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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  Elementary real and complex functions
      5.8  Elementary limits and convergence
      5.9  Elementary trigonometry
      5.10  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 Stuctures 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 937
            1.2.9  Logical 'nand' (Sheffer stroke)   wnan 1298
            1.2.10  Logical 'xor'   wxo 1315
            1.2.11  True and false constants   wtru 1327
            *1.2.12  Truth tables   truantru 1347
            1.2.13  Auxiliary theorems for Alan Sare's virtual deduction tool, part 1   ee22 1373
            *1.2.14  Half-adders and full adders in propositional calculus   whad 1389
      1.3  Other axiomatizations of classical propositional calculus
            1.3.1  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1415
            1.3.2  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1434
            *1.3.3  Derive Nicod's axiom from the standard axioms   nic-dfim 1445
            1.3.4  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1451
            1.3.5  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1470
            1.3.6  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1474
            1.3.7  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1489
            1.3.8  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1512
            1.3.9  Derive the Lukasiewicz axioms from the The Russell-Bernays Axioms   rb-bijust 1525
            *1.3.10  Stoic logic indemonstrables (Chrysippus of Soli)   mpto1 1544
      *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 1551
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1557
            1.4.3  Axiom scheme ax-5 (Quantified Implication)   ax-5 1568
            1.4.4  Axiom scheme ax-17 (Distinctness) - first use of $d   ax-17 1629
            1.4.5  Equality predicate; define substitution   cv 1654
            1.4.6  Axiom scheme ax-9 (Existence)   ax-9 1670
            1.4.7  Axiom scheme ax-8 (Equality)   ax-8 1691
            1.4.8  Membership predicate   wcel 1727
            1.4.9  Axiom scheme ax-13 (Left Equality for Binary Predicate)   ax-13 1729
            1.4.10  Axiom scheme ax-14 (Right Equality for Binary Predicate)   ax-14 1731
            *1.4.11  Logical redundancy of ax-6 , ax-7 , ax-11 , ax-12   ax9dgen 1733
      *1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-6 (Quantified Negation)   ax-6 1746
            1.5.2  Axiom scheme ax-7 (Quantifier Commutation)   ax-7 1751
            1.5.3  Axiom scheme ax-11 (Substitution)   ax-11 1763
            1.5.4  Axiom scheme ax-12 (Quantified Equality)   ax-12 1954
      *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 2232
            *1.6.2  Rederive new axioms from old: ax5 , ax6 , ax9from9o , ax11 , ax12from12o   ax4 2242
            *1.6.3  Legacy theorems using obsolete axioms   ax17o 2254
      1.7  Existential uniqueness
      1.8  Other axiomatizations related to classical predicate calculus
            1.8.1  Predicate calculus with all distinct variables   ax-7d 2392
            *1.8.2  Aristotelian logic: Assertic syllogisms   barbara 2398
            *1.8.3  Intuitionistic logic   axia1 2422
*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 2437
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2442
            2.1.3  Class form not-free predicate   wnfc 2579
            2.1.4  Negated equality and membership   wne 2619
                  2.1.4.1  Negated equality   neii 2623
                  2.1.4.2  Negated membership   neli 2717
            2.1.5  Restricted quantification   wral 2725
            2.1.6  The universal class   cvv 2978
            *2.1.7  Conditional equality (experimental)   wcdeq 3166
            2.1.8  Russell's Paradox   ru 3182
            2.1.9  Proper substitution of classes for sets   wsbc 3183
            2.1.10  Proper substitution of classes for sets into classes   csb 3283
            2.1.11  Define basic set operations and relations   cdif 3319
            2.1.12  Subclasses and subsets   df-ss 3336
            2.1.13  The difference, union, and intersection of two classes   difeq1 3460
                  2.1.13.1  The difference of two classes   difeq1 3460
                  2.1.13.2  The union of two classes   elun 3490
                  2.1.13.3  The intersection of two classes   elin 3532
                  2.1.13.4  Combinations of difference, union, and intersection of two classes   unabs 3572
                  2.1.13.5  Class abstractions with difference, union, and intersection of two classes   symdif2 3608
                  2.1.13.6  Restricted uniqueness with difference, union, and intersection   reuss2 3622
            2.1.14  The empty set   c0 3629
            *2.1.15  "Weak deduction theorem" for set theory   cif 3781
            2.1.16  Power classes   cpw 3842
            2.1.17  Unordered and ordered pairs   csn 3857
            2.1.18  The union of a class   cuni 4060
            2.1.19  The intersection of a class   cint 4096
            2.1.20  Indexed union and intersection   ciun 4139
            2.1.21  Disjointness   wdisj 4230
            2.1.22  Binary relations   wbr 4260
            2.1.23  Ordered-pair class abstractions (class builders)   copab 4317
            2.1.24  Transitive classes   wtr 4353
      2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 4371
            2.2.2  Derive the Axiom of Separation   axsep 4380
            2.2.3  Derive the Null Set Axiom   zfnuleu 4386
            2.2.4  Theorems requiring subset and intersection existence   nalset 4396
            2.2.5  Theorems requiring empty set existence   class2set 4423
      2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4434
            2.3.2  Derive the Axiom of Pairing   zfpair 4458
            2.3.3  Ordered pair theorem   opnz 4489
            2.3.4  Ordered-pair class abstractions (cont.)   opabid 4518
            2.3.5  Power class of union and intersection   pwin 4546
            2.3.6  Epsilon and identity relations   cep 4551
            2.3.7  Partial and complete ordering   wpo 4560
            2.3.8  Founded and well-ordering relations   wfr 4597
            2.3.9  Ordinals   word 4639
      2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4760
            2.4.2  Ordinals (continued)   ordon 4822
            2.4.3  Transfinite induction   tfi 4892
            2.4.4  The natural numbers (i.e. finite ordinals)   com 4904
            2.4.5  Peano's postulates   peano1 4923
            2.4.6  Finite induction (for finite ordinals)   find 4929
            2.4.7  Relations   cxp 4935
            2.4.8  Definite description binder (inverted iota)   cio 5480
            2.4.9  Functions   wfun 5513
            2.4.10  Operations   co 6147
            2.4.11  "Maps to" notation   elmpt2cl 6356
            2.4.12  Function operation   cof 6371
            2.4.13  First and second members of an ordered pair   c1st 6415
            *2.4.14  Special "Maps to" operations   mpt2xopn0yelv 6532
            2.4.15  Function transposition   ctpos 6546
            2.4.16  Curry and uncurry   ccur 6585
            2.4.17  Proper subset relation   crpss 6589
            2.4.18  Iota properties   fvopab5 6602
            2.4.19  Cantor's Theorem   canth 6607
            2.4.20  Undefined values and restricted iota (description binder)   cund 6609
            2.4.21  Functions on ordinals; strictly monotone ordinal functions   iunon 6659
            2.4.22  "Strong" transfinite recursion   crecs 6691
            2.4.23  Recursive definition generator   crdg 6726
            2.4.24  Finite recursion   frfnom 6751
            2.4.25  Abian's "most fundamental" fixed point theorem   abianfplem 6774
            2.4.26  Ordinal arithmetic   c1o 6776
            2.4.27  Natural number arithmetic   nna0 6906
            2.4.28  Equivalence relations and classes   wer 6961
            2.4.29  The mapping operation   cmap 7077
            2.4.30  Infinite Cartesian products   cixp 7122
            2.4.31  Equinumerosity   cen 7165
            2.4.32  Schroeder-Bernstein Theorem   sbthlem1 7276
            2.4.33  Equinumerosity (cont.)   xpf1o 7328
            2.4.34  Pigeonhole Principle   phplem1 7345
