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Table of Contents Summary
PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      1.1  Pre-logic
      1.2  Propositional calculus
      1.3  Other axiomatizations related to classical propositional calculus
      1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
      1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
      1.6  Existential uniqueness
      1.7  Other axiomatizations related to classical predicate calculus
PART 2  ZF (ZERMELO-FRAENKEL) SET THEORY
      2.1  ZF Set Theory - start with the Axiom of Extensionality
      2.2  ZF Set Theory - add the Axiom of Replacement
      2.3  ZF Set Theory - add the Axiom of Power Sets
      2.4  ZF Set Theory - add the Axiom of Union
      2.5  ZF Set Theory - add the Axiom of Regularity
      2.6  ZF Set Theory - add the Axiom of Infinity
PART 3  ZFC (ZERMELO-FRAENKEL WITH CHOICE) SET THEORY
      3.1  ZFC Set Theory - add Countable Choice and Dependent Choice
      3.2  ZFC Set Theory - add the Axiom of Choice
      3.3  ZFC Axioms with no distinct variable requirements
      3.4  The Generalized Continuum Hypothesis
PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      4.1  Inaccessibles
      4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
PART 5  REAL AND COMPLEX NUMBERS
      5.1  Construction and axiomatization of real and complex numbers
      5.2  Derive the basic properties from the field axioms
      5.3  Real and complex numbers - basic operations
      5.4  Integer sets
      5.5  Order sets
      5.6  Elementary integer functions
      5.7  Words over a set
      5.8  Reflexive and transitive closures of relations
      5.9  Elementary real and complex functions
      5.10  Elementary limits and convergence
      5.11  Elementary trigonometry
      5.12  Cardinality of real and complex number subsets
PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
      6.2  Elementary prime number theory
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
      7.2  Moore spaces
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
      8.2  Arrows (disjointified hom-sets)
      8.3  Examples of categories
      8.4  Categorical constructions
PART 9  BASIC ORDER THEORY
      9.1  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 LINEAR ALGEBRA
      11.1  Vectors and free modules
      11.2  Matrices
      11.3  The determinant
      11.4  Polynomial matrices
      11.5  The characteristic polynomial
PART 12  BASIC TOPOLOGY
      12.1  Topology
      12.2  Filters and filter bases
      12.3  Uniform Structures and Spaces
      12.4  Metric spaces
      12.5  Complex metric vector spaces
PART 13  BASIC REAL AND COMPLEX ANALYSIS
      13.1  Continuity
      13.2  Integrals
      13.3  Derivatives
PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      14.1  Polynomials
      14.2  Sequences and series
      14.3  Basic trigonometry
      14.4  Basic number theory
PART 15  ELEMENTARY GEOMETRY
      15.1  Definition and Tarski's Axioms of Geometry
      15.2  Tarskian Geometry
      15.3  Properties of geometries
      15.4  Geometry in Hilbert spaces
PART 16  GRAPH THEORY (UNDER CONSTRUCTION)
      16.1  Undirected graphs - preliminaries
PART 17  GRAPH THEORY (DEPRECATED)
      17.1  Undirected graphs - basics
      17.2  Eulerian paths and the Konigsberg Bridge problem
      17.3  The Friendship Theorem
PART 18  GUIDES AND MISCELLANEA
      18.1  Guides (conventions, explanations, and examples)
      18.2  Humor
      18.3  (Future - to be reviewed and classified)
PART 19  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      19.1  Additional material on group theory (deprecated)
      19.2  Complex vector spaces
      19.3  Normed complex vector spaces
      19.4  Operators on complex vector spaces
      19.5  Inner product (pre-Hilbert) spaces
      19.6  Complex Banach spaces
      19.7  Complex Hilbert spaces
PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      20.1  Axiomatization of complex pre-Hilbert spaces
      20.2  Inner product and norms
      20.3  Cauchy sequences and completeness axiom
      20.4  Subspaces and projections
      20.5  Properties of Hilbert subspaces
      20.6  Operators on Hilbert spaces
      20.7  States on a Hilbert lattice and Godowski's equation
      20.8  Cover relation, atoms, exchange axiom, and modular symmetry
PART 21  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      21.1  Mathboxes for user contributions
      21.2  Mathbox for Stefan Allan
      21.3  Mathbox for Thierry Arnoux
      21.4  Mathbox for Jonathan Ben-Naim
      21.5  Mathbox for Mario Carneiro
      21.6  Mathbox for Filip Cernatescu
      21.7  Mathbox for Paul Chapman
      21.8  Mathbox for Scott Fenton
      21.9  Mathbox for Jeff Hankins
      21.10  Mathbox for Anthony Hart
      21.11  Mathbox for Chen-Pang He
      21.12  Mathbox for Jeff Hoffman
      21.13  Mathbox for Asger C. Ipsen
      21.14  Mathbox for BJ
      21.15  Mathbox for Jim Kingdon
      21.16  Mathbox for ML
      21.17  Mathbox for Wolf Lammen
      21.18  Mathbox for Brendan Leahy
      21.19  Mathbox for Jeff Madsen
      21.20  Mathbox for Giovanni Mascellani
      21.21  Mathbox for Rodolfo Medina
      21.22  Mathbox for Norm Megill
      21.23  Mathbox for OpenAI
      21.24  Mathbox for Stefan O'Rear
      21.25  Mathbox for Jon Pennant
      21.26  Mathbox for Richard Penner
      21.27  Mathbox for Stanislas Polu
      21.28  Mathbox for Steve Rodriguez
      21.29  Mathbox for Andrew Salmon
      21.30  Mathbox for Alan Sare
      21.31  Mathbox for Glauco Siliprandi
      21.32  Mathbox for Saveliy Skresanov
      21.33  Mathbox for Jarvin Udandy
      21.34  Mathbox for Alexander van der Vekens
      21.35  Mathbox for Emmett Weisz
      21.36  Mathbox for David A. Wheeler
      21.37  Mathbox for Kunhao Zheng

Detailed Table of Contents
(* means the section header has a description)
*PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      *1.1  Pre-logic
            *1.1.1  Inferences for assisting proof development   a1ii 1
      *1.2  Propositional calculus
            1.2.1  Recursively define primitive wffs for propositional calculus   wn 3
            *1.2.2  The axioms of propositional calculus   ax-mp 5
            *1.2.3  Logical implication   mp2 9
            *1.2.4  Logical negation   con4 111
            *1.2.5  Logical equivalence   wb 195
            *1.2.6  Logical disjunction and conjunction   wo 382
            *1.2.7  Miscellaneous theorems of propositional calculus   pm5.62 960
            *1.2.8  The conditional operator for propositions   wif 1006
            *1.2.9  The weak deduction theorem   elimh 1024
            1.2.10  Abbreviated conjunction and disjunction of three wff's   w3o 1030
            1.2.11  Logical 'nand' (Sheffer stroke)   wnan 1439
            1.2.12  Logical 'xor'   wxo 1456
            1.2.13  True and false constants   wal 1473
                  *1.2.13.1  Universal quantifier for use by df-tru   wal 1473
                  *1.2.13.2  Equality predicate for use by df-tru   cv 1474
                  1.2.13.3  Define the true and false constants   wtru 1476
            *1.2.14  Truth tables   truantru 1497
            *1.2.15  Half adder and full adder in propositional calculus   whad 1523
                  1.2.15.1  Full adder: sum   whad 1523
                  1.2.15.2  Full adder: carry   wcad 1536
      1.3  Other axiomatizations related to classical propositional calculus
            *1.3.1  Minimal implicational calculus   minimp 1551
            1.3.2  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1557
            1.3.3  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1574
            *1.3.4  Derive Nicod's axiom from the standard axioms   nic-dfim 1585
            1.3.5  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1591
            1.3.6  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1610
            1.3.7  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1614
            1.3.8  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1629
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1652
            1.3.10  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1665
            *1.3.11  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1684
      *1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
            *1.4.1  Universal quantifier (continued); define "exists" and "not free"   wex 1695
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1713
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1728
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1827
            *1.4.5  Equality predicate (continued)   weq 1861
            1.4.6  Define proper substitution   wsb 1867
            1.4.7  Axiom scheme ax-6 (Existence)   ax-6 1875
            1.4.8  Axiom scheme ax-7 (Equality)   ax-7 1922
            1.4.9  Membership predicate   wcel 1977
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 1979
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 1986
            *1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 1992
      *1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2006
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2021
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2034
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2234
      1.6  Existential uniqueness
      1.7  Other axiomatizations related to classical predicate calculus
            *1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2551
            *1.7.2  Intuitionistic logic   axia1 2575
*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 2590
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2596
            2.1.3  Class form not-free predicate   wnfc 2738
            2.1.4  Negated equality and membership   wne 2780
                  2.1.4.1  Negated equality   neii 2784
                  2.1.4.2  Negated membership   neli 2885
            2.1.5  Restricted quantification   wral 2896
            2.1.6  The universal class   cvv 3173
            *2.1.7  Conditional equality (experimental)   wcdeq 3385
            2.1.8  Russell's Paradox   ru 3401
            2.1.9  Proper substitution of classes for sets   wsbc 3402
            2.1.10  Proper substitution of classes for sets into classes   csb 3499
            2.1.11  Define basic set operations and relations   cdif 3537
            2.1.12  Subclasses and subsets   df-ss 3554
            2.1.13  The difference, union, and intersection of two classes   difeq1 3683
                  2.1.13.1  The difference of two classes   difeq1 3683
                  2.1.13.2  The union of two classes   elun 3715
                  2.1.13.3  The intersection of two classes   elin 3758
                  2.1.13.4  The symmetric difference of two classes   csymdif 3805
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 3816
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unab 3853
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuss2 3866
            2.1.14  The empty set   c0 3874
            *2.1.15  "Weak deduction theorem" for set theory   cif 4036
            2.1.16  Power classes   cpw 4108
            2.1.17  Unordered and ordered pairs   snjust 4124
            2.1.18  The union of a class   cuni 4372
            2.1.19  The intersection of a class   cint 4410
            2.1.20  Indexed union and intersection   ciun 4455
            2.1.21  Disjointness   wdisj 4553
            2.1.22  Binary relations   wbr 4583
            2.1.23  Ordered-pair class abstractions (class builders)   copab 4642
            2.1.24  Transitive classes   wtr 4680
      2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 4699
            2.2.2  Derive the Axiom of Separation   axsep 4708
            2.2.3  Derive the Null Set Axiom   zfnuleu 4714
            2.2.4  Theorems requiring subset and intersection existence   nalset 4723
            2.2.5  Theorems requiring empty set existence   class2set 4758
      2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4769
            2.3.2  Derive the Axiom of Pairing   zfpair 4831
            2.3.3  Ordered pair theorem   opnz 4868
