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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 of 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  Predicate calculus with equality: Older axiomatization (1 rule, 14 schemes)
      1.7  Existential uniqueness
      1.8  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  Elementary real and complex functions
      5.9  Elementary limits and convergence
      5.10  Elementary trigonometry
      5.11  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  Presets 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  Abelian groups
      10.4  Rings
      10.5  Division rings and fields
      10.6  Left modules
      10.7  Vector spaces
      10.8  Ideals
      10.9  Associative algebras
      10.10  Abstract multivariate polynomials
      10.11  The complex numbers as an algebraic extensible structure
      10.12  Generalized pre-Hilbert and Hilbert spaces
PART 11  BASIC TOPOLOGY
      11.1  Topology
      11.2  Filters and filter bases
      11.3  Uniform Structures and Spaces
      11.4  Metric spaces
      11.5  Complex metric vector spaces
PART 12  BASIC REAL AND COMPLEX ANALYSIS
      12.1  Continuity
      12.2  Integrals
      12.3  Derivatives
PART 13  BASIC REAL AND COMPLEX FUNCTIONS
      13.1  Polynomials
      13.2  Sequences and series
      13.3  Basic trigonometry
      13.4  Basic number theory
PART 14  GRAPH THEORY
      14.1  Undirected graphs - basics
      14.2  Eulerian paths and the Konigsberg Bridge problem
PART 15  GUIDES AND MISCELLANEA
      15.1  Guides (conventions, explanations, and examples)
      15.2  Humor
      15.3  (Future - to be reviewed and classified)
PART 16  ADDITIONAL MATERIAL ON GROUPS, RINGS, AND FIELDS (DEPRECATED)
      16.1  Additional material on group theory
      16.2  Additional material on rings and fields
PART 17  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      17.1  Complex vector spaces
      17.2  Normed complex vector spaces
      17.3  Operators on complex vector spaces
      17.4  Inner product (pre-Hilbert) spaces
      17.5  Complex Banach spaces
      17.6  Complex Hilbert spaces
PART 18  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      18.1  Axiomatization of complex pre-Hilbert spaces
      18.2  Inner product and norms
      18.3  Cauchy sequences and completeness axiom
      18.4  Subspaces and projections
      18.5  Properties of Hilbert subspaces
      18.6  Operators on Hilbert spaces
      18.7  States on a Hilbert lattice and Godowski's equation
      18.8  Cover relation, atoms, exchange axiom, and modular symmetry
PART 19  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      19.1  Mathboxes for user contributions
      19.2  Mathbox for Stefan Allan
      19.3  Mathbox for Thierry Arnoux
      19.4  Mathbox for Mario Carneiro
      19.5  Mathbox for Paul Chapman
      19.6  Mathbox for Drahflow
      19.7  Mathbox for Scott Fenton
      19.8  Mathbox for Anthony Hart
      19.9  Mathbox for Chen-Pang He
      19.10  Mathbox for Jeff Hoffman
      19.11  Mathbox for Wolf Lammen
      19.12  Mathbox for Brendan Leahy
      19.13  Mathbox for Jeff Hankins
      19.14  Mathbox for Jeff Madsen
      19.15  Mathbox for Giovanni Mascellani
      19.16  Mathbox for Rodolfo Medina
      19.17  Mathbox for Stefan O'Rear
      19.18  Mathbox for Steve Rodriguez
      19.19  Mathbox for Andrew Salmon
      19.20  Mathbox for Glauco Siliprandi
      19.21  Mathbox for Saveliy Skresanov
      19.22  Mathbox for Jarvin Udandy
      19.23  Mathbox for Alexander van der Vekens
      19.24  Mathbox for David A. Wheeler
      19.25  Mathbox for Alan Sare
      19.26  Mathbox for Jonathan Ben-Naim
      19.27  Mathbox for BJ
      19.28  Mathbox for Norm Megill

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   dummylink 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   con4d 100
            *1.2.5  Logical equivalence   wb 178
            *1.2.6  Logical disjunction and conjunction   wo 359
            1.2.7  Miscellaneous theorems of propositional calculus   pm5.21nd 870
            1.2.8  Abbreviated conjunction and disjunction of three wff's   w3o 938
            1.2.9  Logical 'nand' (Sheffer stroke)   wnan 1304
            1.2.10  Logical 'xor'   wxo 1321
            1.2.11  True and false constants   wtru 1334
            *1.2.12  Truth tables   truantru 1354
            1.2.13  Auxiliary theorems for Alan Sare's virtual deduction tool, part 1   ee22 1380
            *1.2.14  Half-adders and full adders in propositional calculus   whad 1396
      1.3  Other axiomatizations of classical propositional calculus
            1.3.1  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1423
            1.3.2  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1442
            *1.3.3  Derive Nicod's axiom from the standard axioms   nic-dfim 1453
            1.3.4  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1459
            1.3.5  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1478
            1.3.6  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1482
            1.3.7  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1497
            1.3.8  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1520
            1.3.9  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1533
            *1.3.10  Stoic logic indemonstrables (Chrysippus of Soli)   mpto1 1552
      *1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
            1.4.1  Universal quantifier; define "exists" and "not free"   wal 1559
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1565
            1.4.3  Axiom scheme ax-5 (Quantified Implication)   ax-5 1576
            1.4.4  Axiom scheme ax-17 (Distinctness) - first use of $d   ax-17 1639
            1.4.5  Equality predicate; define substitution   cv 1664
            1.4.6  Axiom scheme ax-9 (Existence)   ax-9 1680
            1.4.7  Axiom scheme ax-8 (Equality)   ax-8 1701
            1.4.8  Membership predicate   wcel 1737
            1.4.9  Axiom scheme ax-13 (Left Equality for Binary Predicate)   ax-13 1739
            1.4.10  Axiom scheme ax-14 (Right Equality for Binary Predicate)   ax-14 1741
            *1.4.11  Logical redundancy of ax-6 , ax-7 , ax-11 , ax-12   ax9dgen 1743
      *1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-6 (Quantified Negation)   ax-6 1756
            1.5.2  Axiom scheme ax-7 (Quantifier Commutation)   ax-7 1761
            1.5.3  Axiom scheme ax-11 (Substitution)   ax-11 1773
            1.5.4  Axiom scheme ax-12 (Quantified Equality)   ax-12 1966
      *1.6  Predicate calculus with equality: Older axiomatization (1 rule, 14 schemes)
            *1.6.1  Obsolete schemes ax-5o ax-4 ax-6o ax-9o ax-10o ax-10 ax-11o ax-12o ax-15 ax-16   ax-4 2257
            *1.6.2  Rederive new axioms from old: ax5 , ax6 , ax9from9o , ax11 , ax12from12o   ax4 2267
            *1.6.3  Legacy theorems using obsolete axioms   ax17o 2279
      1.7  Existential uniqueness
      1.8  Other axiomatizations related to classical predicate calculus
            1.8.1  Predicate calculus with all distinct variables   ax-7d 2427
            *1.8.2  Aristotelian logic: Assertic syllogisms   barbara 2433
            *1.8.3  Intuitionistic logic   axia1 2457
*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 2472
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2477
            2.1.3  Class form not-free predicate   wnfc 2614
            2.1.4  Negated equality and membership   wne 2654
                  2.1.4.1  Negated equality   neii 2658
                  2.1.4.2  Negated membership   neli 2753
            2.1.5  Restricted quantification   wral 2761
            2.1.6  The universal class   cvv 3016
            *2.1.7  Conditional equality (experimental)   wcdeq 3205
            2.1.8  Russell's Paradox   ru 3221
            2.1.9  Proper substitution of classes for sets   wsbc 3222
            2.1.10  Proper substitution of classes for sets into classes   csb 3323
            2.1.11  Define basic set operations and relations   cdif 3360
            2.1.12  Subclasses and subsets   df-ss 3377
            2.1.13  The difference, union, and intersection of two classes   difeq1 3501
                  2.1.13.1  The difference of two classes   difeq1 3501
                  2.1.13.2  The union of two classes   elun 3531
                  2.1.13.3  The intersection of two classes   elin 3573
                  2.1.13.4  Combinations of difference, union, and intersection of two classes   unabs 3613
                  2.1.13.5  Class abstractions with difference, union, and intersection of two classes   symdif2 3649
                  2.1.13.6  Restricted uniqueness with difference, union, and intersection   reuss2 3663
            2.1.14  The empty set   c0 3670
            *2.1.15  "Weak deduction theorem" for set theory   cif 3822
            2.1.16  Power classes   cpw 3888
            2.1.17  Unordered and ordered pairs   csn 3903
            2.1.18  The union of a class   cuni 4108
            2.1.19  The intersection of a class   cint 4144