            2.4.35  Finite sets   onomeneq 7355
            2.4.36  Finite intersections   cfi 7474
            2.4.37  Hall's marriage theorem   marypha1lem 7497
            2.4.38  Supremum   csup 7504
            2.4.39  Ordinal isomorphism, Hartog's theorem   coi 7537
            2.4.40  Hartogs function, order types, weak dominance   char 7583
      2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 7619
            2.5.2  Axiom of Infinity equivalents   inf0 7635
      2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 7652
            2.6.2  Existence of omega (the set of natural numbers)   omex 7657
            2.6.3  Cantor normal form   ccnf 7675
            2.6.4  Transitive closure   trcl 7723
            2.6.5  Rank   cr1 7747
            2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 7870
            2.6.7  Cardinal numbers   ccrd 7883
            2.6.8  Axiom of Choice equivalents   wac 8057
            2.6.9  Cardinal number arithmetic   ccda 8108
            2.6.10  The Ackermann bijection   ackbij2lem1 8160
            2.6.11  Cofinality (without Axiom of Choice)   cflem 8187
            2.6.12  Eight inequivalent definitions of finite set   sornom 8218
            2.6.13  Hereditarily size-limited sets without Choice   itunifval 8357
*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 8376
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 8387
      3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 8400
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 8435
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 8482
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 8510
            3.2.5  Cofinality using Axiom of Choice   alephreg 8518
      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 8556
            3.4.2  Derivation of the Axiom of Choice   gchaclem 8614
*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 8618
            4.1.2  Weak universes   cwun 8636
            4.1.3  Tarski's classes   ctsk 8684
            4.1.4  Grothendieck's universes   cgru 8726
      4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 8759
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 8762
            4.2.3  Tarski map function   ctskm 8773
*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 8780
            5.1.2  Final derivation of real and complex number postulates   axaddf 9081
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 9107
      5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 9132
            5.2.2  Infinity and the extended real number system   cpnf 9178
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 9208
            5.2.4  Ordering on reals   lttr 9213
            5.2.5  Initial properties of the complex numbers   mul12 9293
      5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 9340
            5.3.2  Subtraction   cmin 9352
            5.3.3  Multiplication   muladd 9527
            5.3.4  Ordering on reals (cont.)   gt0ne0 9554
            5.3.5  Reciprocals   ixi 9712
            5.3.6  Division   cdiv 9738
            5.3.7  Ordering on reals (cont.)   elimgt0 9907
            5.3.8  Completeness Axiom and Suprema   fimaxre 10016
            5.3.9  Imaginary and complex number properties   inelr 10051
            5.3.10  Function operation analogue theorems   ofsubeq0 10058
      5.4  Integer sets
            5.4.1  Natural numbers (as a subset of complex numbers)   cn 10061
            5.4.2  Principle of mathematical induction   nnind 10079
            *5.4.3  Decimal representation of numbers   c2 10110
            *5.4.4  Some properties of specific numbers   0p1e1 10154
            5.4.5  The Archimedean property   nnunb 10278
            5.4.6  Nonnegative integers (as a subset of complex numbers)   cn0 10282
            5.4.7  Integers (as a subset of complex numbers)   cz 10343
            5.4.8  Decimal arithmetic   cdc 10443
            5.4.9  Upper partititions of integers   cuz 10549
            5.4.10  Well-ordering principle for bounded-below sets of integers   uzwo3 10630
            5.4.11  Rational numbers (as a subset of complex numbers)   cq 10635
            5.4.12  Existence of the set of complex numbers   rpnnen1lem1 10661
      5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 10673
            5.5.2  Infinity and the extended real number system (cont.)   cxne 10768
            5.5.3  Supremum on the extended reals   xrsupexmnf 10944
            5.5.4  Real number intervals   cioo 10977
            5.5.5  Finite intervals of integers   cfz 11104
            5.5.6  Half-open integer ranges   cfzo 11196
      5.6  Elementary integer functions
            5.6.1  The floor (greatest integer) function   cfl 11262
            5.6.2  The modulo (remainder) operation   cmo 11311
            5.6.3  The infinite sequence builder "seq"   om2uz0i 11348
            5.6.4  Integer powers   cexp 11443
            5.6.5  Ordered pair theorem for nonnegative integers   nn0le2msqi 11621
            5.6.6  Factorial function   cfa 11627
            5.6.7  The binomial coefficient operation   cbc 11654
            5.6.8  The ` # ` (finite set size) function   chash 11679
                  5.6.8.1  Finite induction on the size of the first component of a binary relation   brfi1indlem 11775
            5.6.9  Words over a set   cword 11778
            5.6.10  Longer string literals   cs2 11866
      5.7  Elementary real and complex functions
            5.7.1  The "shift" operation   cshi 11942
            5.7.2  Real and imaginary parts; conjugate   ccj 11962
            5.7.3  Square root; absolute value   csqr 12099
      5.8  Elementary limits and convergence
            5.8.1  Superior limit (lim sup)   clsp 12325
            5.8.2  Limits   cli 12339
            5.8.3  Finite and infinite sums   csu 12540
            5.8.4  The binomial theorem   binomlem 12669
            5.8.5  The inclusion/exclusion principle   incexclem 12677
            5.8.6  Infinite sums (cont.)   isumshft 12680
            5.8.7  Miscellaneous converging and diverging sequences   divrcnv 12693
            5.8.8  Arithmetic series   arisum 12700
            5.8.9  Geometric series   expcnv 12704
            5.8.10  Ratio test for infinite series convergence   cvgrat 12721
            5.8.11  Mertens' theorem   mertenslem1 12722
      5.9  Elementary trigonometry
            5.9.1  The exponential, sine, and cosine functions   ce 12725
            5.9.2  _e is irrational   eirrlem 12864
      5.10  Cardinality of real and complex number subsets
            5.10.1  Countability of integers and rationals   xpnnen 12869
            5.10.2  The reals are uncountable   rpnnen2lem1 12875
*PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqr2irrlem 12908
            6.1.2  Some Number sets are chains of proper subsets   nthruc 12911
            6.1.3  The divides relation   cdivides 12913
            6.1.4  The division algorithm   divalglem0 12974
            6.1.5  Bit sequences   cbits 12992
            6.1.6  The greatest common divisor operator   cgcd 13067
            6.1.7  Bézout's identity   bezoutlem1 13099
            6.1.8  Algorithms   nn0seqcvgd 13122
            6.1.9  Euclid's Algorithm   eucalgval2 13133
      6.2  Elementary prime number theory
            6.2.1  Elementary properties   cprime 13140
            6.2.2  Properties of the canonical representation of a rational   cnumer 13186
            6.2.3  Euler's theorem   codz 13213
            6.2.4  Pythagorean Triples   coprimeprodsq 13244
            6.2.5  The prime count function   cpc 13271
            6.2.6  Pocklington's theorem   prmpwdvds 13333
            6.2.7  Infinite primes theorem   unbenlem 13337
            6.2.8  Sum of prime reciprocals   prmreclem1 13345
            6.2.9  Fundamental theorem of arithmetic   1arithlem1 13352
            6.2.10  Lagrange's four-square theorem   cgz 13358
            6.2.11  Van der Waerden's theorem   cvdwa 13394
            6.2.12  Ramsey's theorem   cram 13428
            6.2.13  Decimal arithmetic (cont.)   dec2dvds 13460
            6.2.14  Specific prime numbers   4nprm 13488
            6.2.15  Very large primes   1259lem1 13511
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
            *7.1.1  Basic definitions   cstr 13526
            7.1.2  Slot definitions   cplusg 13590