            2.3.4  Ordered-pair class abstractions (cont.)   opabid 4907
            2.3.5  Power class of union and intersection   pwin 4942
            2.3.6  Epsilon and identity relations   cep 4947
            2.3.7  Partial and complete ordering   wpo 4957
            2.3.8  Founded and well-ordering relations   wfr 4994
            2.3.9  Relations   cxp 5036
            2.3.10  The Predecessor Class   cpred 5596
            2.3.11  Well-founded induction   tz6.26 5628
            2.3.12  Ordinals   word 5639
            2.3.13  Definite description binder (inverted iota)   cio 5766
            2.3.14  Functions   wfun 5798
            2.3.15  Cantor's Theorem   canth 6508
            2.3.16  Restricted iota (description binder)   crio 6510
            2.3.17  Operations   co 6549
            2.3.18  "Maps to" notation   mpt2ndm0 6773
            2.3.19  Function operation   cof 6793
            2.3.20  Proper subset relation   crpss 6834
      2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 6847
            2.4.2  Ordinals (continued)   ordon 6874
            2.4.3  Transfinite induction   tfi 6945
            2.4.4  The natural numbers (i.e. finite ordinals)   com 6957
            2.4.5  Peano's postulates   peano1 6977
            2.4.6  Finite induction (for finite ordinals)   find 6983
            2.4.7  First and second members of an ordered pair   c1st 7057
            *2.4.8  The support of functions   csupp 7182
            *2.4.9  Special "Maps to" operations   opeliunxp2f 7223
            2.4.10  Function transposition   ctpos 7238
            2.4.11  Curry and uncurry   ccur 7278
            2.4.12  Undefined values   cund 7285
            2.4.13  Well-founded recursion   cwrecs 7293
            2.4.14  Functions on ordinals; strictly monotone ordinal functions   iunon 7323
            2.4.15  "Strong" transfinite recursion   crecs 7354
            2.4.16  Recursive definition generator   crdg 7392
            2.4.17  Finite recursion   frfnom 7417
            2.4.18  Ordinal arithmetic   c1o 7440
            2.4.19  Natural number arithmetic   nna0 7571
            2.4.20  Equivalence relations and classes   wer 7626
            2.4.21  The mapping operation   cmap 7744
            2.4.22  Infinite Cartesian products   cixp 7794
            2.4.23  Equinumerosity   cen 7838
            2.4.24  Schroeder-Bernstein Theorem   sbthlem1 7955
            2.4.25  Equinumerosity (cont.)   xpf1o 8007
            2.4.26  Pigeonhole Principle   phplem1 8024
            2.4.27  Finite sets   onomeneq 8035
            2.4.28  Finitely supported functions   cfsupp 8158
            2.4.29  Finite intersections   cfi 8199
            2.4.30  Hall's marriage theorem   marypha1lem 8222
            2.4.31  Supremum and infimum   csup 8229
            2.4.32  Ordinal isomorphism, Hartog's theorem   coi 8297
            2.4.33  Hartogs function, order types, weak dominance   char 8344
      2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 8380
            2.5.2  Axiom of Infinity equivalents   inf0 8401
      2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 8418
            2.6.2  Existence of omega (the set of natural numbers)   omex 8423
            2.6.3  Cantor normal form   ccnf 8441
            2.6.4  Transitive closure   trcl 8487
            2.6.5  Rank   cr1 8508
            2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 8631
            2.6.7  Cardinal numbers   ccrd 8644
            2.6.8  Axiom of Choice equivalents   wac 8821
            2.6.9  Cardinal number arithmetic   ccda 8872
            2.6.10  The Ackermann bijection   ackbij2lem1 8924
            2.6.11  Cofinality (without Axiom of Choice)   cflem 8951
            2.6.12  Eight inequivalent definitions of finite set   sornom 8982
            2.6.13  Hereditarily size-limited sets without Choice   itunifval 9121
*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 9140
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 9151
      3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 9164
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 9199
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 9247
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 9275
            3.2.5  Cofinality using Axiom of Choice   alephreg 9283
      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 9321
            3.4.2  Derivation of the Axiom of Choice   gchaclem 9379
*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 9383
            4.1.2  Weak universes   cwun 9401
            4.1.3  Tarski classes   ctsk 9449
            4.1.4  Grothendieck universes   cgru 9491
      4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 9524
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 9527
            4.2.3  Tarski map function   ctskm 9538
*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 9545
            5.1.2  Final derivation of real and complex number postulates   axaddf 9845
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 9871
      5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 9896
            5.2.2  Infinity and the extended real number system   cpnf 9950
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 9988
            5.2.4  Ordering on reals   lttr 9993
            5.2.5  Initial properties of the complex numbers   mul12 10081
      5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 10132
            5.3.2  Subtraction   cmin 10145
            5.3.3  Multiplication   kcnktkm1cn 10340
            5.3.4  Ordering on reals (cont.)   gt0ne0 10372
            5.3.5  Reciprocals   ixi 10535
            5.3.6  Division   cdiv 10563
            5.3.7  Ordering on reals (cont.)   elimgt0 10738
            5.3.8  Completeness Axiom and Suprema   fimaxre 10847
            5.3.9  Imaginary and complex number properties   inelr 10887
            5.3.10  Function operation analogue theorems   ofsubeq0 10894
      5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 10897
            5.4.2  Principle of mathematical induction   nnind 10915
            *5.4.3  Decimal representation of numbers   c2 10947
            *5.4.4  Some properties of specific numbers   neg1cn 11001
            5.4.5  Simple number properties   halfcl 11134
            5.4.6  The Archimedean property   nnunb 11165
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 11169
            *5.4.8  Extended nonnegative integers   cxnn0 11240
            5.4.9  Integers (as a subset of complex numbers)   cz 11254
            5.4.10  Decimal arithmetic   cdc 11369
            5.4.11  Upper sets of integers   cuz 11563
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 11659
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 11664
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 11690
      5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 11708
            5.5.2  Infinity and the extended real number system (cont.)   cxne 11819
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 12007
            5.5.4  Real number intervals   cioo 12046
            5.5.5  Finite intervals of integers   cfz 12197
            *5.5.6  Finite intervals of nonnegative integers   elfz2nn0 12300
            5.5.7  Half-open integer ranges   cfzo 12334
      5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 12453
            5.6.2  The modulo (remainder) operation   cmo 12530
            5.6.3  Miscellaneous theorems about integers   om2uz0i 12608
            5.6.4  Strong induction over upper sets of integers   uzsinds 12648
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 12651
            5.6.6  The infinite sequence builder "seq"   cseq 12663
            5.6.7  Integer powers   cexp 12722
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 12916
            5.6.9  Factorial function   cfa 12922
            5.6.10  The binomial coefficient operation   cbc 12951
            5.6.11  The ` # ` (set size) function   chash 12979
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 13107
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 13129
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   brfi1indlem 13133
      *5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 13146
            5.7.2  Last symbol of a word   lsw 13204
            5.7.3  Concatenations of words   ccatfn 13210
            5.7.4  Singleton words   ids1 13230
            5.7.5  Concatenations with singleton words   ccatws1cl 13249
            5.7.6  Subwords   swrdval 13269
            5.7.7  Subwords of subwords   swrdswrdlem 13311
            5.7.8  Subwords and concatenations   wrdcctswrd 13317
            5.7.9  Subwords of concatenations   swrdccatfn 13333
            5.7.10  Splicing words (substring replacement)   splval 13353
            5.7.11  Reversing words   revval 13360
            5.7.12  Repeated symbol words   reps 13368
            *5.7.13  Cyclical shifts of words   ccsh 13385
            5.7.14  Mapping words by a function   wrdco 13428
            5.7.15  Longer string literals   cs2 13437
      *5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 13559
            5.8.2  Basic properties of closures   cleq1lem 13569
            5.8.3  Definitions and basic properties of transitive closures   ctcl 13572
            5.8.4  Exponentiation of relations   crelexp 13608
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 13643
            *5.8.6  Principle of transitive induction.   relexpindlem 13651
      5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 13654
            5.9.2  Signum (sgn or sign) function   csgn 13674
            5.9.3  Real and imaginary parts; conjugate   ccj 13684
            5.9.4  Square root; absolute value   csqrt 13821
      5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 14049
            5.10.2  Limits   cli 14063
            5.10.3  Finite and infinite sums   csu 14264
            5.10.4  The binomial theorem   binomlem 14400
            5.10.5  The inclusion/exclusion principle   incexclem 14407
            5.10.6  Infinite sums (cont.)   isumshft 14410
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 14423
            5.10.8  Arithmetic series   arisum 14431
            5.10.9  Geometric series   expcnv 14435
            5.10.10  Ratio test for infinite series convergence   cvgrat 14454
            5.10.11  Mertens' theorem   mertenslem1 14455
            5.10.12  Finite and infinite products   prodf 14458
                  5.10.12.1  Product sequences   prodf 14458
                  5.10.12.2  Non-trivial convergence   ntrivcvg 14468
                  5.10.12.3  Complex products   cprod 14474
                  5.10.12.4  Finite products   fprod 14510
                  5.10.12.5  Infinite products   iprodclim 14568
            5.10.13  Falling and Rising Factorial   cfallfac 14574
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 14616
      5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 14631
            5.11.2  _e is irrational   eirrlem 14771
      5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 14778
            5.12.2  The reals are uncountable   rpnnen2lem1 14782
*PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqr2irrlem 14816
            6.1.2  Some Number sets are chains of proper subsets   nthruc 14819
            6.1.3  The divides relation   cdvds 14821
            *6.1.4  Even and odd numbers   evenelz 14898
            6.1.5  The division algorithm   divalglem0 14954
            6.1.6  Bit sequences   cbits 14979
            6.1.7  The greatest common divisor operator   cgcd 15054
            6.1.8  Bézout's identity   bezoutlem1 15094
            6.1.9  Algorithms   nn0seqcvgd 15121
            6.1.10  Euclid's Algorithm   eucalgval2 15132
            *6.1.11  The least common multiple   clcm 15139
            *6.1.12  Coprimality and Euclid's lemma   coprmgcdb 15200
            6.1.13  Cancellability of congruences   congr 15216
      6.2  Elementary prime number theory
            *6.2.1  Elementary properties   cprime 15223
            *6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 15261
            6.2.3  Properties of the canonical representation of a rational   cnumer 15279