            2.1.20  Indexed union and intersection   ciun 4187
            2.1.21  Disjointness   wdisj 4278
            2.1.22  Binary relations   wbr 4308
            2.1.23  Ordered-pair class abstractions (class builders)   copab 4365
            2.1.24  Transitive classes   wtr 4401
      2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 4419
            2.2.2  Derive the Axiom of Separation   axsep 4428
            2.2.3  Derive the Null Set Axiom   zfnuleu 4434
            2.2.4  Theorems requiring subset and intersection existence   nalset 4444
            2.2.5  Theorems requiring empty set existence   class2set 4471
      2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4482
            2.3.2  Derive the Axiom of Pairing   zfpair 4506
            2.3.3  Ordered pair theorem   opnz 4537
            2.3.4  Ordered-pair class abstractions (cont.)   opabid 4567
            2.3.5  Power class of union and intersection   pwin 4595
            2.3.6  Epsilon and identity relations   cep 4600
            2.3.7  Partial and complete ordering   wpo 4609
            2.3.8  Founded and well-ordering relations   wfr 4646
            2.3.9  Ordinals   word 4688
      2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4809
            2.4.2  Ordinals (continued)   ordon 4871
            2.4.3  Transfinite induction   tfi 4941
            2.4.4  The natural numbers (i.e. finite ordinals)   com 4953
            2.4.5  Peano's postulates   peano1 4972
            2.4.6  Finite induction (for finite ordinals)   find 4978
            2.4.7  Relations   cxp 4984
            2.4.8  Definite description binder (inverted iota)   cio 5532
            2.4.9  Functions   wfun 5565
            2.4.10  Operations   co 6203
            2.4.11  "Maps to" notation   elmpt2cl 6412
            2.4.12  Function operation   cof 6427
            2.4.13  First and second members of an ordered pair   c1st 6471
            *2.4.14  Special "Maps to" operations   mpt2xopn0yelv 6590
            2.4.15  Function transposition   ctpos 6604
            2.4.16  Curry and uncurry   ccur 6643
            2.4.17  Proper subset relation   crpss 6647
            2.4.18  Iota properties   fvopab5 6660
            2.4.19  Cantor's Theorem   canth 6665
            2.4.20  Undefined values and restricted iota (description binder)   cund 6667
            2.4.21  Functions on ordinals; strictly monotone ordinal functions   iunon 6717
            2.4.22  "Strong" transfinite recursion   crecs 6749
            2.4.23  Recursive definition generator   crdg 6784
            2.4.24  Finite recursion   frfnom 6809
            2.4.25  Abian's "most fundamental" fixed point theorem   abianfplem 6832
            2.4.26  Ordinal arithmetic   c1o 6834
            2.4.27  Natural number arithmetic   nna0 6964
            2.4.28  Equivalence relations and classes   wer 7019
            2.4.29  The mapping operation   cmap 7135
            2.4.30  Infinite Cartesian products   cixp 7180
            2.4.31  Equinumerosity   cen 7223
            2.4.32  Schroeder-Bernstein Theorem   sbthlem1 7334
            2.4.33  Equinumerosity (cont.)   xpf1o 7386
            2.4.34  Pigeonhole Principle   phplem1 7403
            2.4.35  Finite sets   onomeneq 7413
            2.4.36  Finite intersections   cfi 7532
            2.4.37  Hall's marriage theorem   marypha1lem 7555
            2.4.38  Supremum   csup 7562
            2.4.39  Ordinal isomorphism, Hartog's theorem   coi 7595
            2.4.40  Hartogs function, order types, weak dominance   char 7641
      2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 7677
            2.5.2  Axiom of Infinity equivalents   inf0 7693
      2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 7710
            2.6.2  Existence of omega (the set of natural numbers)   omex 7715
            2.6.3  Cantor normal form   ccnf 7733
            2.6.4  Transitive closure   trcl 7781
            2.6.5  Rank   cr1 7805
            2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 7928
            2.6.7  Cardinal numbers   ccrd 7941
            2.6.8  Axiom of Choice equivalents   wac 8115
            2.6.9  Cardinal number arithmetic   ccda 8166
            2.6.10  The Ackermann bijection   ackbij2lem1 8218
            2.6.11  Cofinality (without Axiom of Choice)   cflem 8245
            2.6.12  Eight inequivalent definitions of finite set   sornom 8276
            2.6.13  Hereditarily size-limited sets without Choice   itunifval 8415
*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 8434
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 8445
      3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 8458
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 8493
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 8540
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 8568
            3.2.5  Cofinality using Axiom of Choice   alephreg 8576
      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 8614
            3.4.2  Derivation of the Axiom of Choice   gchaclem 8672
*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 8676
            4.1.2  Weak universes   cwun 8694
            4.1.3  Tarski's classes   ctsk 8742
            4.1.4  Grothendieck's universes   cgru 8784
      4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 8817
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 8820
            4.2.3  Tarski map function   ctskm 8831
*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 8838
            5.1.2  Final derivation of real and complex number postulates   axaddf 9139
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 9165
      5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 9190
            5.2.2  Infinity and the extended real number system   cpnf 9236
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 9266
            5.2.4  Ordering on reals   lttr 9271
            5.2.5  Initial properties of the complex numbers   mul12 9353
      5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 9400
            5.3.2  Subtraction   cmin 9413
            5.3.3  Multiplication   muladd 9591
            5.3.4  Ordering on reals (cont.)   gt0ne0 9618
            5.3.5  Reciprocals   ixi 9778
            5.3.6  Division   cdiv 9804
            5.3.7  Ordering on reals (cont.)   elimgt0 9973
            5.3.8  Completeness Axiom and Suprema   fimaxre 10082
            5.3.9  Imaginary and complex number properties   inelr 10117
            5.3.10  Function operation analogue theorems   ofsubeq0 10124
      5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 10127
            5.4.2  Principle of mathematical induction   nnind 10145
            *5.4.3  Decimal representation of numbers   c2 10176
            *5.4.4  Some properties of specific numbers   0p1e1 10220
            5.4.5  The Archimedean property   nnunb 10345
            5.4.6  Nonnegative integers (as a subset of complex numbers)   cn0 10349
            5.4.7  Integers (as a subset of complex numbers)   cz 10414
            5.4.8  Decimal arithmetic   cdc 10519
            5.4.9  Upper sets of integers   cuz 10625
            5.4.10  Well-ordering principle for bounded-below sets of integers   uzwo3 10710
            5.4.11  Rational numbers (as a subset of complex numbers)   cq 10715
            5.4.12  Existence of the set of complex numbers   rpnnen1lem1 10741
      5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 10753
            5.5.2  Infinity and the extended real number system (cont.)   cxne 10848
            5.5.3  Supremum on the extended reals   xrsupexmnf 11024
            5.5.4  Real number intervals   cioo 11057
            5.5.5  Finite intervals of integers   cfz 11184
            5.5.6  Half-open integer ranges   cfzo 11291
      5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 11380
            5.6.2  The modulo (remainder) operation   cmo 11448
            5.6.3  The infinite sequence builder "seq"   om2uz0i 11510
            5.6.4  Integer powers   cexp 11605
            5.6.5  Ordered pair theorem for nonnegative integers   nn0le2msqi 11783
            5.6.6  Factorial function   cfa 11789
            5.6.7  The binomial coefficient operation   cbc 11816
            5.6.8  The ` # ` (finite set size) function   chash 11841
                  5.6.8.1  Finite induction on the size of the first component of a binary relation   brfi1indlem 11951
      *5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 11954
            5.7.2  Last symbol of a word   lsw 11999
            5.7.3  Concatenations of words   ccatfn 12005
            5.7.4  Singleton words   ids1 12022
            5.7.5  Concatenations with singleton words   ccatws1cl 12036
            5.7.6  Subwords   swrdval 12046
            5.7.7  Subwords of subwords   swrdswrdlem 12086
            5.7.8  Subwords and concatenations   wrdcctswrd 12092
            5.7.9  Subwords of concatenations   swrdccatfn 12105
            5.7.10  Splicing words (substring replacement)   splval 12125
            5.7.11  Reversing words   revval 12132
            5.7.12  Repeated symbol words   reps 12140