            7.1.3  Definition of the structure product   crest 13709
            7.1.4  Definition of the structure quotient   cordt 13782
      7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 13892
            7.2.2  Independent sets in a Moore system   mrisval 13916
            7.2.3  Algebraic closure systems   isacs 13937
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
            8.1.1  Categories   ccat 13950
            8.1.2  Opposite category   coppc 13998
            8.1.3  Monomorphisms and epimorphisms   cmon 14015
            8.1.4  Sections, inverses, isomorphisms   csect 14031
            8.1.5  Subcategories   cssc 14068
            8.1.6  Functors   cfunc 14112
            8.1.7  Full & faithful functors   cful 14160
            8.1.8  Natural transformations and the functor category   cnat 14199
      8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 14269
      8.3  Examples of categories
            8.3.1  The category of sets   csetc 14291
            8.3.2  The category of categories   ccatc 14310
      8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 14326
            8.4.2  Functor evaluation   cevlf 14367
            8.4.3  Hom functor   chof 14406
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 14458
            9.2.2  Lattices   clat 14563
            9.2.3  The dual of an ordered set   codu 14646
            9.2.4  Subset order structures   cipo 14669
            9.2.5  Distributive lattices   latmass 14706
            9.2.6  Posets and lattices as relations   cps 14716
            9.2.7  Directed sets, nets   cdir 14746
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
            10.1.1  Definition and basic properties   cmnd 14757
            10.1.2  Monoid homomorphisms and submonoids   cmhm 14809
            *10.1.3  Ordered group sum operation   gsumvallem1 14844
            10.1.4  Free monoids   cfrmd 14865
      10.2  Groups
            10.2.1  Definition and basic properties   df-grp 14885
            10.2.2  Subgroups and Quotient groups   csubg 15011
            10.2.3  Elementary theory of group homomorphisms   cghm 15076
            10.2.4  Isomorphisms of groups   cgim 15117
            10.2.5  Group actions   cga 15139
            10.2.6  Symmetry groups and Cayley's Theorem   csymg 15165
            10.2.7  Centralizers and centers   ccntz 15187
            10.2.8  The opposite group   coppg 15214
            10.2.9  p-Groups and Sylow groups; Sylow's theorems   cod 15236
            10.2.10  Direct products   clsm 15341
            10.2.11  Free groups   cefg 15411
      10.3  Abelian groups
            10.3.1  Definition and basic properties   ccmn 15485
            10.3.2  Cyclic groups   ccyg 15560
            10.3.3  Group sum operation   gsumval3a 15585
            10.3.4  Internal direct products   cdprd 15627
            10.3.5  The Fundamental Theorem of Abelian Groups   ablfacrplem 15696
      10.4  Rings
            10.4.1  Multiplicative Group   cmgp 15721
            10.4.2  Definition and basic properties   crg 15733
            10.4.3  Opposite ring   coppr 15800
            10.4.4  Divisibility   cdsr 15816
            10.4.5  Ring homomorphisms   crh 15890
      10.5  Division rings and fields
            10.5.1  Definition and basic properties   cdr 15908
            10.5.2  Subrings of a ring   csubrg 15937
            10.5.3  Absolute value (abstract algebra)   cabv 15977
            10.5.4  Star rings   cstf 16004
      10.6  Left modules
            10.6.1  Definition and basic properties   clmod 16023
            10.6.2  Subspaces and spans in a left module   clss 16081
            10.6.3  Homomorphisms and isomorphisms of left modules   clmhm 16168
            10.6.4  Subspace sum; bases for a left module   clbs 16219
      10.7  Vector spaces
            10.7.1  Definition and basic properties   clvec 16247
      10.8  Ideals
            10.8.1  The subring algebra; ideals   csra 16313
            10.8.2  Two-sided ideals and quotient rings   c2idl 16375
            10.8.3  Principal ideal rings. Divisibility in the integers   clpidl 16385
            10.8.4  Nonzero rings   cnzr 16401
            10.8.5  Left regular elements. More kinds of rings   crlreg 16412
      10.9  Associative algebras
            10.9.1  Definition and basic properties   casa 16442
      10.10  Abstract multivariate polynomials
            10.10.1  Definition and basic properties   cmps 16479
            10.10.2  Polynomial evaluation   evlslem4 16637
            10.10.3  Univariate polynomials   cps1 16642
      10.11  The complex numbers as an algebraic extensible structure
            10.11.1  Definition and basic properties   cpsmet 16758
            10.11.2  Algebraic constructions based on the complexes   czrh 16851
      10.12  Generalized pre-Hilbert and Hilbert spaces
            10.12.1  Definition and basic properties   cphl 16928
            10.12.2  Orthocomplements and closed subspaces   cocv 16960
            10.12.3  Orthogonal projection and orthonormal bases   cpj 17000
PART 11  BASIC TOPOLOGY
      11.1  Topology
            11.1.1  Topological spaces   ctop 17031
            11.1.2  TopBases for topologies   isbasisg 17085
            11.1.3  Examples of topologies   distop 17133
            11.1.4  Closure and interior   ccld 17153
            11.1.5  Neighborhoods   cnei 17234
            11.1.6  Limit points and perfect sets   clp 17271
            11.1.7  Subspace topologies   restrcl 17294
            11.1.8  Order topology   ordtbaslem 17325
            11.1.9  Limits and continuity in topological spaces   ccn 17361
            11.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 17443
            11.1.11  Compactness   ccmp 17522
            11.1.12  Bolzano-Weierstrass theorem   bwth 17546
            11.1.13  Connectedness   ccon 17547
            11.1.14  First- and second-countability   c1stc 17573
            11.1.15  Local topological properties   clly 17600
            11.1.16  Compactly generated spaces   ckgen 17638
            11.1.17  Product topologies   ctx 17665
            11.1.18  Continuous function-builders   cnmptid 17766
            11.1.19  Quotient maps and quotient topology   ckq 17798
            11.1.20  Homeomorphisms   chmeo 17858
      11.2  Filters and filter bases
            11.2.1  Filter bases   elmptrab 17932
            11.2.2  Filters   cfil 17950
            11.2.3  Ultrafilters   cufil 18004
            11.2.4  Filter limits   cfm 18038
            11.2.5  Extension by continuity   ccnext 18163
            11.2.6  Topological groups   ctmd 18173
            11.2.7  Infinite group sum on topological groups   ctsu 18228
            11.2.8  Topological rings, fields, vector spaces   ctrg 18258
      11.3  Uniform Stuctures and Spaces
            11.3.1  Uniform structures   cust 18302
            11.3.2  The topology induced by an uniform structure   cutop 18333
            11.3.3  Uniform Spaces   cuss 18356
            11.3.4  Uniform continuity   cucn 18378
            11.3.5  Cauchy filters in uniform spaces   ccfilu 18389
            11.3.6  Complete uniform spaces   ccusp 18400
      11.4  Metric spaces
            11.4.1  Pseudometric spaces   ispsmet 18408
            11.4.2  Basic metric space properties   cxme 18420
            11.4.3  Metric space balls   blfvalps 18486
            11.4.4  Open sets of a metric space   mopnval 18541
            11.4.5  Continuity in metric spaces   metcnp3 18643
            11.4.6  The uniform structure generated by a metric   metuvalOLD 18652
            11.4.7  Examples of metric spaces   dscmet 18693
            11.4.8  Normed algebraic structures   cnm 18697
            11.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 18812
            11.4.10  Topology on the reals   qtopbaslem 18865
            11.4.11  Topological definitions using the reals   cii 18978
            11.4.12  Path homotopy   chtpy 19065
            11.4.13  The fundamental group   cpco 19098
      11.5  Complex metric vector spaces
            11.5.1  Complex left modules   cclm 19160
            11.5.2  Complex pre-Hilbert space   ccph 19202
            11.5.3  Convergence and completeness   ccfil 19278
            11.5.4  Baire's Category Theorem   bcthlem1 19350
            11.5.5  Banach spaces and complex Hilbert spaces   ccms 19358