            6.2.4  Euler's theorem   codz 15306
            6.2.5  Arithmetic modulo a prime number   modprm1div 15340
            6.2.6  Pythagorean Triples   coprimeprodsq 15351
            6.2.7  The prime count function   cpc 15379
            6.2.8  Pocklington's theorem   prmpwdvds 15446
            6.2.9  Infinite primes theorem   unbenlem 15450
            6.2.10  Sum of prime reciprocals   prmreclem1 15458
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 15465
            6.2.12  Lagrange's four-square theorem   cgz 15471
            6.2.13  Van der Waerden's theorem   cvdwa 15507
            6.2.14  Ramsey's theorem   cram 15541
            *6.2.15  Primorial function   cprmo 15573
            *6.2.16  Prime gaps   prmgaplem1 15591
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 15605
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 15638
            6.2.19  Specific prime numbers   prmlem0 15650
            6.2.20  Very large primes   1259lem1 15676
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
            *7.1.1  Basic definitions   cstr 15691
            7.1.2  Slot definitions   cplusg 15768
            7.1.3  Definition of the structure product   crest 15904
            7.1.4  Definition of the structure quotient   cordt 15982
      7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 16089
            7.2.2  Independent sets in a Moore system   mrisval 16113
            7.2.3  Algebraic closure systems   isacs 16135
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
            8.1.1  Categories   ccat 16148
            8.1.2  Opposite category   coppc 16194
            8.1.3  Monomorphisms and epimorphisms   cmon 16211
            8.1.4  Sections, inverses, isomorphisms   csect 16227
            *8.1.5  Isomorphic objects   ccic 16278
            8.1.6  Subcategories   cssc 16290
            8.1.7  Functors   cfunc 16337
            8.1.8  Full & faithful functors   cful 16385
            8.1.9  Natural transformations and the functor category   cnat 16424
            8.1.10  Initial, terminal and zero objects of a category   cinito 16461
      8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 16526
      8.3  Examples of categories
            8.3.1  The category of sets   csetc 16548
            8.3.2  The category of categories   ccatc 16567
            *8.3.3  The category of extensible structures   fncnvimaeqv 16583
      8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 16631
            8.4.2  Functor evaluation   cevlf 16672
            8.4.3  Hom functor   chof 16711
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 16763
            9.2.2  Lattices   clat 16868
            9.2.3  The dual of an ordered set   codu 16951
            9.2.4  Subset order structures   cipo 16974
            9.2.5  Distributive lattices   latmass 17011
            9.2.6  Posets and lattices as relations   cps 17021
            9.2.7  Directed sets, nets   cdir 17051
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
            *10.1.1  Magmas   cplusf 17062
            *10.1.2  Identity elements   mgmidmo 17082
            *10.1.3  Ordered sums in a magma   gsumvalx 17093
            *10.1.4  Semigroups   csgrp 17106
            *10.1.5  Definition and basic properties of monoids   cmnd 17117
            10.1.6  Monoid homomorphisms and submonoids   cmhm 17156
            *10.1.7  Ordered sums in a monoid   gsumvallem2 17195
            10.1.8  Free monoids   cfrmd 17207
            10.1.9  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 17228
      10.2  Groups
            10.2.1  Definition and basic properties   cgrp 17245
            *10.2.2  Group multiple operation   cmg 17363
            10.2.3  Subgroups and Quotient groups   csubg 17411
            10.2.4  Elementary theory of group homomorphisms   cghm 17480
            10.2.5  Isomorphisms of groups   cgim 17522
            10.2.6  Group actions   cga 17545
            10.2.7  Centralizers and centers   ccntz 17571
            10.2.8  The opposite group   coppg 17598
            10.2.9  Symmetric groups   csymg 17620
                  *10.2.9.1  Definition and basic properties   csymg 17620
                  10.2.9.2  Cayley's theorem   cayleylem1 17655
                  10.2.9.3  Permutations fixing one element   symgfix2 17659
                  *10.2.9.4  Transpositions in the symmetric group   cpmtr 17684
                  10.2.9.5  The sign of a permutation   cpsgn 17732
            10.2.10  p-Groups and Sylow groups; Sylow's theorems   cod 17767
            10.2.11  Direct products   clsm 17872
            10.2.12  Free groups   cefg 17942
      10.3  Abelian groups
            10.3.1  Definition and basic properties   ccmn 18016
            10.3.2  Cyclic groups   ccyg 18102
            10.3.3  Group sum operation   gsumval3a 18127
            10.3.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 18202
            10.3.5  Internal direct products   cdprd 18215
            10.3.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 18287
      10.4  Rings
            10.4.1  Multiplicative Group   cmgp 18312
            10.4.2  Ring unit   cur 18324
                  10.4.2.1  Semirings   csrg 18328
                  *10.4.2.2  The binomial theorem for semirings   srgbinomlem1 18363
            10.4.3  Definition and basic properties of unital rings   crg 18370
            10.4.4  Opposite ring   coppr 18445
            10.4.5  Divisibility   cdsr 18461
            10.4.6  Ring homomorphisms   crh 18535
      10.5  Division rings and fields
            10.5.1  Definition and basic properties   cdr 18570
            10.5.2  Subrings of a ring   csubrg 18599
            10.5.3  Absolute value (abstract algebra)   cabv 18639
            10.5.4  Star rings   cstf 18666
      10.6  Left modules
            10.6.1  Definition and basic properties   clmod 18686
            10.6.2  Subspaces and spans in a left module   clss 18753
            10.6.3  Homomorphisms and isomorphisms of left modules   clmhm 18840
            10.6.4  Subspace sum; bases for a left module   clbs 18895
      10.7  Vector spaces
            10.7.1  Definition and basic properties   clvec 18923
      10.8  Ideals
            10.8.1  The subring algebra; ideals   csra 18989
            10.8.2  Two-sided ideals and quotient rings   c2idl 19052
            10.8.3  Principal ideal rings. Divisibility in the integers   clpidl 19062
            10.8.4  Nonzero rings and zero rings   cnzr 19078
            10.8.5  Left regular elements. More kinds of rings   crlreg 19100
      10.9  Associative algebras
            10.9.1  Definition and basic properties   casa 19130
      10.10  Abstract multivariate polynomials
            10.10.1  Definition and basic properties   cmps 19172
            10.10.2  Polynomial evaluation   ces 19325
            *10.10.3  Additional definitions for (multivariate) polynomials   cmhp 19358
            *10.10.4  Univariate polynomials   cps1 19366
            10.10.5  Univariate polynomial evaluation   ces1 19499
      10.11  The complex numbers as an algebraic extensible structure
            10.11.1  Definition and basic properties   cpsmet 19551
            *10.11.2  Ring of integers   zring 19637
            10.11.3  Algebraic constructions based on the complex numbers   czrh 19667
            10.11.4  Signs as subgroup of the complex numbers   cnmsgnsubg 19742
            10.11.5  Embedding of permutation signs into a ring   zrhpsgnmhm 19749
            10.11.6  The ordered field of real numbers   crefld 19769
      10.12  Generalized pre-Hilbert and Hilbert spaces
            10.12.1  Definition and basic properties   cphl 19788
            10.12.2  Orthocomplements and closed subspaces   cocv 19823
            10.12.3  Orthogonal projection and orthonormal bases   cpj 19863
*PART 11  BASIC LINEAR ALGEBRA
      11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 19894
            *11.1.2  Free modules   cfrlm 19909
            *11.1.3  Standard basis (unit vectors)   cuvc 19940
            *11.1.4  Independent sets and families   clindf 19962
            11.1.5  Characterization of free modules   lmimlbs 19994
      *11.2  Matrices
            *11.2.1  The matrix multiplication   cmmul 20008
            *11.2.2  Square matrices   cmat 20032
            *11.2.3  The matrix algebra   matmulr 20063
            *11.2.4  Matrices of dimension 0 and 1   mat0dimbas0 20091
            *11.2.5  The subalgebras of diagonal and scalar matrices   cdmat 20113
            *11.2.6  Multiplication of a matrix with a "column vector"   cmvmul 20165
            11.2.7  Replacement functions for a square matrix   cmarrep 20181
            11.2.8  Submatrices   csubma 20201
      11.3  The determinant
            11.3.1  Definition and basic properties   cmdat 20209
            11.3.2  Determinants of 2 x 2 -matrices   m2detleiblem1 20249
            11.3.3  The matrix adjugate/adjunct   cmadu 20257
            *11.3.4  Laplace expansion of determinants (special case)   symgmatr01lem 20278
            11.3.5  Inverse matrix   invrvald 20301
            *11.3.6  Cramer's rule   slesolvec 20304
      *11.4  Polynomial matrices
            11.4.1  Basic properties   pmatring 20317
            *11.4.2  Constant polynomial matrices   ccpmat 20327
            *11.4.3  Collecting coefficients of polynomial matrices   cdecpmat 20386
            *11.4.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 20416
      *11.5  The characteristic polynomial
            *11.5.1  Definition and basic properties   cchpmat 20450
            *11.5.2  The characteristic factor function G   fvmptnn04if 20473
            *11.5.3  The Cayley-Hamilton theorem   cpmadurid 20491
PART 12  BASIC TOPOLOGY
      12.1  Topology
            12.1.1  Topological spaces   ctop 20517
            12.1.2  TopBases for topologies   isbasisg 20562
            12.1.3  Examples of topologies   distop 20610
            12.1.4  Closure and interior   ccld 20630
            12.1.5  Neighborhoods   cnei 20711
            12.1.6  Limit points and perfect sets   clp 20748
            12.1.7  Subspace topologies   restrcl 20771
            12.1.8  Order topology   ordtbaslem 20802
            12.1.9  Limits and continuity in topological spaces   ccn 20838
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 20920
            12.1.11  Compactness   ccmp 20999
            12.1.12  Bolzano-Weierstrass theorem   bwth 21023
            12.1.13  Connectedness   ccon 21024
            12.1.14  First- and second-countability   c1stc 21050
            12.1.15  Local topological properties   clly 21077
            12.1.16  Refinements   cref 21115
            12.1.17  Compactly generated spaces   ckgen 21146
            12.1.18  Product topologies   ctx 21173
            12.1.19  Continuous function-builders   cnmptid 21274
            12.1.20  Quotient maps and quotient topology   ckq 21306
            12.1.21  Homeomorphisms   chmeo 21366
      12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 21440
            12.2.2  Filters   cfil 21459
            12.2.3  Ultrafilters   cufil 21513
            12.2.4  Filter limits   cfm 21547
            12.2.5  Extension by continuity   ccnext 21673
            12.2.6  Topological groups   ctmd 21684
            12.2.7  Infinite group sum on topological groups   ctsu 21739
            12.2.8  Topological rings, fields, vector spaces   ctrg 21769
      12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 21813
            12.3.2  The topology induced by an uniform structure   cutop 21844
            12.3.3  Uniform Spaces   cuss 21867
            12.3.4  Uniform continuity   cucn 21889
            12.3.5  Cauchy filters in uniform spaces   ccfilu 21900
            12.3.6  Complete uniform spaces   ccusp 21911
      12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 21919
            12.4.2  Basic metric space properties   cxme 21932
            12.4.3  Metric space balls   blfvalps 21998
            12.4.4  Open sets of a metric space   mopnval 22053
            12.4.5  Continuity in metric spaces   metcnp3 22155