            *5.7.13  Cyclical shifts of words   ccsh 12157
            5.7.14  Mapping words by a function   wrdco 12191
            5.7.15  Longer string literals   cs2 12200
      5.8  Elementary real and complex functions
            5.8.1  The "shift" operation   cshi 12287
            5.8.2  Signum (sgn or sign) function   csgn 12307
            5.8.3  Real and imaginary parts; conjugate   ccj 12317
            5.8.4  Square root; absolute value   csqr 12454
      5.9  Elementary limits and convergence
            5.9.1  Superior limit (lim sup)   clsp 12680
            5.9.2  Limits   cli 12694
            5.9.3  Finite and infinite sums   csu 12895
            5.9.4  The binomial theorem   binomlem 13024
            5.9.5  The inclusion/exclusion principle   incexclem 13031
            5.9.6  Infinite sums (cont.)   isumshft 13034
            5.9.7  Miscellaneous converging and diverging sequences   divrcnv 13047
            5.9.8  Arithmetic series   arisum 13054
            5.9.9  Geometric series   expcnv 13058
            5.9.10  Ratio test for infinite series convergence   cvgrat 13075
            5.9.11  Mertens' theorem   mertenslem1 13076
      5.10  Elementary trigonometry
            5.10.1  The exponential, sine, and cosine functions   ce 13079
            5.10.2  _e is irrational   eirrlem 13218
      5.11  Cardinality of real and complex number subsets
            5.11.1  Countability of integers and rationals   xpnnen 13223
            5.11.2  The reals are uncountable   rpnnen2lem1 13229
*PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqr2irrlem 13262
            6.1.2  Some Number sets are chains of proper subsets   nthruc 13265
            6.1.3  The divides relation   cdivides 13267
            6.1.4  The division algorithm   divalglem0 13328
            6.1.5  Bit sequences   cbits 13346
            6.1.6  The greatest common divisor operator   cgcd 13421
            6.1.7  Bézout's identity   bezoutlem1 13453
            6.1.8  Algorithms   nn0seqcvgd 13476
            6.1.9  Euclid's Algorithm   eucalgval2 13487
      6.2  Elementary prime number theory
            6.2.1  Elementary properties   cprime 13494
            6.2.2  Properties of the canonical representation of a rational   cnumer 13542
            6.2.3  Euler's theorem   codz 13569
            6.2.4  Arithmetic modulo a prime number   modprm1div 13600
            6.2.5  Pythagorean Triples   coprimeprodsq 13607
            6.2.6  The prime count function   cpc 13634
            6.2.7  Pocklington's theorem   prmpwdvds 13696
            6.2.8  Infinite primes theorem   unbenlem 13700
            6.2.9  Sum of prime reciprocals   prmreclem1 13708
            6.2.10  Fundamental theorem of arithmetic   1arithlem1 13715
            6.2.11  Lagrange's four-square theorem   cgz 13721
            6.2.12  Van der Waerden's theorem   cvdwa 13757
            6.2.13  Ramsey's theorem   cram 13791
            6.2.14  Decimal arithmetic (cont.)   dec2dvds 13823
            6.2.15  Cyclical shifts of words (cont.)   cshwsidrepsw 13851
            6.2.16  Specific prime numbers   4nprm 13863
            6.2.17  Very large primes   1259lem1 13886
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
            *7.1.1  Basic definitions   cstr 13901
            7.1.2  Slot definitions   cplusg 13965
            7.1.3  Definition of the structure product   crest 14084
            7.1.4  Definition of the structure quotient   cordt 14157
      7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 14267
            7.2.2  Independent sets in a Moore system   mrisval 14291
            7.2.3  Algebraic closure systems   isacs 14312
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
            8.1.1  Categories   ccat 14325
            8.1.2  Opposite category   coppc 14373
            8.1.3  Monomorphisms and epimorphisms   cmon 14390
            8.1.4  Sections, inverses, isomorphisms   csect 14406
            8.1.5  Subcategories   cssc 14443
            8.1.6  Functors   cfunc 14487
            8.1.7  Full & faithful functors   cful 14535
            8.1.8  Natural transformations and the functor category   cnat 14574
      8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 14644
      8.3  Examples of categories
            8.3.1  The category of sets   csetc 14666
            8.3.2  The category of categories   ccatc 14685
      8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 14701
            8.4.2  Functor evaluation   cevlf 14742
            8.4.3  Hom functor   chof 14781
PART 9  BASIC ORDER THEORY
      9.1  Presets and directed sets using extensible structures
      9.2  Posets and lattices using extensible structures
            9.2.1  Posets   cpo 14833
            9.2.2  Lattices   clat 14938
            9.2.3  The dual of an ordered set   codu 15021
            9.2.4  Subset order structures   cipo 15044
            9.2.5  Distributive lattices   latmass 15081
            9.2.6  Posets and lattices as relations   cps 15091
            9.2.7  Directed sets, nets   cdir 15121
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
            10.1.1  Definition and basic properties   cmnd 15132
            10.1.2  Monoid homomorphisms and submonoids   cmhm 15184
            *10.1.3  Ordered group sum operation   gsumvallem1 15219
            10.1.4  Free monoids   cfrmd 15240
      10.2  Groups
            10.2.1  Definition and basic properties   df-grp 15260
            10.2.2  Subgroups and Quotient groups   csubg 15386
            10.2.3  Elementary theory of group homomorphisms   cghm 15451
            10.2.4  Isomorphisms of groups   cgim 15492
            10.2.5  Group actions   cga 15514
            10.2.6  Symmetry groups and Cayley's Theorem   csymg 15540
            10.2.7  Centralizers and centers   ccntz 15562
            10.2.8  The opposite group   coppg 15589
            10.2.9  p-Groups and Sylow groups; Sylow's theorems   cod 15611
            10.2.10  Direct products   clsm 15716
            10.2.11  Free groups   cefg 15786
      10.3  Abelian groups
            10.3.1  Definition and basic properties   ccmn 15860
            10.3.2  Cyclic groups   ccyg 15935
            10.3.3  Group sum operation   gsumval3a 15960
            10.3.4  Internal direct products   cdprd 16002
            10.3.5  The Fundamental Theorem of Abelian Groups   ablfacrplem 16071
      10.4  Rings
            10.4.1  Multiplicative Group   cmgp 16096
            10.4.2  Definition and basic properties   crg 16108
            10.4.3  Opposite ring   coppr 16175
            10.4.4  Divisibility   cdsr 16191
            10.4.5  Ring homomorphisms   crh 16265
      10.5  Division rings and fields
            10.5.1  Definition and basic properties   cdr 16283
            10.5.2  Subrings of a ring   csubrg 16312
            10.5.3  Absolute value (abstract algebra)   cabv 16352
            10.5.4  Star rings   cstf 16379
      10.6  Left modules
            10.6.1  Definition and basic properties   clmod 16398
            10.6.2  Subspaces and spans in a left module   clss 16456
            10.6.3  Homomorphisms and isomorphisms of left modules   clmhm 16543
            10.6.4  Subspace sum; bases for a left module   clbs 16594
      10.7  Vector spaces
            10.7.1  Definition and basic properties   clvec 16622
      10.8  Ideals
            10.8.1  The subring algebra; ideals   csra 16688
            10.8.2  Two-sided ideals and quotient rings   c2idl 16750
            10.8.3  Principal ideal rings. Divisibility in the integers   clpidl 16760
            10.8.4  Nonzero rings   cnzr 16776
            10.8.5  Left regular elements. More kinds of rings   crlreg 16787
      10.9  Associative algebras
            10.9.1  Definition and basic properties   casa 16817
      10.10  Abstract multivariate polynomials
            10.10.1  Definition and basic properties   cmps 16854
            10.10.2  Polynomial evaluation   evlslem4 17012
            10.10.3  Univariate polynomials   cps1 17017
      10.11  The complex numbers as an algebraic extensible structure
            10.11.1  Definition and basic properties   cpsmet 17133
            10.11.2  Algebraic constructions based on the complexes   czrh 17226
      10.12  Generalized pre-Hilbert and Hilbert spaces
            10.12.1  Definition and basic properties   cphl 17303
            10.12.2  Orthocomplements and closed subspaces   cocv 17335
            10.12.3  Orthogonal projection and orthonormal bases   cpj 17375
PART 11  BASIC TOPOLOGY
      11.1  Topology
            11.1.1  Topological spaces   ctop 17406
            11.1.2  TopBases for topologies   isbasisg 17460
            11.1.3  Examples of topologies   distop 17508
            11.1.4  Closure and interior   ccld 17528
            11.1.5  Neighborhoods   cnei 17609
            11.1.6  Limit points and perfect sets   clp 17646
            11.1.7  Subspace topologies   restrcl 17669
            11.1.8  Order topology   ordtbaslem 17700
            11.1.9  Limits and continuity in topological spaces   ccn 17736
            11.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 17818