            11.5.6  Minimizing Vector Theorem   minveclem1 19398
            11.5.7  Projection Theorem   pjthlem1 19411
PART 12  BASIC REAL AND COMPLEX ANALYSIS
      12.1  Continuity
            12.1.1  Intermediate value theorem   pmltpclem1 19418
      12.2  Integrals
            12.2.1  Lebesgue measure   covol 19432
            12.2.2  Lebesgue integration   cmbf 19579
                  12.2.2.1  Lesbesgue integral   cmbf 19579
                  12.2.2.2  Lesbesgue directed integral   cdit 19806
      12.3  Derivatives
            12.3.1  Real and complex differentiation   climc 19822
                  12.3.1.1  Derivatives of functions of one complex or real variable   climc 19822
                  12.3.1.2  Results on real differentiation   dvferm1lem 19941
PART 13  BASIC REAL AND COMPLEX FUNCTIONS
      13.1  Polynomials
            13.1.1  Abstract polynomials, continued   evlslem6 20007
            13.1.2  Polynomial degrees   cmdg 20049
            13.1.3  The division algorithm for univariate polynomials   cmn1 20121
            13.1.4  Elementary properties of complex polynomials   cply 20176
            13.1.5  The division algorithm for polynomials   cquot 20280
            13.1.6  Algebraic numbers   caa 20304
            13.1.7  Liouville's approximation theorem   aalioulem1 20322
      13.2  Sequences and series
            13.2.1  Taylor polynomials and Taylor's theorem   ctayl 20342
            13.2.2  Uniform convergence   culm 20365
            13.2.3  Power series   pserval 20399
      13.3  Basic trigonometry
            13.3.1  The exponential, sine, and cosine functions (cont.)   efcn 20432
            13.3.2  Properties of pi = 3.14159...   pilem1 20440
            13.3.3  Mapping of the exponential function   efgh 20516
            13.3.4  The natural logarithm on complex numbers   clog 20525
            13.3.5  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 20716
            13.3.6  Solutions of quadratic, cubic, and quartic equations   quad2 20752
            13.3.7  Inverse trigonometric functions   casin 20775
            13.3.8  The Birthday Problem   log2ublem1 20859
            13.3.9  Areas in R^2   carea 20867
            13.3.10  More miscellaneous converging sequences   rlimcnp 20877
            13.3.11  Inequality of arithmetic and geometric means   cvxcl 20896
            13.3.12  Euler-Mascheroni constant   cem 20903
      13.4  Basic number theory
            13.4.1  Wilson's theorem   wilthlem1 20924
            13.4.2  The Fundamental Theorem of Algebra   ftalem1 20928
            13.4.3  The Basel problem (ζ(2) = π2/6)   basellem1 20936
            13.4.4  Number-theoretical functions   ccht 20946
            13.4.5  Perfect Number Theorem   mersenne 21084
            13.4.6  Characters of Z/nZ   cdchr 21089
            13.4.7  Bertrand's postulate   bcctr 21132
            13.4.8  Legendre symbol   clgs 21151
            13.4.9  Quadratic reciprocity   lgseisenlem1 21206
            13.4.10  All primes 4n+1 are the sum of two squares   2sqlem1 21220
            13.4.11  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 21236
            13.4.12  The Prime Number Theorem   mudivsum 21297
            13.4.13  Ostrowski's theorem   abvcxp 21382
*PART 14  GRAPH THEORY
      14.1  Undirected graphs - basics
            14.1.1  Undirected hypergraphs   cuhg 21407
            14.1.2  Undirected multigraphs   cumg 21420
            14.1.3  Undirected simple graphs   cuslg 21437
                  14.1.3.1  Undirected simple graphs - basics   cuslg 21437
                  14.1.3.2  Undirected simple graphs - examples   usgraexvlem 21487
                  14.1.3.3  Finite undirected simple graphs   fiusgraedgfi 21494
            14.1.4  Neighbors, complete graphs and universal vertices   cnbgra 21503
                  14.1.4.1  Neighbors   nbgraop 21509
                  14.1.4.2  Complete graphs   iscusgra 21538
                  14.1.4.3  Universal vertices   isuvtx 21570
            14.1.5  Walks, paths and cycles   cwalk 21579
                  14.1.5.1  Walks and trails   wlks 21599
                  14.1.5.2  Paths and simple paths   pths 21639
                  14.1.5.3  Circuits and cycles   crcts 21682
                  14.1.5.4  Connected graphs   cconngra 21729
            14.1.6  Vertex Degree   cvdg 21737
      14.2  Eulerian paths and the Konigsberg Bridge problem
            14.2.1  Eulerian paths   ceup 21757
            14.2.2  The Konigsberg Bridge problem   vdeg0i 21777
PART 15  GUIDES AND MISCELLANEA
      15.1  Guides (conventions, explanations, and examples)
            *15.1.1  Conventions   conventions 21783
            15.1.2  Natural deduction   natded 21784
            *15.1.3  Natural deduction examples   ex-natded5.2 21785
            15.1.4  Definitional examples   ex-or 21802
      15.2  Humor
            15.2.1  April Fool's theorem   avril1 21830
      15.3  (Future - to be reviewed and classified)
            15.3.1  Planar incidence geometry   cplig 21836
            15.3.2  Algebra preliminaries   crpm 21841
            15.3.3  Transitive closure   ctcl 21843
*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 21847
            16.1.2  Definition and basic properties of Abelian groups   cablo 21942
            16.1.3  Subgroups   csubgo 21962
            16.1.4  Operation properties   cass 21973
            16.1.5  Group-like structures   cmagm 21979
            16.1.6  Examples of Abelian groups   ablosn 22008
            16.1.7  Group homomorphism and isomorphism   cghom 22018
      16.2  Additional material on rings and fields
            16.2.1  Definition and basic properties   crngo 22036
            16.2.2  Examples of rings   cnrngo 22064
            16.2.3  Division Rings   cdrng 22066
            16.2.4  Star Fields   csfld 22069
            16.2.5  Fields and Rings   ccm2 22071
PART 17  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      17.1  Complex vector spaces
            17.1.1  Definition and basic properties   cvc 22097
            17.1.2  Examples of complex vector spaces   cncvc 22135
      17.2  Normed complex vector spaces
            17.2.1  Definition and basic properties   cnv 22136
            17.2.2  Examples of normed complex vector spaces   cnnv 22241
            17.2.3  Induced metric of a normed complex vector space   imsval 22250
            17.2.4  Inner product   cdip 22269
            17.2.5  Subspaces   css 22293
      17.3  Operators on complex vector spaces
            17.3.1  Definitions and basic properties   clno 22314
      17.4  Inner product (pre-Hilbert) spaces
            17.4.1  Definition and basic properties   ccphlo 22386
            17.4.2  Examples of pre-Hilbert spaces   cncph 22393
            17.4.3  Properties of pre-Hilbert spaces   isph 22396
      17.5  Complex Banach spaces
            17.5.1  Definition and basic properties   ccbn 22437
            17.5.2  Examples of complex Banach spaces   cnbn 22444
            17.5.3  Uniform Boundedness Theorem   ubthlem1 22445
            17.5.4  Minimizing Vector Theorem   minvecolem1 22449
      17.6  Complex Hilbert spaces
            17.6.1  Definition and basic properties   chlo 22460
            17.6.2  Standard axioms for a complex Hilbert space   hlex 22473
            17.6.3  Examples of complex Hilbert spaces   cnchl 22491
            17.6.4  Subspaces   ssphl 22492
            17.6.5  Hellinger-Toeplitz Theorem   htthlem 22493
*PART 18  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      18.1  Axiomatization of complex pre-Hilbert spaces
            18.1.1  Basic Hilbert space definitions   chil 22495
            18.1.2  Preliminary ZFC lemmas   df-hnorm 22544
            *18.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 22557
            *18.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 22575
            18.1.5  Vector operations   hvmulex 22587
            18.1.6  Inner product postulates for a Hilbert space   ax-hfi 22654
      18.2  Inner product and norms
            18.2.1  Inner product   his5 22661
            18.2.2  Norms   dfhnorm2 22697
            18.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 22735
            18.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 22754
      18.3  Cauchy sequences and completeness axiom
