ZZ `-module   lmod1lem1 44440
                  20.40.20.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 44460
            20.40.21  Complexity theory   suppdm 44463
                  20.40.21.1  Auxiliary theorems   suppdm 44463
                  20.40.21.2  The modulo (remainder) operation (extension)   fldivmod 44476
                  20.40.21.3  Even and odd integers   nn0onn0ex 44481
                  20.40.21.4  The natural logarithm on complex numbers (extension)   logcxp0 44493
                  20.40.21.5  Division of functions   cfdiv 44495
                  20.40.21.6  Upper bounds   cbigo 44505
                  20.40.21.7  Logarithm to an arbitrary base (extension)   rege1logbrege0 44516
                  *20.40.21.8  The binary logarithm   fldivexpfllog2 44523
                  20.40.21.9  Binary length   cblen 44527
                  *20.40.21.10  Digits   cdig 44553
                  20.40.21.11  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 44573
                  20.40.21.12  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 44582
            20.40.22  Elementary geometry (extension)   fv1prop 44584
                  20.40.22.1  Auxiliary theorems   fv1prop 44584
                  20.40.22.2  Real euclidean space of dimension 2   rrx2pxel 44596
                  20.40.22.3  Spheres and lines in real Euclidean spaces   cline 44612
      20.41  Mathbox for Emmett Weisz
            *20.41.1  Miscellaneous Theorems   nfintd 44674
            20.41.2  Set Recursion   csetrecs 44684
                  *20.41.2.1  Basic Properties of Set Recursion   csetrecs 44684
                  20.41.2.2  Examples and properties of set recursion   elsetrecslem 44699
            *20.41.3  Construction of Games and Surreal Numbers   cpg 44709
      *20.42  Mathbox for David A. Wheeler
            20.42.1  Natural deduction   sbidd 44715
            *20.42.2  Greater than, greater than or equal to.   cge-real 44717
            *20.42.3  Hyperbolic trigonometric functions   csinh 44727
            *20.42.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 44738
            *20.42.5  Identities for "if"   ifnmfalse 44760
            *20.42.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 44761
            *20.42.7  Logarithm laws generalized to an arbitrary base - log_   clog- 44762
            *20.42.8  Formally define terms such as Reflexivity   wreflexive 44764
            *20.42.9  Algebra helpers   comraddi 44768
            *20.42.10  Algebra helper examples   i2linesi 44777
            *20.42.11  Formal methods "surprises"   alimp-surprise 44779
            *20.42.12  Allsome quantifier   walsi 44785
            *20.42.13  Miscellaneous   5m4e1 44796
            20.42.14  Theorems about algebraic numbers   aacllem 44800
      20.43  Mathbox for Kunhao Zheng
            20.43.1  Weighted AM-GM inequality   amgmwlem 44801

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