            11.1.11  Compactness   ccmp 17897
            11.1.12  Bolzano-Weierstrass theorem   bwth 17921
            11.1.13  Connectedness   ccon 17922
            11.1.14  First- and second-countability   c1stc 17948
            11.1.15  Local topological properties   clly 17975
            11.1.16  Compactly generated spaces   ckgen 18013
            11.1.17  Product topologies   ctx 18040
            11.1.18  Continuous function-builders   cnmptid 18141
            11.1.19  Quotient maps and quotient topology   ckq 18173
            11.1.20  Homeomorphisms   chmeo 18233
      11.2  Filters and filter bases
            11.2.1  Filter bases   elmptrab 18307
            11.2.2  Filters   cfil 18325
            11.2.3  Ultrafilters   cufil 18379
            11.2.4  Filter limits   cfm 18413
            11.2.5  Extension by continuity   ccnext 18538
            11.2.6  Topological groups   ctmd 18548
            11.2.7  Infinite group sum on topological groups   ctsu 18603
            11.2.8  Topological rings, fields, vector spaces   ctrg 18633
      11.3  Uniform Structures and Spaces
            11.3.1  Uniform structures   cust 18677
            11.3.2  The topology induced by an uniform structure   cutop 18708
            11.3.3  Uniform Spaces   cuss 18731
            11.3.4  Uniform continuity   cucn 18753
            11.3.5  Cauchy filters in uniform spaces   ccfilu 18764
            11.3.6  Complete uniform spaces   ccusp 18775
      11.4  Metric spaces
            11.4.1  Pseudometric spaces   ispsmet 18783
            11.4.2  Basic metric space properties   cxme 18795
            11.4.3  Metric space balls   blfvalps 18861
            11.4.4  Open sets of a metric space   mopnval 18916
            11.4.5  Continuity in metric spaces   metcnp3 19018
            11.4.6  The uniform structure generated by a metric   metuvalOLD 19027
            11.4.7  Examples of metric spaces   dscmet 19068
            11.4.8  Normed algebraic structures   cnm 19072
            11.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 19187
            11.4.10  Topology on the reals   qtopbaslem 19240
            11.4.11  Topological definitions using the reals   cii 19353
            11.4.12  Path homotopy   chtpy 19440
            11.4.13  The fundamental group   cpco 19473
      11.5  Complex metric vector spaces
            11.5.1  Complex left modules   cclm 19535
            11.5.2  Complex pre-Hilbert space   ccph 19577
            11.5.3  Convergence and completeness   ccfil 19653
            11.5.4  Baire's Category Theorem   bcthlem1 19725
            11.5.5  Banach spaces and complex Hilbert spaces   ccms 19733
            11.5.6  Minimizing Vector Theorem   minveclem1 19773
            11.5.7  Projection Theorem   pjthlem1 19786
PART 12  BASIC REAL AND COMPLEX ANALYSIS
      12.1  Continuity
            12.1.1  Intermediate value theorem   pmltpclem1 19793
      12.2  Integrals
            12.2.1  Lebesgue measure   covol 19807
            12.2.2  Lebesgue integration   cmbf 19954
                  12.2.2.1  Lesbesgue integral   cmbf 19954
                  12.2.2.2  Lesbesgue directed integral   cdit 20181
      12.3  Derivatives
            12.3.1  Real and complex differentiation   climc 20197
                  12.3.1.1  Derivatives of functions of one complex or real variable   climc 20197
                  12.3.1.2  Results on real differentiation   dvferm1lem 20316
PART 13  BASIC REAL AND COMPLEX FUNCTIONS
      13.1  Polynomials
            13.1.1  Abstract polynomials, continued   evlslem6 20382
            13.1.2  Polynomial degrees   cmdg 20424
            13.1.3  The division algorithm for univariate polynomials   cmn1 20496
            13.1.4  Elementary properties of complex polynomials   cply 20551
            13.1.5  The division algorithm for polynomials   cquot 20655
            13.1.6  Algebraic numbers   caa 20679
            13.1.7  Liouville's approximation theorem   aalioulem1 20697
      13.2  Sequences and series
            13.2.1  Taylor polynomials and Taylor's theorem   ctayl 20717
            13.2.2  Uniform convergence   culm 20740
            13.2.3  Power series   pserval 20774
      13.3  Basic trigonometry
            13.3.1  The exponential, sine, and cosine functions (cont.)   efcn 20807
            13.3.2  Properties of pi = 3.14159...   pilem1 20815
            13.3.3  Mapping of the exponential function   efgh 20891
            13.3.4  The natural logarithm on complex numbers   clog 20900
            13.3.5  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 21091
            13.3.6  Solutions of quadratic, cubic, and quartic equations   quad2 21127
            13.3.7  Inverse trigonometric functions   casin 21150
            13.3.8  The Birthday Problem   log2ublem1 21234
            13.3.9  Areas in R^2   carea 21242
            13.3.10  More miscellaneous converging sequences   rlimcnp 21252
            13.3.11  Inequality of arithmetic and geometric means   cvxcl 21271
            13.3.12  Euler-Mascheroni constant   cem 21278
      13.4  Basic number theory
            13.4.1  Wilson's theorem   wilthlem1 21299
            13.4.2  The Fundamental Theorem of Algebra   ftalem1 21303
            13.4.3  The Basel problem (ζ(2) = π2/6)   basellem1 21311
            13.4.4  Number-theoretical functions   ccht 21321
            13.4.5  Perfect Number Theorem   mersenne 21459
            13.4.6  Characters of Z/nZ   cdchr 21464
            13.4.7  Bertrand's postulate   bcctr 21507
            13.4.8  Legendre symbol   clgs 21526
            13.4.9  Quadratic reciprocity   lgseisenlem1 21581
            13.4.10  All primes 4n+1 are the sum of two squares   2sqlem1 21595
            13.4.11  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 21611
            13.4.12  The Prime Number Theorem   mudivsum 21672
            13.4.13  Ostrowski's theorem   abvcxp 21757
*PART 14  GRAPH THEORY
      14.1  Undirected graphs - basics
            14.1.1  Undirected hypergraphs   cuhg 21782
            14.1.2  Undirected multigraphs   cumg 21795
            14.1.3  Undirected simple graphs   cuslg 21812
                  14.1.3.1  Undirected simple graphs - basics   cuslg 21812
                  14.1.3.2  Undirected simple graphs - examples   usgraexvlem 21862
                  14.1.3.3  Finite undirected simple graphs   fiusgraedgfi 21869
            14.1.4  Neighbors, complete graphs and universal vertices   cnbgra 21878
                  14.1.4.1  Neighbors   nbgraop 21884
                  14.1.4.2  Complete graphs   iscusgra 21913
                  14.1.4.3  Universal vertices   isuvtx 21945
            14.1.5  Walks, paths and cycles   cwalk 21954
                  14.1.5.1  Walks and trails   wlks 21974
                  14.1.5.2  Paths and simple paths   pths 22014
                  14.1.5.3  Circuits and cycles   crcts 22057
                  14.1.5.4  Connected graphs   cconngra 22104
            14.1.6  Vertex Degree   cvdg 22112
      14.2  Eulerian paths and the Konigsberg Bridge problem
            14.2.1  Eulerian paths   ceup 22132
            14.2.2  The Konigsberg Bridge problem   vdeg0i 22152
PART 15  GUIDES AND MISCELLANEA
      15.1  Guides (conventions, explanations, and examples)
            *15.1.1  Conventions   conventions 22158
            15.1.2  Natural deduction   natded 22159
            *15.1.3  Natural deduction examples   ex-natded5.2 22160
            15.1.4  Definitional examples   ex-or 22177
      15.2  Humor
            15.2.1  April Fool's theorem   avril1 22205
      15.3  (Future - to be reviewed and classified)
            15.3.1  Planar incidence geometry   cplig 22211
            15.3.2  Algebra preliminaries   crpm 22216
            15.3.3  Transitive closure   ctcl 22218
*PART 16  ADDITIONAL MATERIAL ON GROUPS, RINGS, AND FIELDS (DEPRECATED)
      16.1  Additional material on group theory
            16.1.1  Definitions and basic properties for groups   cgr 22222
            16.1.2  Definition and basic properties of Abelian groups   cablo 22317
            16.1.3  Subgroups   csubgo 22337
            16.1.4  Operation properties   cass 22348
            16.1.5  Group-like structures   cmagm 22354
            16.1.6  Examples of Abelian groups   ablosn 22383
            16.1.7  Group homomorphism and isomorphism   cghom 22393
      16.2  Additional material on rings and fields
            16.2.1  Definition and basic properties   crngo 22411
            16.2.2  Examples of rings   cnrngo 22439
            16.2.3  Division Rings   cdrng 22441
            16.2.4  Star Fields   csfld 22444
            16.2.5  Fields and Rings   ccm2 22446
PART 17  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      17.1  Complex vector spaces
            17.1.1  Definition and basic properties   cvc 22472
            17.1.2  Examples of complex vector spaces   cncvc 22510
      17.2  Normed complex vector spaces
            17.2.1  Definition and basic properties   cnv 22511
            17.2.2  Examples of normed complex vector spaces   cnnv 22616