            18.3.1  Cauchy sequences and limits   hcau 22759
            18.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 22769
            18.3.3  Completeness postulate for a Hilbert space   ax-hcompl 22777
            18.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 22778
      18.4  Subspaces and projections
            18.4.1  Subspaces   df-sh 22782
            18.4.2  Closed subspaces   df-ch 22797
            18.4.3  Orthocomplements   df-oc 22827
            18.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 22883
            18.4.5  Projection theorem   pjhthlem1 22966
            18.4.6  Projectors   df-pjh 22970
      18.5  Properties of Hilbert subspaces
            18.5.1  Orthomodular law   omlsilem 22977
            18.5.2  Projectors (cont.)   pjhtheu2 22991
            18.5.3  Hilbert lattice operations   sh0le 23015
            18.5.4  Span (cont.) and one-dimensional subspaces   spansn0 23116
            18.5.5  Commutes relation for Hilbert lattice elements   df-cm 23158
            18.5.6  Foulis-Holland theorem   fh1 23193
            18.5.7  Quantum Logic Explorer axioms   qlax1i 23202
            18.5.8  Orthogonal subspaces   chscllem1 23212
            18.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 23229
            18.5.10  Projectors (cont.)   pjorthi 23244
            18.5.11  Mayet's equation E_3   mayete3i 23303
      18.6  Operators on Hilbert spaces
            *18.6.1  Operator sum, difference, and scalar multiplication   df-hosum 23306
            18.6.2  Zero and identity operators   df-h0op 23324
            18.6.3  Operations on Hilbert space operators   hoaddcl 23334
            18.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 23415
            18.6.5  Linear and continuous functionals and norms   df-nmfn 23421
            18.6.6  Adjoint   df-adjh 23425
            18.6.7  Dirac bra-ket notation   df-bra 23426
            18.6.8  Positive operators   df-leop 23428
            18.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 23429
            18.6.10  Theorems about operators and functionals   nmopval 23432
            18.6.11  Riesz lemma   riesz3i 23638
            18.6.12  Adjoints (cont.)   cnlnadjlem1 23643
            18.6.13  Quantum computation error bound theorem   unierri 23680
            18.6.14  Dirac bra-ket notation (cont.)   branmfn 23681
            18.6.15  Positive operators (cont.)   leopg 23698
            18.6.16  Projectors as operators   pjhmopi 23722
      18.7  States on a Hilbert lattice and Godowski's equation
            18.7.1  States on a Hilbert lattice   df-st 23787
            18.7.2  Godowski's equation   golem1 23847
      18.8  Cover relation, atoms, exchange axiom, and modular symmetry
            18.8.1  Covers relation; modular pairs   df-cv 23855
            18.8.2  Atoms   df-at 23914
            18.8.3  Superposition principle   superpos 23930
            18.8.4  Atoms, exchange and covering properties, atomicity   chcv1 23931
            18.8.5  Irreducibility   chirredlem1 23966
            18.8.6  Atoms (cont.)   atcvat3i 23972
            18.8.7  Modular symmetry   mdsymlem1 23979
PART 19  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      19.1  Mathboxes for user contributions
            19.1.1  Mathbox guidelines   mathbox 24018
      19.2  Mathbox for Stefan Allan
      19.3  Mathbox for Thierry Arnoux
            19.3.1  Propositional Calculus - misc additions   bian1d 24023
            19.3.2  Predicate Calculus   spc2ed 24034
                  19.3.2.1  Predicate Calculus - misc additions   spc2ed 24034
                  19.3.2.2  Restricted quantification - misc additions   reximddv 24037
                  19.3.2.3  Substitution (without distinct variables) - misc additions   clelsb3f 24043
                  19.3.2.4  Existential "at most one" - misc additions   moel 24046
                  19.3.2.5  Existential uniqueness - misc additions   2reuswap2 24051
                  19.3.2.6  Restricted "at most one" - misc additions   rmoxfrdOLD 24055
            19.3.3  General Set Theory   ceqsexv2d 24061
                  19.3.3.1  Class abstractions (a.k.a. class builders)   ceqsexv2d 24061
                  19.3.3.2  Image Sets   abrexdomjm 24069
                  19.3.3.3  Set relations and operations - misc additions   eqri 24075
                  19.3.3.4  Unordered pairs   elpreq 24081
                  19.3.3.5  Conditional operator - misc additions   ifeqeqx 24083
                  19.3.3.6  Indexed union - misc additions   iuneq12daf 24089
                  19.3.3.7  Disjointness - misc additions   disjnf 24097
            19.3.4  Relations and Functions   dfrel4 24118
                  19.3.4.1  Relations - misc additions   dfrel4 24118
                  19.3.4.2  Functions - misc additions   fdmrn 24128
                  19.3.4.3  Operations - misc additions   mpt2mptxf 24181
                  19.3.4.4  Isomorphisms - misc. add.   gtiso 24182
                  19.3.4.5  Disjointness (additional proof requiring functions)   disjdsct 24184
                  19.3.4.6  First and second members of an ordered pair - misc additions   df1stres 24185
                  19.3.4.7  Supremum - misc additions   supval3 24192
                  19.3.4.8  Finite Sets   unifi3 24194
                  19.3.4.9  Countable Sets   nnct 24195
            19.3.5  Real and Complex Numbers   addeq0 24225
                  19.3.5.1  Complex addition - misc. additions   addeq0 24225
                  19.3.5.2  Ordering on reals - misc additions   lt2addrd 24226
                  19.3.5.3  Extended reals - misc additions   xgepnf 24232
                  19.3.5.4  Real number intervals - misc additions   icossicc 24245
                  19.3.5.5  Finite intervals of integers - misc additions   fzssnn 24263
                  19.3.5.6  Half-open integer ranges - misc additions   iundisjfi 24269
                  19.3.5.7  The ` # ` (finite set size) function - misc additions   hashresfn 24273
                  19.3.5.8  The greatest common divisor operator - misc. add   numdenneg 24277
                  19.3.5.9  Integers   nn0indd 24279
                  19.3.5.10  Division in the extended real number system   cxdiv 24283
            19.3.6  Structure builders   ress0g 24302
                  19.3.6.1  Structure builder restriction operator   ress0g 24302
                  19.3.6.2  The opposite group   oppgle 24306
                  19.3.6.3  Posets   ressprs 24308
                  19.3.6.4  Complete lattices   clatp0cl 24324
                  19.3.6.5  Extended reals Structure - misc additions   ax-xrssca 24326
                  19.3.6.6  The extended non-negative real numbers commutative monoid   xrge0base 24338
            19.3.7  Algebra   abliso 24352
                  19.3.7.1  Monoids Homomorphisms   abliso 24352
                  19.3.7.2  Ordered monoids and groups   comnd 24353
                  19.3.7.3  Signum in an ordered monoid   csgns 24381
                  19.3.7.4  The Archimedean property for generic ordered algebraic structures   cinftm 24386
                  19.3.7.5  Semirings   csrg 24409
                  19.3.7.6  Semiring left modules   cslmd 24434
                  19.3.7.7  Finitely supported group sums - misc additions   sumpr 24460
                  19.3.7.8  Rings - misc additions   rngurd 24478
                  19.3.7.9  Ordered rings and fields   corng 24485
                  19.3.7.10  Ring homomorphisms - misc additions   rhmdvdsr 24508
                  19.3.7.11  The ordered commutative ring of integers   zzsbase 24515
                  19.3.7.12  Scalar restriction operation   cresv 24521
                  19.3.7.13  The commutative ring of gaussian integers   gzcrng 24536
                  19.3.7.14  The archimedean ordered field of real numbers   rebase 24537
            19.3.8  Topology   cmetid 24553
                  19.3.8.1  Pseudometrics   cmetid 24553
                  19.3.8.2  Continuity - misc additions   hauseqcn 24565
                  19.3.8.3  Topology of the closed unit   unitsscn 24566
                  19.3.8.4  Topology of ` ( RR X. RR ) `   unicls 24573
                  19.3.8.5  Order topology - misc. additions   cnvordtrestixx 24583