            12.4.6  The uniform structure generated by a metric   metuval 22164
            12.4.7  Examples of metric spaces   dscmet 22187
            *12.4.8  Normed algebraic structures   cnm 22191
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 22319
            12.4.10  Topology on the reals   qtopbaslem 22372
            12.4.11  Topological definitions using the reals   cii 22486
            12.4.12  Path homotopy   chtpy 22574
            12.4.13  The fundamental group   cpco 22608
      12.5  Complex metric vector spaces
            12.5.1  Complex left modules   cclm 22670
            *12.5.2  Complex vector spaces   ccvs 22731
            *12.5.3  Normed complex vector spaces   isncvsngp 22757
            12.5.4  Complex pre-Hilbert space   ccph 22774
            12.5.5  Convergence and completeness   ccfil 22858
            12.5.6  Baire's Category Theorem   bcthlem1 22929
            12.5.7  Banach spaces and complex Hilbert spaces   ccms 22937
                  12.5.7.1  The complete ordered field of the real numbers   retopn 22975
            12.5.8  Euclidean spaces   crrx 22979
            12.5.9  Minimizing Vector Theorem   minveclem1 23003
            12.5.10  Projection Theorem   pjthlem1 23016
PART 13  BASIC REAL AND COMPLEX ANALYSIS
      13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 23024
      13.2  Integrals
            13.2.1  Lebesgue measure   covol 23038
            13.2.2  Lebesgue integration   cmbf 23189
                  13.2.2.1  Lesbesgue integral   cmbf 23189
                  13.2.2.2  Lesbesgue directed integral   cdit 23416
      13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 23432
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 23432
                  13.3.1.2  Results on real differentiation   dvferm1lem 23551
PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 23617
            14.1.2  The division algorithm for univariate polynomials   cmn1 23689
            14.1.3  Elementary properties of complex polynomials   cply 23744
            14.1.4  The division algorithm for polynomials   cquot 23849
            14.1.5  Algebraic numbers   caa 23873
            14.1.6  Liouville's approximation theorem   aalioulem1 23891
      14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 23911
            14.2.2  Uniform convergence   culm 23934
            14.2.3  Power series   pserval 23968
      14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 24001
            14.3.2  Properties of pi = 3.14159...   pilem1 24009
            14.3.3  Mapping of the exponential function   efgh 24091
            14.3.4  The natural logarithm on complex numbers   clog 24105
            *14.3.5  Logarithms to an arbitrary base   clogb 24302
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 24331
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 24366
            14.3.8  Inverse trigonometric functions   casin 24389
            14.3.9  The Birthday Problem   log2ublem1 24473
            14.3.10  Areas in R^2   carea 24482
            14.3.11  More miscellaneous converging sequences   rlimcnp 24492
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 24511
            14.3.13  Euler-Mascheroni constant   cem 24518
            14.3.14  Zeta function   czeta 24539
            14.3.15  Gamma function   clgam 24542
      14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 24594
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 24599
            14.4.3  The Basel problem (ζ(2) = π2/6)   basellem1 24607
            14.4.4  Number-theoretical functions   ccht 24617
            14.4.5  Perfect Number Theorem   mersenne 24752
            14.4.6  Characters of Z/nZ   cdchr 24757
            14.4.7  Bertrand's postulate   bcctr 24800
            *14.4.8  Quadratic residues and the Legendre symbol   clgs 24819
            *14.4.9  Gauss' Lemma   gausslemma2dlem0a 24881
            14.4.10  Quadratic reciprocity   lgseisenlem1 24900
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 24942
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 24958
            14.4.13  The Prime Number Theorem   mudivsum 25019
            14.4.14  Ostrowski's theorem   abvcxp 25104
*PART 15  ELEMENTARY GEOMETRY
      15.1  Definition and Tarski's Axioms of Geometry
      15.2  Tarskian Geometry
            15.2.1  Congruence   tgcgrcomimp 25172
            15.2.2  Betweenness   tgbtwntriv2 25182
            15.2.3  Dimension   tglowdim1 25195
            15.2.4  Betweenness and Congruence   tgifscgr 25203
            15.2.5  Congruence of a series of points   ccgrg 25205
            15.2.6  Motions   cismt 25227
            15.2.7  Colinearity   tglng 25241
            15.2.8  Connectivity of betweenness   tgbtwnconn1lem1 25267
            15.2.9  Less-than relation in geometric congruences   cleg 25277
            15.2.10  Rays   chlg 25295
            15.2.11  Lines   btwnlng1 25314
            15.2.12  Point inversions   cmir 25347
            15.2.13  Right angles   crag 25388
            15.2.14  Half-planes   islnopp 25431
            15.2.15  Midpoints and Line Mirroring   cmid 25464
            15.2.16  Congruence of angles   ccgra 25499
            15.2.17  Angle Comparisons   cinag 25526
            15.2.18  Congruence Theorems   tgsas1 25535
            15.2.19  Equilateral triangles   ceqlg 25545
      15.3  Properties of geometries
            15.3.1  Isomorphisms between geometries   f1otrgds 25549
      15.4  Geometry in Hilbert spaces
            15.4.1  Geometry in the complex plane   cchhllem 25567
            15.4.2  Geometry in Euclidean spaces   cee 25568
                  15.4.2.1  Definition of the Euclidean space   cee 25568
                  15.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 25593
                  15.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 25657
PART 16  GRAPH THEORY (UNDER CONSTRUCTION)
      16.1  Undirected graphs - preliminaries
            16.1.1  The edge function extractor for extensible structures   cedgf 25667
            *16.1.2  Vertices and edges   cvtx 25673
                  16.1.2.1  Definitions and basic properties   cvtx 25673
                  16.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 25680
                  16.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdm2val 25688
                  16.1.2.4  Representations of graphs without edges   snstrvtxval 25712
                  16.1.2.5  Degenerated cases of representations of graphs   vtxval0 25714
            16.1.3  Undirected hypergraphs   cuhgr 25722
            16.1.4  Undirected pseudographs and multigraphs   cupgr 25747
            *16.1.5  Loop-free graphs   umgrislfupgrlem 25788
            16.1.6  Edges as subsets of vertices of graphs   cedga 25792
*PART 17  GRAPH THEORY (DEPRECATED)
      17.1  Undirected graphs - basics
            17.1.1  Undirected hypergraphs   cuhg 25819
            17.1.2  Undirected multigraphs   cumg 25841
            17.1.3  Undirected simple graphs   cuslg 25858
                  17.1.3.1  Undirected simple graphs - basics   cuslg 25858
                  17.1.3.2  Undirected simple graphs - examples   usgraex0elv 25924
                  17.1.3.3  Finite undirected simple graphs   fiusgraedgfi 25936
            17.1.4  Neighbors, complete graphs and universal vertices   cnbgra 25946
                  17.1.4.1  Neighbors   nbgraop 25952
                  17.1.4.2  Complete graphs   iscusgra 25985
                  17.1.4.3  Universal vertices   isuvtx 26016
            17.1.5  Walks, paths and cycles   cwalk 26026
                  17.1.5.1  Walks and trails   relwlk 26046
                  17.1.5.2  Paths and simple paths   pths 26096
                  17.1.5.3  Circuits and cycles   crcts 26150
                  17.1.5.4  Connected graphs   cconngra 26197
                  17.1.5.5  Walks as words   cwwlk 26205
                  17.1.5.6  Closed walks   cclwlk 26275
                  17.1.5.7  Walks/paths of length 2 as ordered triples   c2wlkot 26381
            17.1.6  Vertex degree   cvdg 26420
            17.1.7  Regular graphs   crgra 26449
                  17.1.7.1  Definition and basic properties   crgra 26449
                  17.1.7.2  Walks in regular graphs   rusgranumwwlkl1 26473
      17.2  Eulerian paths and the Konigsberg Bridge problem
            17.2.1  Eulerian paths   ceup 26489
            17.2.2  The Konigsberg Bridge problem   vdeg0i 26509
      *17.3  The Friendship Theorem
            17.3.1  Friendship graphs - basics   cfrgra 26515
            17.3.2  The friendship theorem for small graphs   frgra1v 26525
            17.3.3  Theorems according to Mertzios and Unger   2pthfrgrarn 26536
            *17.3.4  Huneke's Proof of the Friendship Theorem   frgrancvvdeqlem1 26557
PART 18  GUIDES AND MISCELLANEA
      18.1  Guides (conventions, explanations, and examples)
            *18.1.1  Conventions   conventions 26650
            18.1.2  Natural deduction   natded 26652
            *18.1.3  Natural deduction examples   ex-natded5.2 26653
            18.1.4  Definitional examples   ex-or 26670
            18.1.5  Other examples   aevdemo 26709
      18.2  Humor
            18.2.1  April Fool's theorem   avril1 26711
      18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 26718
            18.3.2  Algebra preliminaries   crpm 26723
            *18.3.3  Aliases kept to prevent broken links   dummylink 26725
*PART 19  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      *19.1  Additional material on group theory (deprecated)
            19.1.1  Definitions and basic properties for groups   cgr 26727
            19.1.2  Abelian groups   cablo 26782
      19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 26797
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 26820
      19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 26823
            19.3.2  Examples of normed complex vector spaces   cnnv 26916
            19.3.3  Induced metric of a normed complex vector space   imsval 26924
            19.3.4  Inner product   cdip 26939
            19.3.5  Subspaces   css 26960
      19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 26979
      19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 27051
            19.5.2  Examples of pre-Hilbert spaces   cncph 27058
            19.5.3  Properties of pre-Hilbert spaces   isph 27061
      19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 27102
            19.6.2  Examples of complex Banach spaces   cnbn 27109
            19.6.3  Uniform Boundedness Theorem   ubthlem1 27110
            19.6.4  Minimizing Vector Theorem   minvecolem1 27114
      19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 27125
            19.7.2  Standard axioms for a complex Hilbert space   hlex 27138
            19.7.3  Examples of complex Hilbert spaces   cnchl 27156
            19.7.4  Subspaces   ssphl 27157
            19.7.5  Hellinger-Toeplitz Theorem   htthlem 27158
*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chil 27160
            20.1.2  Preliminary ZFC lemmas   df-hnorm 27209
            *20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 27222
            *20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 27240
            20.1.5  Vector operations   hvmulex 27252
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 27320
      20.2  Inner product and norms
            20.2.1  Inner product   his5 27327
            20.2.2  Norms   dfhnorm2 27363
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 27401
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 27420
      20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 27425
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 27435
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 27443
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 27444
      20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 27448
            20.4.2  Closed subspaces   df-ch 27462
            20.4.3  Orthocomplements   df-oc 27493
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 27551
            20.4.5  Projection theorem   pjhthlem1 27634
            20.4.6  Projectors   df-pjh 27638