            17.2.3  Induced metric of a normed complex vector space   imsval 22625
            17.2.4  Inner product   cdip 22644
            17.2.5  Subspaces   css 22668
      17.3  Operators on complex vector spaces
            17.3.1  Definitions and basic properties   clno 22689
      17.4  Inner product (pre-Hilbert) spaces
            17.4.1  Definition and basic properties   ccphlo 22761
            17.4.2  Examples of pre-Hilbert spaces   cncph 22768
            17.4.3  Properties of pre-Hilbert spaces   isph 22771
      17.5  Complex Banach spaces
            17.5.1  Definition and basic properties   ccbn 22812
            17.5.2  Examples of complex Banach spaces   cnbn 22819
            17.5.3  Uniform Boundedness Theorem   ubthlem1 22820
            17.5.4  Minimizing Vector Theorem   minvecolem1 22824
      17.6  Complex Hilbert spaces
            17.6.1  Definition and basic properties   chlo 22835
            17.6.2  Standard axioms for a complex Hilbert space   hlex 22848
            17.6.3  Examples of complex Hilbert spaces   cnchl 22866
            17.6.4  Subspaces   ssphl 22867
            17.6.5  Hellinger-Toeplitz Theorem   htthlem 22868
*PART 18  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      18.1  Axiomatization of complex pre-Hilbert spaces
            18.1.1  Basic Hilbert space definitions   chil 22870
            18.1.2  Preliminary ZFC lemmas   df-hnorm 22919
            *18.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 22932
            *18.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 22950
            18.1.5  Vector operations   hvmulex 22962
            18.1.6  Inner product postulates for a Hilbert space   ax-hfi 23029
      18.2  Inner product and norms
            18.2.1  Inner product   his5 23036
            18.2.2  Norms   dfhnorm2 23072
            18.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 23110
            18.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 23129
      18.3  Cauchy sequences and completeness axiom
            18.3.1  Cauchy sequences and limits   hcau 23134
            18.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 23144
            18.3.3  Completeness postulate for a Hilbert space   ax-hcompl 23152
            18.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 23153
      18.4  Subspaces and projections
            18.4.1  Subspaces   df-sh 23157
            18.4.2  Closed subspaces   df-ch 23172
            18.4.3  Orthocomplements   df-oc 23202
            18.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 23258
            18.4.5  Projection theorem   pjhthlem1 23341
            18.4.6  Projectors   df-pjh 23345
      18.5  Properties of Hilbert subspaces
            18.5.1  Orthomodular law   omlsilem 23352
            18.5.2  Projectors (cont.)   pjhtheu2 23366
            18.5.3  Hilbert lattice operations   sh0le 23390
            18.5.4  Span (cont.) and one-dimensional subspaces   spansn0 23491
            18.5.5  Commutes relation for Hilbert lattice elements   df-cm 23533
            18.5.6  Foulis-Holland theorem   fh1 23568
            18.5.7  Quantum Logic Explorer axioms   qlax1i 23577
            18.5.8  Orthogonal subspaces   chscllem1 23587
            18.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 23604
            18.5.10  Projectors (cont.)   pjorthi 23619
            18.5.11  Mayet's equation E_3   mayete3i 23678
      18.6  Operators on Hilbert spaces
            *18.6.1  Operator sum, difference, and scalar multiplication   df-hosum 23681
            18.6.2  Zero and identity operators   df-h0op 23699
            18.6.3  Operations on Hilbert space operators   hoaddcl 23709
            18.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 23790
            18.6.5  Linear and continuous functionals and norms   df-nmfn 23796
            18.6.6  Adjoint   df-adjh 23800
            18.6.7  Dirac bra-ket notation   df-bra 23801
            18.6.8  Positive operators   df-leop 23803
            18.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 23804
            18.6.10  Theorems about operators and functionals   nmopval 23807
            18.6.11  Riesz lemma   riesz3i 24013
            18.6.12  Adjoints (cont.)   cnlnadjlem1 24018
            18.6.13  Quantum computation error bound theorem   unierri 24055
            18.6.14  Dirac bra-ket notation (cont.)   branmfn 24056
            18.6.15  Positive operators (cont.)   leopg 24073
            18.6.16  Projectors as operators   pjhmopi 24097
      18.7  States on a Hilbert lattice and Godowski's equation
            18.7.1  States on a Hilbert lattice   df-st 24162
            18.7.2  Godowski's equation   golem1 24222
      18.8  Cover relation, atoms, exchange axiom, and modular symmetry
            18.8.1  Covers relation; modular pairs   df-cv 24230
            18.8.2  Atoms   df-at 24289
            18.8.3  Superposition principle   superpos 24305
            18.8.4  Atoms, exchange and covering properties, atomicity   chcv1 24306
            18.8.5  Irreducibility   chirredlem1 24341
            18.8.6  Atoms (cont.)   atcvat3i 24347
            18.8.7  Modular symmetry   mdsymlem1 24354
PART 19  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      19.1  Mathboxes for user contributions
            19.1.1  Mathbox guidelines   mathbox 24393
      19.2  Mathbox for Stefan Allan
      19.3  Mathbox for Thierry Arnoux
            19.3.1  Propositional Calculus - misc additions   bian1d 24398
            19.3.2  Predicate Calculus   spc2ed 24409
                  19.3.2.1  Predicate Calculus - misc additions   spc2ed 24409
                  19.3.2.2  Restricted quantification - misc additions   reximddv 24412
                  19.3.2.3  Substitution (without distinct variables) - misc additions   clelsb3f 24417
                  19.3.2.4  Existential "at most one" - misc additions   moel 24420
                  19.3.2.5  Existential uniqueness - misc additions   2reuswap2 24425
                  19.3.2.6  Restricted "at most one" - misc additions   rmoxfrdOLD 24429
            19.3.3  General Set Theory   ceqsexv2d 24435
                  19.3.3.1  Class abstractions (a.k.a. class builders)   ceqsexv2d 24435
                  19.3.3.2  Image Sets   abrexdomjm 24441
                  19.3.3.3  Set relations and operations - misc additions   eqri 24447
                  19.3.3.4  Unordered pairs   elpreq 24453
                  19.3.3.5  Conditional operator - misc additions   ifeqeqx 24455
                  19.3.3.6  Indexed union - misc additions   iuneq12daf 24464
                  19.3.3.7  Disjointness - misc additions   disjnf 24472
            19.3.4  Relations and Functions   dfrel4 24493
                  19.3.4.1  Relations - misc additions   dfrel4 24493
                  19.3.4.2  Functions - misc additions   fdmrn 24502
                  19.3.4.3  Operations - misc additions   mpt2mptxf 24555
                  19.3.4.4  Isomorphisms - misc. add.   gtiso 24556
                  19.3.4.5  Disjointness (additional proof requiring functions)   disjdsct 24558
                  19.3.4.6  First and second members of an ordered pair - misc additions   df1stres 24559
                  19.3.4.7  Supremum - misc additions   supssd 24564
                  19.3.4.8  Finite Sets   unifi3 24565
                  19.3.4.9  Countable Sets   nnct 24566
            19.3.5  Real and Complex Numbers   addeq0 24595
                  19.3.5.1  Complex addition - misc. additions   addeq0 24595
                  19.3.5.2  Ordering on reals - misc additions   lt2addrd 24596
                  19.3.5.3  Extended reals - misc additions   xgepnf 24603
                  19.3.5.4  Real number intervals - misc additions   icossicc 24616
                  19.3.5.5  Finite intervals of integers - misc additions   fzssnn 24634
                  19.3.5.6  Half-open integer ranges - misc additions   iundisjfi 24640
                  19.3.5.7  The ` # ` (finite set size) function - misc additions   hashunif 24644
                  19.3.5.8  The greatest common divisor operator - misc. add   numdenneg 24646
                  19.3.5.9  Integers   nn0indd 24648
                  19.3.5.10  Division in the extended real number system   cxdiv 24652
            19.3.6  Structure builders   ress0g 24671
                  19.3.6.1  Structure builder restriction operator   ress0g 24671
                  19.3.6.2  The opposite group   oppgle 24675
                  19.3.6.3  Posets   ressprs 24677
                  19.3.6.4  Complete lattices   clatp0cl 24693
                  19.3.6.5  Extended reals Structure - misc additions   ax-xrssca 24695
                  19.3.6.6  The extended non-negative real numbers commutative monoid   xrge0base 24707
            19.3.7  Algebra   abliso 24721
                  19.3.7.1  Monoids Homomorphisms   abliso 24721
                  19.3.7.2  Ordered monoids and groups   comnd 24722
                  19.3.7.3  Signum in an ordered monoid   csgns 24750
                  19.3.7.4  The Archimedean property for generic ordered algebraic structures   cinftm 24755