                  19.3.8.6  Continuity in topological spaces - misc. additions   mndpluscn 24596
                  19.3.8.7  Topology of the extended non-negative real numbers ordered monoid   xrge0hmph 24602
                  19.3.8.8  Limits - misc additions   lmlim 24617
                  19.3.8.9  Univariate polynomials   pl1cn 24625
            19.3.9  Uniform Stuctures and Spaces   chcmp 24626
                  19.3.9.1  Hausdorff uniform completion   chcmp 24626
            19.3.10  Topology and algebraic structures   zzsnm 24628
                  19.3.10.1  The norm on the ring of the integer numbers   zzsnm 24628
                  19.3.10.2  The complete ordered field of the real numbers   recms 24629
                  19.3.10.3  Topological ` ZZ ` -modules   zlm0 24632
                  19.3.10.4  Canonical embedding of the field of the rational numbers into a division ring   cqqh 24642
                  19.3.10.5  Canonical embedding of the real numbers into a complete ordered field   crrh 24663
                  19.3.10.6  Embedding from the extended real numbers into a complete lattice   cxrh 24683
                  19.3.10.7  Canonical embeddings into the ordered field of the real numbers   zrhre 24686
            19.3.11  Real and complex functions   nexple 24689
                  19.3.11.1  Integer powers - misc. additions   nexple 24689
                  *19.3.11.2  Logarithm laws generalized to an arbitrary base - logb   clogb 24690
                  19.3.11.3  Indicator Functions   cind 24710
                  19.3.11.4  Extended sum   cesum 24726
            19.3.12  Mixed Function/Constant operation   cofc 24780
            19.3.13  Abstract measure   csiga 24792
                  19.3.13.1  Sigma-Algebra   csiga 24792
                  19.3.13.2  Generated Sigma-Algebra   csigagen 24823
                  19.3.13.3  The Borel algebra on the real numbers   cbrsiga 24837
                  19.3.13.4  Product Sigma-Algebra   csx 24844
                  19.3.13.5  Measures   cmeas 24851
                  19.3.13.6  The counting measure   cntmeas 24882
                  19.3.13.7  The Lebesgue measure - misc additions   volss 24885
                  19.3.13.8  The Dirac delta measure   cdde 24890
                  19.3.13.9  The 'almost everywhere' relation   cae 24895
                  19.3.13.10  Measurable functions   cmbfm 24907
                  19.3.13.11  Borel Algebra on ` ( RR X. RR ) `   br2base 24926
            19.3.14  Integration   itgeq12dv 24948
                  19.3.14.1  Lebesgue integral - misc additions   itgeq12dv 24948
                  19.3.14.2  Bochner integral   citgm 24949
            19.3.15  Euler's partition theorem   oddpwdc 24973
            19.3.16  Probability   cprb 25002
                  19.3.16.1  Probability Theory   cprb 25002
                  19.3.16.2  Conditional Probabilities   ccprob 25026
                  19.3.16.3  Real Valued Random Variables   crrv 25035
                  19.3.16.4  Preimage set mapping operator   corvc 25050
                  19.3.16.5  Distribution Functions   orvcelval 25063
                  19.3.16.6  Cumulative Distribution Functions   orvclteel 25067
                  19.3.16.7  Probabilities - example   coinfliplem 25073
                  19.3.16.8  Bertrand's Ballot Problem   ballotlemoex 25080
            19.3.17  Signum (sgn or sign) function   csgn 25133
            19.3.18  Descartes's rule of signs   plyrecld 25143
                  19.3.18.1  Intermediate value theorem   plyrecld 25143
                  19.3.18.2  Sign changes in a word over real numbers   signspval 25146
      19.4  Mathbox for Mario Carneiro
            19.4.1  Miscellaneous stuff   quartfull 25152
            19.4.2  Zeta function   czeta 25153
            19.4.3  Gamma function   clgam 25156
            19.4.4  Derangements and the Subfactorial   deranglem 25208
            19.4.5  The Erdős-Szekeres theorem   erdszelem1 25233
            19.4.6  The Kuratowski closure-complement theorem   kur14lem1 25248
            19.4.7  Retracts and sections   cretr 25259
            19.4.8  Path-connected and simply connected spaces   cpcon 25262
            19.4.9  Covering maps   ccvm 25298
            19.4.10  Normal numbers   snmlff 25372
            19.4.11  Godel-sets of formulas   cgoe 25376
            19.4.12  Models of ZF   cgze 25404
            19.4.13  Splitting fields   citr 25418
            19.4.14  p-adic number fields   czr 25434
      19.5  Mathbox for Paul Chapman
            19.5.1  Group homomorphism and isomorphism   ghomgrpilem1 25452
            19.5.2  Real and complex numbers (cont.)   climuzcnv 25464
            19.5.3  Miscellaneous theorems   elfzm12 25468
      *19.6  Mathbox for Drahflow
      19.7  Mathbox for Scott Fenton
            19.7.1  ZFC Axioms in primitive form   axextprim 25503
            19.7.2  Untangled classes   untelirr 25510
            19.7.3  Extra propositional calculus theorems   3orel1 25517
            19.7.4  Misc. Useful Theorems   nepss 25528
            19.7.5  Properties of reals and complexes   sqdivzi 25537
            19.7.6  Product sequences   prodf 25568
            19.7.7  Non-trivial convergence   ntrivcvg 25578
            19.7.8  Complex products   cprod 25584
            19.7.9  Finite products   fprod 25620
            19.7.10  Infinite products   iprodclim 25664
            19.7.11  Falling and Rising Factorial   cfallfac 25673
            19.7.12  Factorial limits   faclimlem1 25715
            19.7.13  Greatest common divisor and divisibility   pdivsq 25721
            19.7.14  Properties of relationships   brtp 25725
            19.7.15  Properties of functions and mappings   funpsstri 25742
            19.7.16  Epsilon induction   setinds 25758
            19.7.17  Ordinal numbers   elpotr 25761
            19.7.18  Defined equality axioms   axextdfeq 25778
            19.7.19  Hypothesis builders   hbntg 25786
            19.7.20  The Predecessor Class   cpred 25791
            19.7.21  (Trans)finite Recursion Theorems   tfisg 25832
            19.7.22  Well-founded induction   tz6.26 25833
            19.7.23  Transitive closure under a relationship   ctrpred 25848
            19.7.24  Founded Induction   frmin 25870
            19.7.25  Ordering Ordinal Sequences   orderseqlem 25880
            19.7.26  Well-founded recursion   cwrecs 25883
            19.7.27  Transfinite recursion via Well-founded recursion   tfrALTlem 25910
            19.7.28  Well-founded zero, successor, and limits   cwsuc 25914
            19.7.29  Founded Recursion   frr3g 25934
            19.7.30  Surreal Numbers   csur 25948
            19.7.31  Surreal Numbers: Ordering   sltsolem1 25976
            19.7.32  Surreal Numbers: Birthday Function   bdayfo 25983
            19.7.33  Surreal Numbers: Density   fvnobday 25990
            19.7.34  Surreal Numbers: Density   nodenselem3 25991
            19.7.35  Surreal Numbers: Upper and Lower Bounds   nobndlem1 26000
            19.7.36  Surreal Numbers: Full-Eta Property   nofulllem1 26010
            19.7.37  Symmetric difference   csymdif 26015
            19.7.38  Quantifier-free definitions   ctxp 26027
            19.7.39  Alternate ordered pairs   caltop 26154
            19.7.40  Tarskian geometry   cee 26180
            19.7.41  Tarski's axioms for geometry   axdimuniq 26205
            19.7.42  Congruence properties   cofs 26269
            19.7.43  Betweenness properties   btwntriv2 26299
            19.7.44  Segment Transportation   ctransport 26316
            19.7.45  Properties relating betweenness and congruence   cifs 26322
            19.7.46  Connectivity of betweenness   btwnconn1lem1 26374
            19.7.47  Segment less than or equal to   csegle 26393
            19.7.48  Outside of relationship   coutsideof 26406
            19.7.49  Lines and Rays   cline2 26421
            19.7.50  Bernoulli polynomials and sums of k-th powers   cbp 26445
            19.7.51  Rank theorems   rankung 26460
            19.7.52  Hereditarily Finite Sets   chf 26466
      19.8  Mathbox for Anthony Hart
            19.8.1  Propositional Calculus   tb-ax1 26481
            19.8.2  Predicate Calculus   quantriv 26503
            19.8.3  Misc. Single Axiom Systems   meran1 26514
            19.8.4  Connective Symmetry   negsym1 26520
      19.9  Mathbox for Chen-Pang He