      20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 27645
            20.5.2  Projectors (cont.)   pjhtheu2 27659
            20.5.3  Hilbert lattice operations   sh0le 27683
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 27784
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 27826
            20.5.6  Foulis-Holland theorem   fh1 27861
            20.5.7  Quantum Logic Explorer axioms   qlax1i 27870
            20.5.8  Orthogonal subspaces   chscllem1 27880
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 27897
            20.5.10  Projectors (cont.)   pjorthi 27912
            20.5.11  Mayet's equation E_3   mayete3i 27971
      20.6  Operators on Hilbert spaces
            *20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 27973
            20.6.2  Zero and identity operators   df-h0op 27991
            20.6.3  Operations on Hilbert space operators   hoaddcl 28001
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 28082
            20.6.5  Linear and continuous functionals and norms   df-nmfn 28088
            20.6.6  Adjoint   df-adjh 28092
            20.6.7  Dirac bra-ket notation   df-bra 28093
            20.6.8  Positive operators   df-leop 28095
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 28096
            20.6.10  Theorems about operators and functionals   nmopval 28099
            20.6.11  Riesz lemma   riesz3i 28305
            20.6.12  Adjoints (cont.)   cnlnadjlem1 28310
            20.6.13  Quantum computation error bound theorem   unierri 28347
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 28348
            20.6.15  Positive operators (cont.)   leopg 28365
            20.6.16  Projectors as operators   pjhmopi 28389
      20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 28454
            20.7.2  Godowski's equation   golem1 28514
      20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 28522
            20.8.2  Atoms   df-at 28581
            20.8.3  Superposition principle   superpos 28597
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 28598
            20.8.5  Irreducibility   chirredlem1 28633
            20.8.6  Atoms (cont.)   atcvat3i 28639
            20.8.7  Modular symmetry   mdsymlem1 28646
PART 21  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 28685
      21.2  Mathbox for Stefan Allan
      21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   bian1d 28690
            21.3.2  Predicate Calculus   spc2ed 28696
                  21.3.2.1  Predicate Calculus - misc additions   spc2ed 28696
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 28700
                  21.3.2.3  Substitution (without distinct variables) - misc additions   clelsb3f 28704
                  21.3.2.4  Existential "at most one" - misc additions   moel 28707
                  21.3.2.5  Existential uniqueness - misc additions   2reuswap2 28712
                  21.3.2.6  Restricted "at most one" - misc additions   rmoxfrdOLD 28716
            21.3.3  General Set Theory   rabrab 28722
                  21.3.3.1  Class abstractions (a.k.a. class builders)   rabrab 28722
                  21.3.3.2  Image Sets   abrexdomjm 28729
                  21.3.3.3  Set relations and operations - misc additions   eqri 28735
                  21.3.3.4  Unordered pairs   elpreq 28744
                  21.3.3.5  Conditional operator - misc additions   ifeqeqx 28745
                  21.3.3.6  Set union   uniinn0 28749
                  21.3.3.7  Indexed union - misc additions   cbviunf 28755
                  21.3.3.8  Disjointness - misc additions   disjnf 28766
            21.3.4  Relations and Functions   xpdisjres 28793
                  21.3.4.1  Relations - misc additions   xpdisjres 28793
                  21.3.4.2  Functions - misc additions   mptexgf 28809
                  21.3.4.3  Operations - misc additions   mpt2mptxf 28860
                  21.3.4.4  Isomorphisms - misc. add.   gtiso 28861
                  21.3.4.5  Disjointness (additional proof requiring functions)   disjdsct 28863
                  21.3.4.6  First and second members of an ordered pair - misc additions   df1stres 28864
                  21.3.4.7  Supremum - misc additions   supssd 28870
                  21.3.4.8  Finite Sets   imafi2 28872
                  21.3.4.9  Countable Sets   snct 28874
            21.3.5  Real and Complex Numbers   addeq0 28898
                  21.3.5.1  Complex operations - misc. additions   addeq0 28898
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 28903
                  21.3.5.3  Extended reals - misc additions   xgepnf 28904
                  21.3.5.4  Real number intervals - misc additions   joiniooico 28926
                  21.3.5.5  Finite intervals of integers - misc additions   nndiffz1 28936
                  21.3.5.6  Half-open integer ranges - misc additions   iundisjfi 28942
                  21.3.5.7  The ` # ` (set size) function - misc additions   hashunif 28949
                  21.3.5.8  The greatest common divisor operator - misc. add   numdenneg 28950
                  21.3.5.9  Integers   nnindf 28952
                  21.3.5.10  Division in the extended real number system   cxdiv 28956
            21.3.6  Prime Number Theory   bhmafibid1 28975
                  21.3.6.1  Fermat's two square theorem   bhmafibid1 28975
            21.3.7  Extensible Structures   ressplusf 28981
                  21.3.7.1  Structure restriction operator   ressplusf 28981
                  21.3.7.2  The opposite group   oppgle 28984
                  21.3.7.3  Posets   ressprs 28986
                  21.3.7.4  Complete lattices   clatp0cl 29002
                  21.3.7.5  Extended reals Structure - misc additions   ax-xrssca 29004
                  21.3.7.6  The extended nonnegative real numbers commutative monoid   xrge0base 29016
            21.3.8  Algebra   abliso 29027
                  21.3.8.1  Monoids Homomorphisms   abliso 29027
                  21.3.8.2  Ordered monoids and groups   comnd 29028
                  21.3.8.3  Signum in an ordered monoid   csgns 29056
                  21.3.8.4  The Archimedean property for generic ordered algebraic structures   cinftm 29061
                  21.3.8.5  Semiring left modules   cslmd 29084
                  21.3.8.6  Finitely supported group sums - misc additions   gsumle 29110
                  21.3.8.7  Rings - misc additions   rngurd 29119
                  21.3.8.8  Ordered rings and fields   corng 29126
                  21.3.8.9  Ring homomorphisms - misc additions   rhmdvdsr 29149
                  21.3.8.10  Scalar restriction operation   cresv 29155
                  21.3.8.11  The commutative ring of gaussian integers   gzcrng 29170
                  21.3.8.12  The archimedean ordered field of real numbers   reofld 29171
            21.3.9  Matrices   symgfcoeu 29176
                  21.3.9.1  The symmetric group   symgfcoeu 29176
                  21.3.9.2  Permutation Signs   psgndmfi 29177
                  21.3.9.3  Submatrices   csmat 29187
                  21.3.9.4  Matrix literals   clmat 29205
                  21.3.9.5  Laplace expansion of determinants   mdetpmtr1 29217
            21.3.10  Topology   fvproj 29227
                  21.3.10.1  Open maps   fvproj 29227
                  21.3.10.2  Topology of the unit circle   qtopt1 29230
                  21.3.10.3  Refinements   reff 29234
                  21.3.10.4  Open cover refinement property   ccref 29237
                  21.3.10.5  Lindelöf spaces   cldlf 29247
                  21.3.10.6  Paracompact spaces   cpcmp 29250
                  21.3.10.7  Pseudometrics   cmetid 29257
                  21.3.10.8  Continuity - misc additions   hauseqcn 29269
                  21.3.10.9  Topology of the closed unit   unitsscn 29270
                  21.3.10.10  Topology of ` ( RR X. RR ) `   unicls 29277
                  21.3.10.11  Order topology - misc. additions   cnvordtrestixx 29287
                  21.3.10.12  Continuity in topological spaces - misc. additions   mndpluscn 29300
                  21.3.10.13  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 29306
                  21.3.10.14  Limits - misc additions   lmlim 29321
                  21.3.10.15  Univariate polynomials   pl1cn 29329
            21.3.11  Uniform Stuctures and Spaces   chcmp 29330
                  21.3.11.1  Hausdorff uniform completion   chcmp 29330
            21.3.12  Topology and algebraic structures   zringnm 29332
                  21.3.12.1  The norm on the ring of the integer numbers   zringnm 29332
                  21.3.12.2  Topological ` ZZ ` -modules   zlm0 29334
                  21.3.12.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 29344
                  21.3.12.4  Canonical embedding of the real numbers into a complete ordered field   crrh 29365
                  21.3.12.5  Embedding from the extended real numbers into a complete lattice   cxrh 29388
                  21.3.12.6  Canonical embeddings into the ordered field of the real numbers   zrhre 29391
                  *21.3.12.7  Topological Manifolds   cmntop 29394
            21.3.13  Real and complex functions   nexple 29399
                  21.3.13.1  Integer powers - misc. additions   nexple 29399
                  21.3.13.2  Indicator Functions   cind 29400
                  21.3.13.3  Extended sum   cesum 29416
            21.3.14  Mixed Function/Constant operation   cofc 29484
            21.3.15  Abstract measure   csiga 29497
                  21.3.15.1  Sigma-Algebra   csiga 29497
                  21.3.15.2  Generated sigma-Algebra   csigagen 29528
                  *21.3.15.3  lambda and pi-Systems, Rings of Sets   ispisys 29542
                  21.3.15.4  The Borel algebra on the real numbers   cbrsiga 29571
                  21.3.15.5  Product Sigma-Algebra   csx 29578
                  21.3.15.6  Measures   cmeas 29585
                  21.3.15.7  The counting measure   cntmeas 29616
                  21.3.15.8  The Lebesgue measure - misc additions   voliune 29619
                  21.3.15.9  The Dirac delta measure   cdde 29622
                  21.3.15.10  The 'almost everywhere' relation   cae 29627
                  21.3.15.11  Measurable functions   cmbfm 29639
                  21.3.15.12  Borel Algebra on ` ( RR X. RR ) `   br2base 29658
                  *21.3.15.13  Caratheodory's extension theorem   coms 29680
            21.3.16  Integration   itgeq12dv 29715
                  21.3.16.1  Lebesgue integral - misc additions   itgeq12dv 29715
                  21.3.16.2  Bochner integral   citgm 29716
            21.3.17  Euler's partition theorem   oddpwdc 29743
            21.3.18  Sequences defined by strong recursion   csseq 29772
            21.3.19  Fibonacci Numbers   cfib 29785
            21.3.20  Probability   cprb 29796
                  21.3.20.1  Probability Theory   cprb 29796
                  21.3.20.2  Conditional Probabilities   ccprob 29820
                  21.3.20.3  Real Valued Random Variables   crrv 29829
                  21.3.20.4  Preimage set mapping operator   corvc 29844
                  21.3.20.5  Distribution Functions   orvcelval 29857
                  21.3.20.6  Cumulative Distribution Functions   orvclteel 29861
                  21.3.20.7  Probabilities - example   coinfliplem 29867
                  21.3.20.8  Bertrand's Ballot Problem   ballotlemoex 29874
            21.3.21  Signum (sgn or sign) function - misc. additions   sgncl 29927
            21.3.22  Words over a set - misc additions   wrdres 29943
                  21.3.22.1  Operations on words   ccatmulgnn0dir 29945
            21.3.23  Polynomials with real coefficients - misc additions   plymul02 29949
            21.3.24  Descartes's rule of signs   signspval 29955
                  21.3.24.1  Sign changes in a word over real numbers   signspval 29955
                  21.3.24.2  Counting sign changes in a word over real numbers   signslema 29965
            21.3.25  Elementary Geometry   cstrkg2d 29995
                  *21.3.25.1  Two-dimension geometry   cstrkg2d 29995
                  21.3.25.2  Outer Five Segment (not used, no need to move to main)   cafs 30000