                  19.3.7.5  Semirings   csrg 24778
                  19.3.7.6  Semiring left modules   cslmd 24803
                  19.3.7.7  Finitely supported group sums - misc additions   sumpr 24829
                  19.3.7.8  Rings - misc additions   rngurd 24847
                  19.3.7.9  Ordered rings and fields   corng 24854
                  19.3.7.10  Ring homomorphisms - misc additions   rhmdvdsr 24877
                  19.3.7.11  The ordered commutative ring of integers   zzsbase 24884
                  19.3.7.12  Scalar restriction operation   cresv 24890
                  19.3.7.13  The commutative ring of gaussian integers   gzcrng 24905
                  19.3.7.14  The archimedean ordered field of real numbers   rebase 24906
            19.3.8  Topology   cmetid 24922
                  19.3.8.1  Pseudometrics   cmetid 24922
                  19.3.8.2  Continuity - misc additions   hauseqcn 24934
                  19.3.8.3  Topology of the closed unit   unitsscn 24935
                  19.3.8.4  Topology of ` ( RR X. RR ) `   unicls 24942
                  19.3.8.5  Order topology - misc. additions   cnvordtrestixx 24952
                  19.3.8.6  Continuity in topological spaces - misc. additions   mndpluscn 24965
                  19.3.8.7  Topology of the extended non-negative real numbers ordered monoid   xrge0hmph 24971
                  19.3.8.8  Limits - misc additions   lmlim 24986
                  19.3.8.9  Univariate polynomials   pl1cn 24994
            19.3.9  Uniform Stuctures and Spaces   chcmp 24995
                  19.3.9.1  Hausdorff uniform completion   chcmp 24995
            19.3.10  Topology and algebraic structures   zzsnm 24997
                  19.3.10.1  The norm on the ring of the integer numbers   zzsnm 24997
                  19.3.10.2  The complete ordered field of the real numbers   recms 24998
                  19.3.10.3  Topological ` ZZ ` -modules   zlm0 25001
                  19.3.10.4  Canonical embedding of the field of the rational numbers into a division ring   cqqh 25011
                  19.3.10.5  Canonical embedding of the real numbers into a complete ordered field   crrh 25032
                  19.3.10.6  Embedding from the extended real numbers into a complete lattice   cxrh 25052
                  19.3.10.7  Canonical embeddings into the ordered field of the real numbers   zrhre 25055
            19.3.11  Real and complex functions   nexple 25058
                  19.3.11.1  Integer powers - misc. additions   nexple 25058
                  *19.3.11.2  Logarithm laws generalized to an arbitrary base - logb   clogb 25059
                  19.3.11.3  Indicator Functions   cind 25077
                  19.3.11.4  Extended sum   cesum 25093
            19.3.12  Mixed Function/Constant operation   cofc 25147
            19.3.13  Abstract measure   csiga 25160
                  19.3.13.1  Sigma-Algebra   csiga 25160
                  19.3.13.2  Generated Sigma-Algebra   csigagen 25191
                  19.3.13.3  The Borel algebra on the real numbers   cbrsiga 25205
                  19.3.13.4  Product Sigma-Algebra   csx 25212
                  19.3.13.5  Measures   cmeas 25219
                  19.3.13.6  The counting measure   cntmeas 25250
                  19.3.13.7  The Lebesgue measure - misc additions   volss 25253
                  19.3.13.8  The Dirac delta measure   cdde 25258
                  19.3.13.9  The 'almost everywhere' relation   cae 25263
                  19.3.13.10  Measurable functions   cmbfm 25275
                  19.3.13.11  Borel Algebra on ` ( RR X. RR ) `   br2base 25294
            19.3.14  Integration   itgeq12dv 25316
                  19.3.14.1  Lebesgue integral - misc additions   itgeq12dv 25316
                  19.3.14.2  Bochner integral   citgm 25317
            19.3.15  Euler's partition theorem   oddpwdc 25341
            19.3.16  Probability   cprb 25370
                  19.3.16.1  Probability Theory   cprb 25370
                  19.3.16.2  Conditional Probabilities   ccprob 25394
                  19.3.16.3  Real Valued Random Variables   crrv 25403
                  19.3.16.4  Preimage set mapping operator   corvc 25418
                  19.3.16.5  Distribution Functions   orvcelval 25431
                  19.3.16.6  Cumulative Distribution Functions   orvclteel 25435
                  19.3.16.7  Probabilities - example   coinfliplem 25441
                  19.3.16.8  Bertrand's Ballot Problem   ballotlemoex 25448
            19.3.17  Signum (sgn or sign) function - misc. additions   sgncl 25501
            19.3.18  Words over a set - misc additions   wrdres 25519
                  19.3.18.1  Operations on words   ccatmulgnn0dir 25521
            19.3.19  Polynomials with real coefficients - misc additions   plymul02 25528
            19.3.20  Descartes's rule of signs   signspval 25534
                  19.3.20.1  Sign changes in a word over real numbers   signspval 25534
                  19.3.20.2  Counting sign changes in a word over real numbers   signslema 25544
      19.4  Mathbox for Mario Carneiro
            19.4.1  Miscellaneous stuff   quartfull 25574
            19.4.2  Zeta function   czeta 25575
            19.4.3  Gamma function   clgam 25578
            19.4.4  Derangements and the Subfactorial   deranglem 25630
            19.4.5  The Erdős-Szekeres theorem   erdszelem1 25655
            19.4.6  The Kuratowski closure-complement theorem   kur14lem1 25670
            19.4.7  Retracts and sections   cretr 25681
            19.4.8  Path-connected and simply connected spaces   cpcon 25684
            19.4.9  Covering maps   ccvm 25720
            19.4.10  Normal numbers   snmlff 25794
            19.4.11  Godel-sets of formulas   cgoe 25798
            19.4.12  Models of ZF   cgze 25826
            19.4.13  Splitting fields   citr 25840
            19.4.14  p-adic number fields   czr 25856
      19.5  Mathbox for Paul Chapman
            19.5.1  Group homomorphism and isomorphism   ghomgrpilem1 25874
            19.5.2  Real and complex numbers (cont.)   climuzcnv 25886
            19.5.3  Miscellaneous theorems   elfzm12 25890
      *19.6  Mathbox for Drahflow
      19.7  Mathbox for Scott Fenton
            19.7.1  ZFC Axioms in primitive form   axextprim 25922
            19.7.2  Untangled classes   untelirr 25929
            19.7.3  Extra propositional calculus theorems   3orel1 25936
            19.7.4  Misc. Useful Theorems   nepss 25944
            19.7.5  Properties of reals and complexes   sqdivzi 25953
            19.7.6  Product sequences   prodf 25984
            19.7.7  Non-trivial convergence   ntrivcvg 25994
            19.7.8  Complex products   cprod 26000
            19.7.9  Finite products   fprod 26036
            19.7.10  Infinite products   iprodclim 26080
            19.7.11  Falling and Rising Factorial   cfallfac 26089
            19.7.12  Factorial limits   faclimlem1 26131
            19.7.13  Greatest common divisor and divisibility   pdivsq 26137
            19.7.14  Properties of relationships   brtp 26141
            19.7.15  Properties of functions and mappings   funpsstri 26158
            19.7.16  Epsilon induction   setinds 26174
            19.7.17  Ordinal numbers   elpotr 26177
            19.7.18  Defined equality axioms   axextdfeq 26194
            19.7.19  Hypothesis builders   hbntg 26202
            19.7.20  The Predecessor Class   cpred 26207
            19.7.21  (Trans)finite Recursion Theorems   tfisg 26248
            19.7.22  Well-founded induction   tz6.26 26249
            19.7.23  Transitive closure under a relationship   ctrpred 26264
            19.7.24  Founded Induction   frmin 26286
            19.7.25  Ordering Ordinal Sequences   orderseqlem 26296
            19.7.26  Well-founded recursion   cwrecs 26299
            19.7.27  Transfinite recursion via Well-founded recursion   tfrALTlem 26326
            19.7.28  Well-founded zero, successor, and limits   cwsuc 26330
            19.7.29  Founded Recursion   frr3g 26350
            19.7.30  Surreal Numbers   csur 26364
            19.7.31  Surreal Numbers: Ordering   sltsolem1 26392
            19.7.32  Surreal Numbers: Birthday Function   bdayfo 26399
            19.7.33  Surreal Numbers: Density   fvnobday 26406
            19.7.34  Surreal Numbers: Density   nodenselem3 26407
            19.7.35  Surreal Numbers: Upper and Lower Bounds   nobndlem1 26416
            19.7.36  Surreal Numbers: Full-Eta Property   nofulllem1 26426
            19.7.37  Symmetric difference   csymdif 26431
            19.7.38  Quantifier-free definitions   ctxp 26443
            19.7.39  Alternate ordered pairs   caltop 26570
            19.7.40  Tarskian geometry   cee 26596
            19.7.41  Tarski's axioms for geometry   axdimuniq 26621
            19.7.42  Congruence properties   cofs 26685
            19.7.43  Betweenness properties   btwntriv2 26715
            19.7.44  Segment Transportation   ctransport 26732
            19.7.45  Properties relating betweenness and congruence   cifs 26738
            19.7.46  Connectivity of betweenness   btwnconn1lem1 26790