            19.9.1  Ordinal topology   ontopbas 26531
      19.10  Mathbox for Jeff Hoffman
            19.10.1  Inferences for finite induction on generic function values   fveleq 26554
            19.10.2  gdc.mm   nnssi2 26558
      *19.11  Mathbox for Wolf Lammen
      19.12  Mathbox for Brendan Leahy
      19.13  Mathbox for Jeff Hankins
            19.13.1  Miscellany   a1i13 26650
            19.13.2  Basic topological facts   topbnd 26679
            19.13.3  Topology of the real numbers   ivthALT 26690
            19.13.4  Refinements   cfne 26691
            19.13.5  Neighborhood bases determine topologies   neibastop1 26740
            19.13.6  Lattice structure of topologies   topmtcl 26744
            19.13.7  Filter bases   fgmin 26751
            19.13.8  Directed sets, nets   tailfval 26753
      19.14  Mathbox for Jeff Madsen
            19.14.1  Logic and set theory   anim12da 26764
            19.14.2  Real and complex numbers; integers   filbcmb 26794
            19.14.3  Sequences and sums   sdclem2 26798
            19.14.4  Topology   subspopn 26810
            19.14.5  Metric spaces   metf1o 26813
            19.14.6  Continuous maps and homeomorphisms   constcncf 26820
            19.14.7  Boundedness   ctotbnd 26827
            19.14.8  Isometries   cismty 26859
            19.14.9  Heine-Borel Theorem   heibor1lem 26870
            19.14.10  Banach Fixed Point Theorem   bfplem1 26883
            19.14.11  Euclidean space   crrn 26886
            19.14.12  Intervals (continued)   ismrer1 26899
            19.14.13  Groups and related structures   exidcl 26903
            19.14.14  Rings   rngonegcl 26913
            19.14.15  Ring homomorphisms   crnghom 26928
            19.14.16  Commutative rings   ccring 26957
            19.14.17  Ideals   cidl 26969
            19.14.18  Prime rings and integral domains   cprrng 27008
            19.14.19  Ideal generators   cigen 27021
      19.15  Mathbox for Giovanni Mascellani
            *19.15.1  Tools for automatic proof building   efald2 27040
            *19.15.2  Tseitin axioms   fald 27063
            *19.15.3  Equality deductions   sbcbi2 27090
            *19.15.4  Miscellanea   sbcom3OLD 27105
      19.16  Mathbox for Rodolfo Medina
            19.16.1  Partitions   prtlem60 27112
      19.17  Mathbox for Stefan O'Rear
            19.17.1  Additional elementary logic and set theory   nelss 27154
            19.17.2  Additional theory of functions   fninfp 27157
            19.17.3  Extensions beyond function theory   gsumvsmul 27167
            19.17.4  Additional topology   elrfi 27170
            19.17.5  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 27174
            19.17.6  Algebraic closure systems   cnacs 27178
            19.17.7  Miscellanea 1. Map utilities   constmap 27189
            19.17.8  Miscellanea for polynomials   ofmpteq 27198
            19.17.9  Multivariate polynomials over the integers   cmzpcl 27200
            19.17.10  Miscellanea for Diophantine sets 1   coeq0 27232
            19.17.11  Diophantine sets 1: definitions   cdioph 27235
            19.17.12  Diophantine sets 2 miscellanea   ellz1 27247
            19.17.13  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 27253
            19.17.14  Diophantine sets 3: construction   diophrex 27256
            19.17.15  Diophantine sets 4 miscellanea   2sbcrex 27265
            19.17.16  Diophantine sets 4: Quantification   rexrabdioph 27274
            19.17.17  Diophantine sets 5: Arithmetic sets   rabdiophlem1 27281
            19.17.18  Diophantine sets 6 miscellanea   fz1ssnn 27291
            19.17.19  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 27292
            19.17.20  Pigeonhole Principle and cardinality helpers   fphpd 27297
            19.17.21  A non-closed set of reals is infinite   rencldnfilem 27301
            19.17.22  Miscellanea for Lagrange's theorem   icodiamlt 27303
            19.17.23  Lagrange's rational approximation theorem   irrapxlem1 27305
            19.17.24  Pell equations 1: A nontrivial solution always exists   pellexlem1 27312
            19.17.25  Pell equations 2: Algebraic number theory of the solution set   csquarenn 27319
            19.17.26  Pell equations 3: characterizing fundamental solution   infmrgelbi 27361
            19.17.27  Logarithm laws generalized to an arbitrary base   reglogcl 27373
            19.17.28  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 27381
            19.17.29  X and Y sequences 1: Definition and recurrence laws   crmx 27383
            19.17.30  Ordering and induction lemmas for the integers   monotuz 27424
            19.17.31  X and Y sequences 2: Order properties   rmxypos 27432
            19.17.32  Congruential equations   congtr 27450
            19.17.33  Alternating congruential equations   acongid 27460
            19.17.34  Additional theorems on integer divisibility   bezoutr 27470
            19.17.35  X and Y sequences 3: Divisibility properties   jm2.18 27479
            19.17.36  X and Y sequences 4: Diophantine representability of Y   jm2.27a 27496
            19.17.37  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 27506
            19.17.38  Uncategorized stuff not associated with a major project   setindtr 27515
            19.17.39  More equivalents of the Axiom of Choice   axac10 27524
            19.17.40  Finitely generated left modules   clfig 27562
            19.17.41  Noetherian left modules I   clnm 27570
            19.17.42  Addenda for structure powers   pwssplit0 27584
            19.17.43  Direct sum of left modules   cdsmm 27594
            19.17.44  Free modules   cfrlm 27609
            19.17.45  Every set admits a group structure iff choice   unxpwdom3 27653
            19.17.46  Independent sets and families   clindf 27671
            19.17.47  Characterization of free modules   lmimlbs 27703
            19.17.48  Noetherian rings and left modules II   clnr 27710
            19.17.49  Hilbert's Basis Theorem   cldgis 27722
            19.17.50  Additional material on polynomials [DEPRECATED]   cmnc 27732
            19.17.51  Degree and minimal polynomial of algebraic numbers   cdgraa 27742
            19.17.52  Algebraic integers I   citgo 27759
            19.17.53  Finite cardinality [SO]   en1uniel 27777
            *19.17.54  Words in monoids and ordered group sum   issubmd 27780
            19.17.55  Transpositions in the symmetric group   cpmtr 27781
            19.17.56  The sign of a permutation   cpsgn 27811
            19.17.57  The matrix algebra   cmmul 27836
            19.17.58  The determinant   cmdat 27880
            19.17.59  Endomorphism algebra   cmend 27886
            19.17.60  Subfields   csdrg 27900
            19.17.61  Cyclic groups and order   idomrootle 27908
            19.17.62  Cyclotomic polynomials   ccytp 27918
            19.17.63  Miscellaneous topology   fgraphopab 27926
      19.18  Mathbox for Steve Rodriguez
            19.18.1  Miscellanea   iso0 27933
            19.18.2  Function operations   caofcan 27937
            19.18.3  Calculus   lhe4.4ex1a 27943
      19.19  Mathbox for Andrew Salmon
            19.19.1  Principia Mathematica * 10   pm10.12 27950
            19.19.2  Principia Mathematica * 11   2alanimi 27964
            19.19.3  Predicate Calculus   sbeqal1 27994
            19.19.4  Principia Mathematica * 13 and * 14   pm13.13a 28004
            19.19.5  Set Theory   elnev 28035
            19.19.6  Arithmetic   addcomgi 28056
            19.19.7  Geometry   cplusr 28057
      19.20  Mathbox for Glauco Siliprandi
            19.20.1  Miscellanea   ssrexf 28079
            19.20.2  Finite multiplication of numbers and finite multiplication of functions   fmul01 28105
            19.20.3  Limits   clim1fr1 28122
            19.20.4  Derivatives   dvsinexp 28135
            19.20.5  Integrals   ioovolcl 28137
            19.20.6  Stone Weierstrass theorem - real version   stoweidlem1 28145
            19.20.7  Wallis' product for π   wallispilem1 28209
            19.20.8  Stirling's approximation formula for ` n ` factorial   stirlinglem1 28218
      19.21  Mathbox for Saveliy Skresanov
            19.21.1  Ceva's theorem   sigarval 28235
      19.22  Mathbox for Jarvin Udandy
      19.23  Mathbox for Alexander van der Vekens