      *21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 30017
            21.4.2  Well founded induction and recursion   bnj110 30182
            21.4.3  The existence of a minimal element in certain classes   bnj69 30332
            21.4.4  Well-founded induction   bnj1204 30334
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 30384
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 30390
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 30394
      21.5  Mathbox for Mario Carneiro
            21.5.1  Predicate calculus with all distinct variables   ax-7d 30395
            21.5.2  Miscellaneous stuff   quartfull 30401
            21.5.3  Derangements and the Subfactorial   deranglem 30402
            21.5.4  The Erdős-Szekeres theorem   erdszelem1 30427
            21.5.5  The Kuratowski closure-complement theorem   kur14lem1 30442
            21.5.6  Retracts and sections   cretr 30453
            21.5.7  Path-connected and simply connected spaces   cpcon 30455
            21.5.8  Covering maps   ccvm 30491
            21.5.9  Normal numbers   snmlff 30565
            21.5.10  Godel-sets of formulas   cgoe 30569
            21.5.11  Models of ZF   cgze 30597
            *21.5.12  Metamath formal systems   cmcn 30611
            21.5.13  Grammatical formal systems   cm0s 30736
            21.5.14  Models of formal systems   cmuv 30754
            21.5.15  Splitting fields   citr 30776
            21.5.16  p-adic number fields   czr 30792
      *21.6  Mathbox for Filip Cernatescu
      21.7  Mathbox for Paul Chapman
            21.7.1  Real and complex numbers (cont.)   climuzcnv 30819
            21.7.2  Miscellaneous theorems   elfzm12 30823
      21.8  Mathbox for Scott Fenton
            21.8.1  ZFC Axioms in primitive form   axextprim 30832
            21.8.2  Untangled classes   untelirr 30839
            21.8.3  Extra propositional calculus theorems   3orel1 30846
            21.8.4  Misc. Useful Theorems   nepss 30854
            21.8.5  Properties of real and complex numbers   sqdivzi 30863
            21.8.6  Infinite products   iprodefisumlem 30879
            21.8.7  Factorial limits   faclimlem1 30882
            21.8.8  Greatest common divisor and divisibility   pdivsq 30888
            21.8.9  Properties of relationships   brtp 30892
            21.8.10  Properties of functions and mappings   funpsstri 30909
            21.8.11  Epsilon induction   setinds 30927
            21.8.12  Ordinal numbers   elpotr 30930
            21.8.13  Defined equality axioms   axextdfeq 30947
            21.8.14  Hypothesis builders   hbntg 30955
            21.8.15  (Trans)finite Recursion Theorems   tfisg 30960
            21.8.16  Transitive closure under a relationship   ctrpred 30961
            21.8.17  Founded Induction   frmin 30983
            21.8.18  Ordering Ordinal Sequences   orderseqlem 30993
            21.8.19  Well-founded zero, successor, and limits   cwsuc 30996
            21.8.20  Founded Recursion   frr3g 31023
            21.8.21  Surreal Numbers   csur 31037
            21.8.22  Surreal Numbers: Ordering   sltsolem1 31067
            21.8.23  Surreal Numbers: Birthday Function   bdayfo 31074
            21.8.24  Surreal Numbers: Density   fvnobday 31081
            21.8.25  Surreal Numbers: Upper and Lower Bounds   nobndlem1 31091
            21.8.26  Surreal Numbers: Full-Eta Property   nofulllem1 31101
            21.8.27  Quantifier-free definitions   ctxp 31106
            21.8.28  Alternate ordered pairs   caltop 31233
            21.8.29  Geometry in the Euclidean space   cofs 31259
                  21.8.29.1  Congruence properties   cofs 31259
                  21.8.29.2  Betweenness properties   btwntriv2 31289
                  21.8.29.3  Segment Transportation   ctransport 31306
                  21.8.29.4  Properties relating betweenness and congruence   cifs 31312
                  21.8.29.5  Connectivity of betweenness   btwnconn1lem1 31364
                  21.8.29.6  Segment less than or equal to   csegle 31383
                  21.8.29.7  Outside of relationship   coutsideof 31396
                  21.8.29.8  Lines and Rays   cline2 31411
            21.8.30  Forward difference   cfwddif 31435
            21.8.31  Rank theorems   rankung 31443
            21.8.32  Hereditarily Finite Sets   chf 31449
      21.9  Mathbox for Jeff Hankins
            21.9.1  Miscellany   a1i14 31464
            21.9.2  Basic topological facts   topbnd 31489
            21.9.3  Topology of the real numbers   ivthALT 31500
            21.9.4  Refinements   cfne 31501
            21.9.5  Neighborhood bases determine topologies   neibastop1 31524
            21.9.6  Lattice structure of topologies   topmtcl 31528
            21.9.7  Filter bases   fgmin 31535
            21.9.8  Directed sets, nets   tailfval 31537
      21.10  Mathbox for Anthony Hart
            21.10.1  Propositional Calculus   tb-ax1 31548
            21.10.2  Predicate Calculus   allt 31570
            21.10.3  Misc. Single Axiom Systems   meran1 31580
            21.10.4  Connective Symmetry   negsym1 31586
      21.11  Mathbox for Chen-Pang He
            21.11.1  Ordinal topology   ontopbas 31597
      21.12  Mathbox for Jeff Hoffman
            21.12.1  Inferences for finite induction on generic function values   fveleq 31620
            21.12.2  gdc.mm   nnssi2 31624
      21.13  Mathbox for Asger C. Ipsen
            21.13.1  Continuous nowhere differentiable functions   dnival 31631
      *21.14  Mathbox for BJ
            *21.14.1  Propositional calculus   bj-mp2c 31701
                  *21.14.1.1  Derived rules of inference   bj-mp2c 31701
                  *21.14.1.2  A syntactic theorem   bj-0 31703
                  21.14.1.3  Minimal implicational calculus   bj-a1k 31705
                  21.14.1.4  Positive calculus   bj-orim2 31711
                  21.14.1.5  Implication and negation   pm4.81ALT 31716
                  *21.14.1.6  Disjunction   bj-jaoi1 31726
                  *21.14.1.7  Logical equivalence   bj-dfbi4 31728
                  21.14.1.8  The conditional operator for propositions   bj-consensus 31732
                  *21.14.1.9  Propositional calculus: miscellaneous   sylancl2 31737
            *21.14.2  Modal logic   bj-axdd2 31749
            *21.14.3  Provability logic   cprvb 31755
            *21.14.4  First-order logic   wnff 31764
                  21.14.4.1  Universal and existential quantifiers, "non-free" predicate   wnff 31764
                  21.14.4.2  Adding ax-gen   bj-nfth 31772
                  21.14.4.3  Adding ax-4   bj-2alim 31779
                  21.14.4.4  Adding ax-5   bj-ax5ea 31805
                  21.14.4.5  Equality and substitution   wssb 31808
                  21.14.4.6  Adding ax-6   bj-extru 31843
                  21.14.4.7  Adding ax-7   bj-cbvexw 31851
                  21.14.4.8  Membership predicate, ax-8 and ax-9   bj-elequ2g 31853
                  21.14.4.9  Adding ax-11   bj-alcomexcom 31857
                  21.14.4.10  Adding ax-12   axc11n11 31859
                  21.14.4.11  Adding ax-13   bj-axc10 31894
                  *21.14.4.12  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 31904
                  *21.14.4.13  Strengthenings of theorems of the main part   bj-sb3b 31992
                  *21.14.4.14  Distinct var metavariables   bj-hbaeb2 31993
                  *21.14.4.15  Around ~ equsal   bj-equsal1t 31997
                  *21.14.4.16  Some Principia Mathematica proofs   stdpc5t 32002
                  21.14.4.17  Alternate definition of substitution   bj-sbsb 32012
                  21.14.4.18  Lemmas for substitution   bj-sbf3 32014
                  21.14.4.19  Existential uniqueness   bj-eu3f 32017
                  *21.14.4.20  First-logic: miscellaneous   bj-nfdiOLD 32019
            21.14.5  Set theory   eliminable1 32033
                  *21.14.5.1  Eliminability of class terms   eliminable1 32033
                  *21.14.5.2  Classes without extensionality   bj-eleq1w 32040
                  *21.14.5.3  The class-form not-free predicate   bj-nfcsym 32079
                  *21.14.5.4  Proposal for the definitions of class membership and class equality   bj-ax8 32080
                  *21.14.5.5  Lemmas for class substitution   bj-sbeqALT 32087
                  21.14.5.6  Removing some dv conditions   bj-exlimmpi 32097
                  *21.14.5.7  Class abstractions   bj-unrab 32114
                  *21.14.5.8  Restricted non-freeness   wrnf 32122
                  *21.14.5.9  Russell's paradox   bj-ru0 32124
                  *21.14.5.10  Some disjointness results   bj-n0i 32127
                  *21.14.5.11  Complements on direct products   bj-xpimasn 32135
                  *21.14.5.12  "Singletonization" and tagging   bj-sels 32143
                  *21.14.5.13  Tuples of classes   bj-cproj 32171
                  *21.14.5.14  Set theory: miscellaneous   bj-vjust2 32206
                  *21.14.5.15  Elementwise intersection (families of sets induced on a subset)   bj-rest00 32215
                  21.14.5.16  Topology (complements)   bj-toptopon2 32234
                  21.14.5.17  Maps-to notation for functions with three arguments   bj-0nelmpt 32250
                  *21.14.5.18  Currying   cfset 32256
            *21.14.6  Extended real and complex numbers, real and complex projectives lines   bj-elid 32262
                  *21.14.6.1  Diagonal in a Cartesian square   bj-elid 32262
                  *21.14.6.2  Extended numbers and projective lines as sets   cinftyexpi 32270
                  *21.14.6.3  Addition and opposite   caddcc 32301
                  *21.14.6.4  Argument, multiplication and inverse   cprcpal 32305
            *21.14.7  Monoids   bj-cmnssmnd 32313
                  *21.14.7.1  Finite sums in monoids   cfinsum 32322
            *21.14.8  Affine, Euclidean, and Cartesian geometry   crrvec 32325
                  *21.14.8.1  Convex hull in real vector spaces   crrvec 32325
                  *21.14.8.2  Complex numbers (supplements)   bj-subcom 32331
                  *21.14.8.3  Barycentric coordinates   bj-bary1lem 32337
      21.15  Mathbox for Jim Kingdon
      21.16  Mathbox for ML
      21.17  Mathbox for Wolf Lammen
            21.17.1  1. Bootstrapping   wl-section-boot 32420
            21.17.2  Implication chains   wl-section-impchain 32444
            21.17.3  An alternative definition of df-nf   wl-section-nf 32462
            21.17.4  An alternative axiom ~ ax-13   ax-wl-13v 32465
            21.17.5  Other stuff   wl-jarri 32467
      21.18  Mathbox for Brendan Leahy
      21.19  Mathbox for Jeff Madsen
            21.19.1  Logic and set theory   anim12da 32676
            21.19.2  Real and complex numbers; integers   filbcmb 32705
            21.19.3  Sequences and sums   sdclem2 32708
            21.19.4  Topology   subspopn 32718
            21.19.5  Metric spaces   metf1o 32721
            21.19.6  Continuous maps and homeomorphisms   constcncf 32728
            21.19.7  Boundedness   ctotbnd 32735
            21.19.8  Isometries   cismty 32767
            21.19.9  Heine-Borel Theorem   heibor1lem 32778
            21.19.10  Banach Fixed Point Theorem   bfplem1 32791
            21.19.11  Euclidean space   crrn 32794
            21.19.12  Intervals (continued)   ismrer1 32807
            21.19.13  Operation properties   cass 32811
            21.19.14  Groups and related structures   cmagm 32817
            21.19.15  Group homomorphism and isomorphism   cghomOLD 32852
            21.19.16  Rings   crngo 32863
            21.19.17  Division Rings   cdrng 32917
            21.19.18  Ring homomorphisms   crnghom 32929
            21.19.19  Commutative rings   ccm2 32958
            21.19.20  Ideals   cidl 32976
            21.19.21  Prime rings and integral domains   cprrng 33015
            21.19.22  Ideal generators   cigen 33028
      21.20  Mathbox for Giovanni Mascellani
            *21.20.1  Tools for automatic proof building   efald2 33047
            *21.20.2  Tseitin axioms   fald 33106
            *21.20.3  Equality deductions   iuneq2f 33133
            *21.20.4  Miscellanea   scottexf 33146
      21.21  Mathbox for Rodolfo Medina