            19.7.47  Segment less than or equal to   csegle 26809
            19.7.48  Outside of relationship   coutsideof 26822
            19.7.49  Lines and Rays   cline2 26837
            19.7.50  Bernoulli polynomials and sums of k-th powers   cbp 26861
            19.7.51  Rank theorems   rankung 26876
            19.7.52  Hereditarily Finite Sets   chf 26882
      19.8  Mathbox for Anthony Hart
            19.8.1  Propositional Calculus   tb-ax1 26897
            19.8.2  Predicate Calculus   quantriv 26919
            19.8.3  Misc. Single Axiom Systems   meran1 26930
            19.8.4  Connective Symmetry   negsym1 26936
      19.9  Mathbox for Chen-Pang He
            19.9.1  Ordinal topology   ontopbas 26947
      19.10  Mathbox for Jeff Hoffman
            19.10.1  Inferences for finite induction on generic function values   fveleq 26970
            19.10.2  gdc.mm   nnssi2 26974
      *19.11  Mathbox for Wolf Lammen
      19.12  Mathbox for Brendan Leahy
      19.13  Mathbox for Jeff Hankins
            19.13.1  Miscellany   a1i13 27068
            19.13.2  Basic topological facts   topbnd 27097
            19.13.3  Topology of the real numbers   ivthALT 27108
            19.13.4  Refinements   cfne 27109
            19.13.5  Neighborhood bases determine topologies   neibastop1 27158
            19.13.6  Lattice structure of topologies   topmtcl 27162
            19.13.7  Filter bases   fgmin 27169
            19.13.8  Directed sets, nets   tailfval 27171
      19.14  Mathbox for Jeff Madsen
            19.14.1  Logic and set theory   anim12da 27182
            19.14.2  Real and complex numbers; integers   filbcmb 27212
            19.14.3  Sequences and sums   sdclem2 27216
            19.14.4  Topology   subspopn 27228
            19.14.5  Metric spaces   metf1o 27231
            19.14.6  Continuous maps and homeomorphisms   constcncf 27238
            19.14.7  Boundedness   ctotbnd 27245
            19.14.8  Isometries   cismty 27277
            19.14.9  Heine-Borel Theorem   heibor1lem 27288
            19.14.10  Banach Fixed Point Theorem   bfplem1 27301
            19.14.11  Euclidean space   crrn 27304
            19.14.12  Intervals (continued)   ismrer1 27317
            19.14.13  Groups and related structures   exidcl 27321
            19.14.14  Rings   rngonegcl 27331
            19.14.15  Ring homomorphisms   crnghom 27346
            19.14.16  Commutative rings   ccring 27375
            19.14.17  Ideals   cidl 27387
            19.14.18  Prime rings and integral domains   cprrng 27426
            19.14.19  Ideal generators   cigen 27439
      19.15  Mathbox for Giovanni Mascellani
            *19.15.1  Tools for automatic proof building   efald2 27458
            *19.15.2  Tseitin axioms   fald 27481
            *19.15.3  Equality deductions   iuneq2f 27508
            *19.15.4  Miscellanea   sbcom3OLD 27521
      19.16  Mathbox for Rodolfo Medina
            19.16.1  Partitions   prtlem60 27528
      19.17  Mathbox for Stefan O'Rear
            19.17.1  Additional elementary logic and set theory   nelss 27570
            19.17.2  Additional theory of functions   fninfp 27573
            19.17.3  Extensions beyond function theory   gsumvsmul 27583
            19.17.4  Additional topology   elrfi 27586
            19.17.5  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 27590
            19.17.6  Algebraic closure systems   cnacs 27594
            19.17.7  Miscellanea 1. Map utilities   constmap 27605
            19.17.8  Miscellanea for polynomials   ofmpteq 27614
            19.17.9  Multivariate polynomials over the integers   cmzpcl 27616
            19.17.10  Miscellanea for Diophantine sets 1   coeq0 27648
            19.17.11  Diophantine sets 1: definitions   cdioph 27651
            19.17.12  Diophantine sets 2 miscellanea   ellz1 27663
            19.17.13  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 27669
            19.17.14  Diophantine sets 3: construction   diophrex 27672
            19.17.15  Diophantine sets 4 miscellanea   2sbcrex 27681
            19.17.16  Diophantine sets 4: Quantification   rexrabdioph 27690
            19.17.17  Diophantine sets 5: Arithmetic sets   rabdiophlem1 27697
            19.17.18  Diophantine sets 6 miscellanea   fz1ssnn 27707
            19.17.19  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 27708
            19.17.20  Pigeonhole Principle and cardinality helpers   fphpd 27713
            19.17.21  A non-closed set of reals is infinite   rencldnfilem 27717
            19.17.22  Miscellanea for Lagrange's theorem   icodiamlt 27719
            19.17.23  Lagrange's rational approximation theorem   irrapxlem1 27721
            19.17.24  Pell equations 1: A nontrivial solution always exists   pellexlem1 27728
            19.17.25  Pell equations 2: Algebraic number theory of the solution set   csquarenn 27735
            19.17.26  Pell equations 3: characterizing fundamental solution   infmrgelbi 27777
            19.17.27  Logarithm laws generalized to an arbitrary base   reglogcl 27789
            19.17.28  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 27797
            19.17.29  X and Y sequences 1: Definition and recurrence laws   crmx 27799
            19.17.30  Ordering and induction lemmas for the integers   monotuz 27840
            19.17.31  X and Y sequences 2: Order properties   rmxypos 27848
            19.17.32  Congruential equations   congtr 27866
            19.17.33  Alternating congruential equations   acongid 27876
            19.17.34  Additional theorems on integer divisibility   bezoutr 27886
            19.17.35  X and Y sequences 3: Divisibility properties   jm2.18 27895
            19.17.36  X and Y sequences 4: Diophantine representability of Y   jm2.27a 27912
            19.17.37  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 27922
            19.17.38  Uncategorized stuff not associated with a major project   setindtr 27931
            19.17.39  More equivalents of the Axiom of Choice   axac10 27940
            19.17.40  Finitely generated left modules   clfig 27978
            19.17.41  Noetherian left modules I   clnm 27986
            19.17.42  Addenda for structure powers   pwssplit0 28000
            19.17.43  Direct sum of left modules   cdsmm 28010
            19.17.44  Free modules   cfrlm 28025
            19.17.45  Every set admits a group structure iff choice   unxpwdom3 28069
            19.17.46  Independent sets and families   clindf 28087
            19.17.47  Characterization of free modules   lmimlbs 28119
            19.17.48  Noetherian rings and left modules II   clnr 28126
            19.17.49  Hilbert's Basis Theorem   cldgis 28138
            19.17.50  Additional material on polynomials [DEPRECATED]   cmnc 28148
            19.17.51  Degree and minimal polynomial of algebraic numbers   cdgraa 28158
            19.17.52  Algebraic integers I   citgo 28175
            19.17.53  Finite cardinality [SO]   en1uniel 28193
            *19.17.54  Words in monoids and ordered group sum   issubmd 28196
            19.17.55  Transpositions in the symmetric group   cpmtr 28197
            19.17.56  The sign of a permutation   cpsgn 28227
            19.17.57  The matrix algebra   cmmul 28252
            19.17.58  The determinant   cmdat 28296
            19.17.59  Endomorphism algebra   cmend 28302
            19.17.60  Subfields   csdrg 28316
            19.17.61  Cyclic groups and order   idomrootle 28324
            19.17.62  Cyclotomic polynomials   ccytp 28334
            19.17.63  Miscellaneous topology   fgraphopab 28342
      19.18  Mathbox for Steve Rodriguez
            19.18.1  Miscellanea   iso0 28349
            19.18.2  Function operations   caofcan 28353
            19.18.3  Calculus   lhe4.4ex1a 28359
      19.19  Mathbox for Andrew Salmon
            19.19.1  Principia Mathematica * 10   pm10.12 28366
            19.19.2  Principia Mathematica * 11   2alanimi 28380
            19.19.3  Predicate Calculus   sbeqal1 28407
            19.19.4  Principia Mathematica * 13 and * 14   pm13.13a 28417
            19.19.5  Set Theory   elnev 28448
            19.19.6  Arithmetic   addcomgi 28469
            19.19.7  Geometry   cplusr 28470
      19.20  Mathbox for Glauco Siliprandi
            19.20.1  Miscellanea   ssrexf 28492
            19.20.2  Finite multiplication of numbers and finite multiplication of functions   fmul01 28518
            19.20.3  Limits   clim1fr1 28534
            19.20.4  Derivatives   dvsinexp 28547
            19.20.5  Integrals   ioovolcl 28549
            19.20.6  Stone Weierstrass theorem - real version   stoweidlem1 28557
            19.20.7  Wallis' product for π   wallispilem1 28621
            19.20.8  Stirling's approximation formula for ` n ` factorial   stirlinglem1 28630
      19.21  Mathbox for Saveliy Skresanov
            19.21.1  Ceva's theorem   sigarval 28647
      19.22  Mathbox for Jarvin Udandy
      19.23  Mathbox for Alexander van der Vekens
            19.23.1  Double restricted existential uniqueness   r19.32 28752
                  19.23.1.1  Restricted quantification (extension)   r19.32 28752