            19.23.1  Double restricted existential uniqueness   r19.32 28340
                  19.23.1.1  Restricted quantification (extension)   r19.32 28340
                  19.23.1.2  The empty set (extension)   raaan2 28348
                  19.23.1.3  Restricted uniqueness and "at most one" quantification   rmoimi 28349
                  19.23.1.4  Analogs to Existential uniqueness (double quantification)   2reurex 28354
            *19.23.2  Alternative definitions of function's and operation's values   wdfat 28366
                  19.23.2.1  Restricted quantification (extension)   ralbinrald 28372
                  19.23.2.2  The universal class (extension)   nvelim 28373
                  19.23.2.3  Introduce the Axiom of Power Sets (extension)   alneu 28374
                  19.23.2.4  Relations (extension)   sbcrel 28376
                  19.23.2.5  Functions (extension)   sbcfung 28383
                  19.23.2.6  Predicate "defined at"   dfateq12d 28389
                  19.23.2.7  Alternative definition of the value of a function   dfafv2 28392
                  19.23.2.8  Alternative definition of the value of an operation   aoveq123d 28438
            *19.23.3  Auxiliary theorems for graph theory   jaoi3 28468
                  19.23.3.1  Logical disjunction and conjunction   jaoi3 28468
                  19.23.3.2  Abbreviated conjunction and disjunction of three wff's   3an4anass 28470
                  19.23.3.3  Negated equality and membership - extension   eqneqall 28471
                  19.23.3.4  Proper substitution of classes for sets into classes - extension   csbprg 28473
                  19.23.3.5  The empty set - extension   rspn0 28474
                  19.23.3.6  "Weak deduction theorem" for set theory - extension   ifeqda 28477
                  19.23.3.7  Power classes - extension   3xpexg 28479
                  19.23.3.8  Unordered and ordered pairs - extension   nelprd 28480
                  19.23.3.9  Indexed union and intersection - extension   iunxprg 28494
                  19.23.3.10  Ordered-pair class abstractions - extension   elopaelxp 28495
                  19.23.3.11  Introduce the Axiom of Union - extension   ralxfrd2 28497
                  19.23.3.12  Relations - extension   resisresindm 28499
                  19.23.3.13  Functions - extension   sbcfng 28501
                  19.23.3.14  Operations - extension   oprabv 28519
                  19.23.3.15  Equinumerosity - extension   resfnfinfin 28525
                  19.23.3.16  Subtraction - extension   cnm1cn 28527
                  19.23.3.17  Multiplication - extension   kcnktkm1cn 28528
                  19.23.3.18  Ordering on reals (cont.) - extension   leaddsuble 28531
                  19.23.3.19  Nonnegative integers (as a subset of complex numbers) - extension   0mnnnnn0 28535
                  19.23.3.20  Upper partititions of integers - extension   1eluzge0 28540
                  19.23.3.21  Finite intervals of integers - extension   ssfz12 28546
                  19.23.3.22  Half-open integer ranges - extension   elfzonn0 28564
                  19.23.3.23  The floor (greatest integer) function - extension   nn0nndivcl 28586
                  19.23.3.24  The modulo (remainder) operation - extension   modvalr 28594
                  19.23.3.25  The ` # ` (finite set size) function - extension   hashimarn 28608
                  19.23.3.26  Words over a set - extension   wrdlen1 28624
                  19.23.3.27  Words over a set - extension (concatenations)   elfzelfzccat 28640
                  19.23.3.28  Words over a set - extension (subwords)   swrdltnd 28646
                  19.23.3.29  Words over a set - extension (subwords of subwords)   swrd0swrd 28663
                  19.23.3.30  Words over a set - extension (subwords of concatenations)   swrdccat3a0 28669
                  19.23.3.31  Finite and infinite sums - extension   fsumz 28689
                  19.23.3.32  Prime numbers: elementary properties - extension   prmgt1 28700
                  *19.23.3.33  Words over a set - extension (cyclic shift)   ccsh 28710
            19.23.4  Graph theory   uhgraedgrnv 28776
                  19.23.4.1  Undirected hypergraphs   uhgraedgrnv 28776
                  19.23.4.2  Undirected simple graphs   usisuhgra 28777
                  19.23.4.3  Neighbors, complete graphs and universal vertices   nbfiusgrafi 28778
                  19.23.4.4  Walks, Paths and Cycles   wlkn0 28782
                  19.23.4.5  Walks as words   cwwlk 28809
                  19.23.4.6  Walks/paths of length 2 as ordered triples   c2wlkot 28836
                  19.23.4.7  Vertex Degree   usgfidegfi 28875
                  19.23.4.8  Regular graphs   crgra 28887
                  *19.23.4.9  Friendship graphs   cfrgra 28902
      *19.24  Mathbox for David A. Wheeler
            19.24.1  Natural deduction   19.8ad 28984
            *19.24.2  Greater than, greater than or equal to.   cge-real 28987
            *19.24.3  Hyperbolic trig functions   csinh 28997
            *19.24.4  Reciprocal trig functions (sec, csc, cot)   csec 29008
            *19.24.5  Identities for "if"   ifnmfalse 29030
            19.24.6  Not-member-of   AnelBC 29031
            *19.24.7  Decimal point   cdp2 29032
            19.24.8  Ceiling function   ccei 29040
            19.24.9  Logarithms generalized to arbitrary base using ` logb `   ene0 29044
            *19.24.10  Logarithm laws generalized to an arbitrary base - log_   clog_ 29047
            *19.24.11  Miscellaneous   5m4e1 29049
      19.25  Mathbox for Alan Sare
            19.25.1  Supplementary "adant" deductions   ad4ant13 29052
            19.25.2  Supplementary unification deductions   biimp 29078
            19.25.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 29097
            19.25.4  What is Virtual Deduction?   wvd1 29175
            19.25.5  Virtual Deduction Theorems   df-vd1 29176
            19.25.6  Theorems proved using virtual deduction   trsspwALT 29446
            19.25.7  Theorems proved using virtual deduction with mmj2 assistance   simplbi2VD 29475
            19.25.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 29542
            19.25.9  Theorems proved using conjunction-form virtual deduction   elpwgdedVD 29546
            19.25.10  Theorems with VD proofs in conventional notation derived from VD proofs   suctrALT3 29553
            *19.25.11  Theorems with a proof in conventional notation automatically derived   notnot2ALT2 29556
      *19.26  Mathbox for Jonathan Ben-Naim
            19.26.1  First order logic and set theory   bnj170 29579
            19.26.2  Well founded induction and recursion   bnj110 29746
            19.26.3  The existence of a minimal element in certain classes   bnj69 29896
            19.26.4  Well-founded induction   bnj1204 29898
            19.26.5  Well-founded recursion, part 1 of 3   bnj60 29948
            19.26.6  Well-founded recursion, part 2 of 3   bnj1500 29954
            19.26.7  Well-founded recursion, part 3 of 3   bnj1522 29958
      19.27  Mathbox for BJ
            19.27.1  First-order logic   sp2 29959
                  *19.27.1.1  Some PM proofs   sp2 29959
      *19.28  Mathbox for Norm Megill
            *19.28.1  Experiments to study ax-7 unbundling   ax-7v 29978
                  19.28.1.1  Theorems derived from ax-7v (suffixes NEW7 and AUX7)   ax-7v 29978
                  19.28.1.2  Theorems derived from ax-7 (suffix OLD7)   ax-7OLD7 30214
            19.28.2  Miscellanea   cnaddcom 30305
            19.28.3  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 30307
            19.28.4  Functionals and kernels of a left vector space (or module)   clfn 30390
            19.28.5  Opposite rings and dual vector spaces   cld 30456
            19.28.6  Ortholattices and orthomodular lattices   cops 30505
            19.28.7  Atomic lattices with covering property   ccvr 30595
            19.28.8  Hilbert lattices   chlt 30683
            19.28.9  Projective geometries based on Hilbert lattices   clln 30823
            19.28.10  Construction of a vector space from a Hilbert lattice   cdlema1N 31123
            19.28.11  Construction of involution and inner product from a Hilbert lattice   clpoN 32813

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