            21.21.1  Partitions   prtlem60 33152
      *21.22  Mathbox for Norm Megill
            *21.22.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 33186
            *21.22.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 33196
            *21.22.3  Legacy theorems using obsolete axioms   ax5ALT 33210
            21.22.4  Experiments with weak deduction theorem   elimhyps 33265
            21.22.5  Miscellanea   cnaddcom 33277
            21.22.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 33279
            21.22.7  Functionals and kernels of a left vector space (or module)   clfn 33362
            21.22.8  Opposite rings and dual vector spaces   cld 33428
            21.22.9  Ortholattices and orthomodular lattices   cops 33477
            21.22.10  Atomic lattices with covering property   ccvr 33567
            21.22.11  Hilbert lattices   chlt 33655
            21.22.12  Projective geometries based on Hilbert lattices   clln 33795
            21.22.13  Construction of a vector space from a Hilbert lattice   cdlema1N 34095
            21.22.14  Construction of involution and inner product from a Hilbert lattice   clpoN 35787
      21.23  Mathbox for OpenAI
      21.24  Mathbox for Stefan O'Rear
            21.24.1  Additional elementary logic and set theory   moxfr 36273
            21.24.2  Additional theory of functions   imaiinfv 36274
            21.24.3  Additional topology   elrfi 36275
            21.24.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 36279
            21.24.5  Algebraic closure systems   cnacs 36283
            21.24.6  Miscellanea 1. Map utilities   constmap 36294
            21.24.7  Miscellanea for polynomials   mptfcl 36301
            21.24.8  Multivariate polynomials over the integers   cmzpcl 36302
            21.24.9  Miscellanea for Diophantine sets 1   coeq0i 36334
            21.24.10  Diophantine sets 1: definitions   cdioph 36336
            21.24.11  Diophantine sets 2 miscellanea   ellz1 36348
            21.24.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 36354
            21.24.13  Diophantine sets 3: construction   diophrex 36357
            21.24.14  Diophantine sets 4 miscellanea   2sbcrex 36366
            21.24.15  Diophantine sets 4: Quantification   rexrabdioph 36376
            21.24.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 36383
            21.24.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 36393
            21.24.18  Pigeonhole Principle and cardinality helpers   fphpd 36398
            21.24.19  A non-closed set of reals is infinite   rencldnfilem 36402
            21.24.20  Lagrange's rational approximation theorem   irrapxlem1 36404
            21.24.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 36411
            21.24.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 36418
            21.24.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 36460
            *21.24.24  Logarithm laws generalized to an arbitrary base   reglogcl 36472
            21.24.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 36480
            21.24.26  X and Y sequences 1: Definition and recurrence laws   crmx 36482
            21.24.27  Ordering and induction lemmas for the integers   monotuz 36524
            21.24.28  X and Y sequences 2: Order properties   rmxypos 36532
            21.24.29  Congruential equations   congtr 36550
            21.24.30  Alternating congruential equations   acongid 36560
            21.24.31  Additional theorems on integer divisibility   coprmdvdsb 36570
            21.24.32  X and Y sequences 3: Divisibility properties   jm2.18 36573
            21.24.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 36590
            21.24.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 36600
            21.24.35  Uncategorized stuff not associated with a major project   setindtr 36609
            21.24.36  More equivalents of the Axiom of Choice   axac10 36618
            21.24.37  Finitely generated left modules   clfig 36655
            21.24.38  Noetherian left modules I   clnm 36663
            21.24.39  Addenda for structure powers   pwssplit4 36677
            21.24.40  Every set admits a group structure iff choice   unxpwdom3 36683
            21.24.41  Noetherian rings and left modules II   clnr 36698
            21.24.42  Hilbert's Basis Theorem   cldgis 36710
            21.24.43  Additional material on polynomials [DEPRECATED]   cmnc 36720
            21.24.44  Degree and minimal polynomial of algebraic numbers   cdgraa 36729
            21.24.45  Algebraic integers I   citgo 36746
            21.24.46  Endomorphism algebra   cmend 36764
            21.24.47  Subfields   csdrg 36784
            21.24.48  Cyclic groups and order   idomrootle 36792
            21.24.49  Cyclotomic polynomials   ccytp 36799
            21.24.50  Miscellaneous topology   fgraphopab 36807
      21.25  Mathbox for Jon Pennant
      21.26  Mathbox for Richard Penner
            21.26.1  Short Studies   ifpan123g 36822
                  21.26.1.1  Additional work on conditional logical operator   ifpan123g 36822
                  21.26.1.2  Sophisms   rp-fakeimass 36876
                  *21.26.1.3  Finite Sets   rp-isfinite5 36882
                  21.26.1.4  Infinite Sets   pwelg 36884
                  *21.26.1.5  Finite intersection property   fipjust 36889
                  21.26.1.6  RP ADDTO: Subclasses and subsets   rababg 36898
                  21.26.1.7  RP ADDTO: The intersection of a class   elintabg 36899
                  21.26.1.8  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 36902
                  21.26.1.9  RP ADDTO: Relations   xpinintabd 36905
                  *21.26.1.10  RP ADDTO: Functions   elmapintab 36921
                  *21.26.1.11  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 36925
                  21.26.1.12  RP ADDTO: First and second members of an ordered pair   elcnvlem 36926
                  21.26.1.13  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 36929
                  21.26.1.14  RP ADDTO: Basic properties of closures   cleq2lem 36933
                  21.26.1.15  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 36956
            21.26.2  Additional statements on relations and subclasses   al3im 36957
                  21.26.2.1  Transitive relations (not to be confused with transitive classes).   trrelind 36976
                  21.26.2.2  Reflexive closures   crcl 36983
                  *21.26.2.3  Finite relationship composition.   relexp2 36988
                  21.26.2.4  Transitive closure of a relation   dftrcl3 37031
                  *21.26.2.5  Adapted from Frege   frege77d 37057
            *21.26.3  Propositions from _Begriffsschrift_   dfxor4 37077
                  *21.26.3.1  _Begriffsschrift_ Chapter I   dfxor4 37077
                  *21.26.3.2  _Begriffsschrift_ Notation hints   rp-imass 37085
                  21.26.3.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 37104
                  21.26.3.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 37143
                  *21.26.3.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 37170
                  21.26.3.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 37201
                  21.26.3.7  _Begriffsschrift_ Chapter II with equivalence of classes (where they are sets)   frege53c 37228
                  *21.26.3.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 37246
                  *21.26.3.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 37253
                  *21.26.3.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 37276
                  *21.26.3.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 37292
            *21.26.4  Exploring Topology via Seifert And Threlfall   enrelmap 37311
                  *21.26.4.1  Equinumerosity of sets of relations and maps   enrelmap 37311
                  *21.26.4.2  Generic Pseudoclosure Spaces, Pseudointeror Spaces, and Pseudoneighborhoods   sscon34b 37337
                  *21.26.4.3  Generic Neighborhood Spaces   gneispa 37448
            *21.26.5  Exploring Higher Homotopy via Kerodon   k0004lem1 37465
                  *21.26.5.1  Simplicial Sets   k0004lem1 37465
      21.27  Mathbox for Stanislas Polu
            21.27.1  IMO Problems   wwlemuld 37474
                  21.27.1.1  IMO 1972 B2   wwlemuld 37474
            *21.27.2  INT Inequalities Proof Generator   int-addcomd 37498
            *21.27.3  N-Digit Addition Proof Generator   unitadd 37520
            21.27.4  AM-GM (for k = 2,3,4)   gsumws3 37521
      21.28  Mathbox for Steve Rodriguez
            21.28.1  Miscellanea   nanorxor 37526
            21.28.2  Ratio test for infinite series convergence and divergence   dvgrat 37533
            21.28.3  Multiples   reldvds 37536
            21.28.4  Function operations   caofcan 37544
            21.28.5  Calculus   lhe4.4ex1a 37550
            21.28.6  The generalized binomial coefficient operation   cbcc 37557
            21.28.7  Binomial series   uzmptshftfval 37567
      21.29  Mathbox for Andrew Salmon
            21.29.1  Principia Mathematica * 10   pm10.12 37579
            21.29.2  Principia Mathematica * 11   2alanimi 37593
            21.29.3  Predicate Calculus   sbeqal1 37620
            21.29.4  Principia Mathematica * 13 and * 14   pm13.13a 37630
            21.29.5  Set Theory   elnev 37661
            21.29.6  Arithmetic   addcomgi 37681
            21.29.7  Geometry   cplusr 37682
      *21.30  Mathbox for Alan Sare
            21.30.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 37704
            21.30.2  Supplementary unification deductions   bi1imp 37708
            21.30.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 37728
            21.30.4  What is Virtual Deduction?   wvd1 37806
            21.30.5  Virtual Deduction Theorems   df-vd1 37807
            21.30.6  Theorems proved using Virtual Deduction   trsspwALT 38067
            21.30.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 38103
            21.30.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 38171
            21.30.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 38175
            21.30.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 38182
            *21.30.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 38185
      21.31  Mathbox for Glauco Siliprandi
            21.31.1  Miscellanea   fnvinran 38196
            21.31.2  Functions   unima 38340
            21.31.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 38426
            21.31.4  Real intervals   gtnelioc 38559
            21.31.5  Finite sums   sumeq2ad 38632
            21.31.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 38647
            21.31.7  Limits   clim1fr1 38668
            21.31.8  Trigonometry   coseq0 38747
            21.31.9  Continuous Functions   mulcncff 38753
            21.31.10  Derivatives   dvsinexp 38798
            21.31.11  Integrals   volioo 38840
            21.31.12  Stone Weierstrass theorem - real version   stoweidlem1 38894
            21.31.13  Wallis' product for π   wallispilem1 38958
            21.31.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 38967
            21.31.15  Dirichlet kernel   dirkerval 38984
            21.31.16  Fourier Series   fourierdlem1 39001
            21.31.17  e is transcendental   elaa2lem 39126
            21.31.18  n-dimensional Euclidean space   rrxtopn 39177
            21.31.19  Basic measure theory   csalg 39204
                  *21.31.19.1  σ-Algebras   csalg 39204
                  21.31.19.2  Sum of nonnegative extended reals   csumge0 39255
                  *21.31.19.3  Measures   cmea 39342
                  *21.31.19.4  Outer measures and Caratheodory's construction   come 39379
                  *21.31.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 39426
                  *21.31.19.6  Measurable functions   csmblfn 39586
      21.32  Mathbox for Saveliy Skresanov
            21.32.1  Ceva's theorem   sigarval 39688
      21.33  Mathbox for Jarvin Udandy
      21.34  Mathbox for Alexander van der Vekens