                  19.23.1.2  The empty set (extension)   raaan2 28760
                  19.23.1.3  Restricted uniqueness and "at most one" quantification   rmoimi 28761
                  19.23.1.4  Analogs to Existential uniqueness (double quantification)   2reurex 28766
            *19.23.2  Alternative definitions of function's and operation's values   wdfat 28778
                  19.23.2.1  Restricted quantification (extension)   ralbinrald 28784
                  19.23.2.2  The universal class (extension)   nvelim 28785
                  19.23.2.3  Introduce the Axiom of Power Sets (extension)   alneu 28786
                  19.23.2.4  Relations (extension)   csbdmgOLD 28788
                  19.23.2.5  Functions (extension)   fveqvfvv 28793
                  19.23.2.6  Predicate "defined at"   dfateq12d 28798
                  19.23.2.7  Alternative definition of the value of a function   dfafv2 28801
                  19.23.2.8  Alternative definition of the value of an operation   aoveq123d 28847
            *19.23.3  Auxiliary theorems for graph theory   ornld 28877
                  19.23.3.1  Logical disjunction and conjunction - extension   ornld 28877
                  19.23.3.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4anass 28878
                  19.23.3.3  Negated equality and membership - extension   elnelall 28879
                  19.23.3.4  Proper substitution of classes for sets into classes - extension   csbprg 28880
                  19.23.3.5  The empty set - extension   rspn0 28881
                  19.23.3.6  Power classes - extension   3xpexg 28884
                  19.23.3.7  Unordered and ordered pairs - extension   nelprd 28885
                  19.23.3.8  Indexed union and intersection - extension   iunxprg 28899
                  19.23.3.9  Ordered-pair class abstractions - extension   elopaelxp 28900
                  19.23.3.10  Introduce the Axiom of Union - extension   ralxfrd2 28902
                  19.23.3.11  Relations - extension   resisresindm 28904
                  19.23.3.12  Functions - extension   f0bi 28906
                  19.23.3.13  Operations - extension   oprabv 28923
                  19.23.3.14  Equinumerosity - extension   resfnfinfin 28929
                  19.23.3.15  Subtraction - extension   cnm1cn 28931
                  19.23.3.16  Multiplication - extension   kcnktkm1cn 28934
                  19.23.3.17  Ordering on reals (cont.) - extension   leltletr 28937
                  19.23.3.18  Some properties of specific numbers - extension   cnm2m1cnm3 28943
                  19.23.3.19  Nonnegative integers (as a subset of complex numbers) - extension   lesubnn0 28945
                  19.23.3.20  Integers (as a subset of complex numbers) - extension   zadd2cl 28952
                  19.23.3.21  Upper sets of integers - extension   2eluzge0 28953
                  19.23.3.22  Finite intervals of integers - extension   ssfz12 28963
                  19.23.3.23  Half-open integer ranges - extension   subsubelfzo0 28976
                  19.23.3.24  The ` # ` (finite set size) function - extension   hash2prv 28994
                  19.23.3.25  Finite and infinite sums - extension   fsumz 29004
                  19.23.3.26  Prime numbers: elementary properties - extension   m1dvdsndvds 29015
            19.23.4  Additional definitions and more theorems about Words   wrdlen1 29018
                  19.23.4.1  Words over a set - extension   wrdlen1 29018
                  19.23.4.2  Last symbol of a word   lswn0 29026
                  19.23.4.3  Words over a set - extension (concatenations and subwords)   ccats1swrdeqrex 29027
                  19.23.4.4  Words over a set ( concatenations with singleton words) - extension   lswccats1fst 29030
            19.23.5  Graph theory   uhgraedgrnv 29041
                  19.23.5.1  Undirected hypergraphs   uhgraedgrnv 29041
                  19.23.5.2  Undirected simple graphs   usisuhgra 29042
                  19.23.5.3  Neighbors, complete graphs and universal vertices   nbfiusgrafi 29043
                  19.23.5.4  Walks, Paths and Cycles   wlkn0 29047
                  19.23.5.5  Walks as words   cwwlk 29079
                  19.23.5.6  Walks/paths of length 2 as ordered triples   c2wlkot 29141
                  19.23.5.7  Closed Walks   cclwlk 29180
                  19.23.5.8  Vertex Degree   usgfidegfi 29295
                  19.23.5.9  Regular graphs   crgra 29307
            19.23.6  The Friendship Theorem   cfrgra 29348
                  *19.23.6.1  Friendship graphs - basics   cfrgra 29348
                  *19.23.6.2  Huneke's Proof of the Friendship Theorem   frgrancvvdeqlem1 29391
      *19.24  Mathbox for David A. Wheeler
            19.24.1  Natural deduction   19.8ad 29484
            *19.24.2  Greater than, greater than or equal to.   cge-real 29487
            *19.24.3  Hyperbolic trig functions   csinh 29497
            *19.24.4  Reciprocal trig functions (sec, csc, cot)   csec 29508
            *19.24.5  Identities for "if"   ifnmfalse 29530
            19.24.6  Not-member-of   AnelBC 29531
            *19.24.7  Decimal point   cdp2 29532
            19.24.8  Logarithms generalized to arbitrary base using ` logb `   ene0 29540
            *19.24.9  Logarithm laws generalized to an arbitrary base - log_   clog_ 29543
            *19.24.10  Algebra helpers   pncan3oi 29545
            *19.24.11  Algebra helper examples   i2linesi 29564
            *19.24.12  Formal methods "surprises"   alimp-surprise 29566
            *19.24.13  Allsome quantifier   walsi 29572
            *19.24.14  Miscellaneous   5m4e1 29583
      19.25  Mathbox for Alan Sare
            19.25.1  Supplementary "adant" deductions   ad4ant13 29587
            19.25.2  Supplementary unification deductions   biimp 29613
            19.25.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 29632
            19.25.4  What is Virtual Deduction?   wvd1 29710
            19.25.5  Virtual Deduction Theorems   df-vd1 29711
            19.25.6  Theorems proved using virtual deduction   trsspwALT 29981
            19.25.7  Theorems proved using virtual deduction with mmj2 assistance   simplbi2VD 30010
            19.25.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 30077
            19.25.9  Theorems proved using conjunction-form virtual deduction   elpwgdedVD 30081
            19.25.10  Theorems with VD proofs in conventional notation derived from VD proofs   suctrALT3 30088
            *19.25.11  Theorems with a proof in conventional notation automatically derived   notnot2ALT2 30091
      *19.26  Mathbox for Jonathan Ben-Naim
            19.26.1  First order logic and set theory   bnj170 30114
            19.26.2  Well founded induction and recursion   bnj110 30281
            19.26.3  The existence of a minimal element in certain classes   bnj69 30431
            19.26.4  Well-founded induction   bnj1204 30433
            19.26.5  Well-founded recursion, part 1 of 3   bnj60 30483
            19.26.6  Well-founded recursion, part 2 of 3   bnj1500 30489
            19.26.7  Well-founded recursion, part 3 of 3   bnj1522 30493
      19.27  Mathbox for BJ
            19.27.1  Propositional calculus   bj-andnotim 30494
            19.27.2  First-order logic   bj-alax1 30497
                  *19.27.2.1  Miscellaneous lemmas   bj-alax1 30497
                  *19.27.2.2  Strengthenings of theorems of the main part   bj-aecom 30499
                  *19.27.2.3  Distinct var metavariables   bj-dvv 30503
                  *19.27.2.4  Around ~ equsal   bj-equsal1t 30504
                  *19.27.2.5  Some PM proofs   stdpc5t 30509
            19.27.3  Set theory   bj-nfcsym 30520
                  19.27.3.1  The class-form not-free predicate   bj-nfcsym 30520
                  *19.27.3.2  Lemmas for substitution   bj-sbeqALT 30521
                  *19.27.3.3  "Singletonization" and tagging   bj-nel0 30527
                  *19.27.3.4  Tuples of classes   bj-12zdisj 30551
      *19.28  Mathbox for Norm Megill
            *19.28.1  Experiments to study ax-7 unbundling   ax-7v 30576
                  19.28.1.1  Theorems derived from ax-7v (suffixes NEW7 and AUX7)   ax-7v 30576
                  19.28.1.2  Theorems derived from ax-7 (suffix OLD7)   ax-7OLD7 30812
            19.28.2  Miscellanea   cnaddcom 30903
            19.28.3  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 30905
            19.28.4  Functionals and kernels of a left vector space (or module)   clfn 30988
            19.28.5  Opposite rings and dual vector spaces   cld 31054
            19.28.6  Ortholattices and orthomodular lattices   cops 31103
            19.28.7  Atomic lattices with covering property   ccvr 31193
            19.28.8  Hilbert lattices   chlt 31281
            19.28.9  Projective geometries based on Hilbert lattices   clln 31421
            19.28.10  Construction of a vector space from a Hilbert lattice   cdlema1N 31721
            19.28.11  Construction of involution and inner product from a Hilbert lattice   clpoN 33411

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