            21.34.1  Double restricted existential uniqueness   r19.32 39816
                  21.34.1.1  Restricted quantification (extension)   r19.32 39816
                  21.34.1.2  The empty set (extension)   raaan2 39824
                  21.34.1.3  Restricted uniqueness and "at most one" quantification   rmoimi 39825
                  21.34.1.4  Analogs to Existential uniqueness (double quantification)   2reurex 39830
            *21.34.2  Alternative definitions of function's and operation's values   wdfat 39842
                  21.34.2.1  Restricted quantification (extension)   ralbinrald 39848
                  21.34.2.2  The universal class (extension)   nvelim 39849
                  21.34.2.3  Introduce the Axiom of Power Sets (extension)   alneu 39850
                  21.34.2.4  Relations (extension)   eldmressn 39852
                  21.34.2.5  Functions (extension)   fveqvfvv 39853
                  21.34.2.6  Predicate "defined at"   dfateq12d 39858
                  21.34.2.7  Alternative definition of the value of a function   dfafv2 39861
                  21.34.2.8  Alternative definition of the value of an operation   aoveq123d 39907
            21.34.3  General auxiliary theorems   1t10e1p1e11 39937
                  21.34.3.1  Miscellanea   1t10e1p1e11 39937
                  21.34.3.2  The modulo (remainder) operation (extension)   m1mod0mod1 39949
                  *21.34.3.3  Partitions of real intervals   ciccp 39951
            21.34.4  Number theory (extension)   cfmtno 39977
                  *21.34.4.1  Fermat numbers   cfmtno 39977
                  *21.34.4.2  Mersenne primes   m2prm 40043
                  21.34.4.3  Proth's theorem   modexp2m1d 40067
            *21.34.5  Even and odd numbers   ceven 40075
                  21.34.5.1  Definitions and basic properties   ceven 40075
                  21.34.5.2  Alternate definitions using the "divides" relation   dfeven2 40100
                  21.34.5.3  Alternate definitions using the "modulo" operation   dfeven3 40108
                  21.34.5.4  Alternate definitions using the "gcd" operation   iseven5 40114
                  21.34.5.5  Theorems of part 5 revised   zneoALTV 40118
                  21.34.5.6  Theorems of part 6 revised   odd2np1ALTV 40123
                  21.34.5.7  Theorems of AV's mathbox revised   0evenALTV 40137
                  21.34.5.8  Additional theorems   epoo 40150
                  21.34.5.9  Perfect Number Theorem (revised)   perfectALTVlem1 40164
                  *21.34.5.10  Goldbach's conjectures   cgbe 40167
            21.34.6  Words over a set (extension)   wrdred1 40240
                  21.34.6.1  Truncated words   wrdred1 40240
                  21.34.6.2  Last symbol of a word (extension)   lswn0 40242
                  21.34.6.3  Concatenations with singleton words (extension)   ccatw2s1cl 40243
                  *21.34.6.4  Prefixes of a word   cpfx 40244
            *21.34.7  Auxiliary theorems for graph theory   elnelall 40302
                  21.34.7.1  Negated equality and membership - extension   elnelall 40302
                  21.34.7.2  Subclasses and subsets - extension   clel5 40303
                  21.34.7.3  The empty set - extension   ralnralall 40307
                  21.34.7.4  Unordered and ordered pairs - extension   elpwdifsn 40312
                  21.34.7.5  Indexed union and intersection - extension   otiunsndisjX 40317
                  21.34.7.6  Ordered-pair class abstractions - extension   opabn1stprc 40318
                  21.34.7.7  Relations - extension   resresdm 40319
                  21.34.7.8  Functions - extension   fvifeq 40321
                  21.34.7.9  Restricted iota - extension   riotaeqimp 40338
                  21.34.7.10  Equinumerosity - extension   resfnfinfin 40339
                  21.34.7.11  Subtraction - extension   cnambpcma 40341
                  21.34.7.12  Ordering on reals (cont.) - extension   leaddsuble 40343
                  21.34.7.13  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 40349
                  21.34.7.14  Upper sets of integers - extension   eluzge0nn0 40350
                  21.34.7.15  Finite intervals of integers - extension   ssfz12 40351
                  21.34.7.16  Half-open integer ranges - extension   subsubelfzo0 40359
                  21.34.7.17  The ` # ` (set size) function - extension   nfile 40369
                  21.34.7.18  Finite and infinite sums - extension   fsummsndifre 40371
            21.34.8  Graph theory (revised, deprecated)   uhgruhgra 40375
                  21.34.8.1  Undirected hypergraphs (deprecated)   uhgruhgra 40375
                  *21.34.8.2  Undirected simple graphs - basics   cuspgr 40378
                  21.34.8.3  Examples for graphs   usgr0e 40462
                  21.34.8.4  Subgraphs   csubgr 40491
                  21.34.8.5  Undirected simple graphs - finite graphs   cfusgr 40535
                  21.34.8.6  Neighbors, complete graphs and universal vertices   cnbgr 40550
                  *21.34.8.7  Vertex degree   cvtxdg 40681
                  *21.34.8.8  Regular graphs   crgr 40755
                  *21.34.8.9  Walks   cewlks 40795
                  21.34.8.10  Walks for loop-free graphs   lfgrwlkprop 40896
                  21.34.8.11  Trails   ctrls 40899
                  21.34.8.12  Paths   cpths 40919
                  21.34.8.13  Closed walks   cclwlks 40976
                  21.34.8.14  Circuits and cycles   ccrcts 40990
                  *21.34.8.15  Walks as words   cwwlks 41028
                  21.34.8.16  Walks/paths of length 2 (as length 3 strings)   21wlkdlem1 41132
                  21.34.8.17  Walks in regular graphs   rusgrnumwwlkl1 41172
                  *21.34.8.18  Closed walks as words   cclwwlks 41183
                  21.34.8.19  Examples for walks, trails and paths   0ewlk 41282
                  21.34.8.20  Connected graphs   cconngr 41353
                  *21.34.8.21  Eulerian paths   ceupth 41364
                  *21.34.8.22  The Königsberg Bridge problem   konigsbergvtx 41414
                  21.34.8.23  Friendship graphs - basics   cfrgr 41428
                  21.34.8.24  The friendship theorem for small graphs   frgr1v 41441
                  21.34.8.25  Theorems according to Mertzios and Unger   2pthfrgrrn 41452
                  *21.34.8.26  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 41469
            21.34.9  Monoids (extension)   ovn0dmfun 41554
                  21.34.9.1  Auxiliary theorems   ovn0dmfun 41554
                  21.34.9.2  Magmas and Semigroups (extension)   plusfreseq 41562
                  21.34.9.3  Magma homomorphisms and submagmas   cmgmhm 41567
                  21.34.9.4  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpt2ismgm 41597
            *21.34.10  Magmas and internal binary operations (alternate approach)   ccllaw 41609
                  *21.34.10.1  Laws for internal binary operations   ccllaw 41609
                  *21.34.10.2  Internal binary operations   cintop 41622
                  21.34.10.3  Alternative definitions for Magmas and Semigroups   cmgm2 41641
            21.34.11  Categories (extension)   idfusubc0 41655
                  21.34.11.1  Subcategories (extension)   idfusubc0 41655
            21.34.12  Rings (extension)   lmod0rng 41658
                  21.34.12.1  Nonzero rings (extension)   lmod0rng 41658
                  *21.34.12.2  Non-unital rings ("rngs")   crng 41664
                  21.34.12.3  Rng homomorphisms   crngh 41675
                  21.34.12.4  Ring homomorphisms (extension)   rhmfn 41708
                  21.34.12.5  Ideals as non-unital rings   lidldomn1 41711
                  21.34.12.6  The non-unital ring of even integers   0even 41721
                  21.34.12.7  A constructed not unital ring   plusgndxnmulrndx 41743
                  *21.34.12.8  The category of non-unital rings   crngc 41749
                  *21.34.12.9  The category of (unital) rings   cringc 41795
                  21.34.12.10  Subcategories of the category of rings   srhmsubclem1 41865
            21.34.13  Basic algebraic structures (extension)   xpprsng 41903
                  21.34.13.1  Auxiliary theorems   xpprsng 41903
                  21.34.13.2  The binomial coefficient operation (extension)   bcpascm1 41922
                  21.34.13.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 41925
                  21.34.13.4  Ordered group sum operation (extension)   gsumpr 41932
                  21.34.13.5  Symmetric groups (extension)   nn0le2is012 41938
                  21.34.13.6  Divisibility (extension)   invginvrid 41942
                  21.34.13.7  The support of functions (extension)   rmsupp0 41943
                  21.34.13.8  Finitely supported functions (extension)   rmsuppfi 41948
                  21.34.13.9  Left modules (extension)   lmodvsmdi 41957
                  21.34.13.10  Associative algebras (extension)   ascl0 41959
                  21.34.13.11  Univariate polynomials (extension)   ply1vr1smo 41963
                  21.34.13.12  Univariate polynomials (examples)   linply1 41975
            21.34.14  Linear algebra (extension)   cdmatalt 41979
                  *21.34.14.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 41979
                  *21.34.14.2  Linear combinations   clinc 41987
                  *21.34.14.3  Linear independency   clininds 42023
                  21.34.14.4  Simple left modules and the ` ZZ `-module   lmod1lem1 42070
                  21.34.14.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 42090
            21.34.15  Complexity theory   offval0 42093
                  21.34.15.1  Auxiliary theorems   offval0 42093
                  21.34.15.2  The modulo (remainder) operation (extension)   fldivmod 42107
                  21.34.15.3  Even and odd integers   nn0onn0ex 42112
                  21.34.15.4  The natural logarithm on complex numbers (extension)   logge0b 42123
                  21.34.15.5  Division of functions   cfdiv 42129
                  21.34.15.6  Upper bounds   cbigo 42139
                  21.34.15.7  Logarithm to an arbitrary base (extension)   rege1logbrege0 42150
                  *21.34.15.8  The binary logarithm   fldivexpfllog2 42157
                  21.34.15.9  Binary length   cblen 42161
                  *21.34.15.10  Digits   cdig 42187
                  21.34.15.11  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 42207
                  21.34.15.12  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 42216
      21.35  Mathbox for Emmett Weisz
            *21.35.1  Miscellaneous Theorems   nfintd 42218
            21.35.2  Set Recursion   csetrecs 42229
                  *21.35.2.1  Basic Properties of Set Recursion   csetrecs 42229
                  21.35.2.2  Examples and properties of set recursion   elsetrecslem 42243
            *21.35.3  Construction of Games and Surreal Numbers   cpg 42251
      *21.36  Mathbox for David A. Wheeler
            21.36.1  Natural deduction   19.8ad 42257
            *21.36.2  Greater than, greater than or equal to.   cge-real 42260
            *21.36.3  Hyperbolic trigonometric functions   csinh 42270
            *21.36.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 42281
            *21.36.5  Identities for "if"   ifnmfalse 42303
            *21.36.6  Decimal point   cdp2 42304
            *21.36.7  Logarithms generalized to arbitrary base using ` logb `   logb2aval 42314
            *21.36.8  Logarithm laws generalized to an arbitrary base - log_   clog- 42315
            *21.36.9  Formally define terms such as Reflexivity   wreflexive 42317
            *21.36.10  Algebra helpers   comraddi 42321
            *21.36.11  Algebra helper examples   i2linesi 42333
            *21.36.12  Formal methods "surprises"   alimp-surprise 42335
            *21.36.13  Allsome quantifier   walsi 42341
            *21.36.14  Miscellaneous   5m4e1 42352
            21.36.15  AA theorems   aacllem 42356
      21.37  Mathbox for Kunhao Zheng
            21.37.1  Weighted AM-GM Inequality   amgmwlem 42357

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