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Table of Contents Summary
PART 1  CLASSICAL FIRST ORDER LOGIC WITH EQUALITY
      1.1  Pre-logic
      1.2  Propositional calculus
      1.3  Other axiomatizations of classical propositional calculus
      1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
      1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
      1.6  Predicate calculus with equality: Older axiomatization (1 rule, 14 schemes)
      1.7  Existential uniqueness
      1.8  Other axiomatizations related to classical predicate calculus
PART 2  ZF (ZERMELO-FRAENKEL) SET THEORY
      2.1  ZF Set Theory - start with the Axiom of Extensionality
      2.2  ZF Set Theory - add the Axiom of Replacement
      2.3  ZF Set Theory - add the Axiom of Power Sets
      2.4  ZF Set Theory - add the Axiom of Union
      2.5  ZF Set Theory - add the Axiom of Regularity
      2.6  ZF Set Theory - add the Axiom of Infinity
PART 3  ZFC (ZERMELO-FRAENKEL WITH CHOICE) SET THEORY
      3.1  ZFC Set Theory - add Countable Choice and Dependent Choice
      3.2  ZFC Set Theory - add the Axiom of Choice
      3.3  ZFC Axioms with no distinct variable requirements
      3.4  The Generalized Continuum Hypothesis
PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      4.1  Inaccessibles
      4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
PART 5  REAL AND COMPLEX NUMBERS
      5.1  Construction and axiomatization of real and complex numbers
      5.2  Derive the basic properties from the field axioms
      5.3  Real and complex numbers - basic operations
      5.4  Integer sets
      5.5  Order sets
      5.6  Elementary integer functions
      5.7  Elementary real and complex functions
      5.8  Elementary limits and convergence
      5.9  Elementary trigonometry
      5.10  Cardinality of real and complex number subsets
PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
      6.2  Elementary prime number theory
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
      7.2  Moore spaces
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
      8.2  Arrows (disjointified hom-sets)
      8.3  Examples of categories
      8.4  Categorical constructions
PART 9  BASIC ORDER THEORY
      9.1  Presets and directed sets using extensible structures
      9.2  Posets and lattices using extensible structures
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
      10.2  Groups
      10.3  Abelian groups
      10.4  Rings
      10.5  Division rings and fields
      10.6  Left modules
      10.7  Vector spaces
      10.8  Ideals
      10.9  Associative algebras
      10.10  Abstract multivariate polynomials
      10.11  The complex numbers as an extensible structure
      10.12  Hilbert spaces
PART 11  BASIC TOPOLOGY
      11.1  Topology
      11.2  Filters and filter bases
      11.3  Metric spaces
      11.4  Complex metric vector spaces
PART 12  BASIC REAL AND COMPLEX ANALYSIS
      12.1  Continuity
      12.2  Integrals
      12.3  Derivatives
PART 13  BASIC REAL AND COMPLEX FUNCTIONS
      13.1  Polynomials
      13.2  Sequences and series
      13.3  Basic trigonometry
      13.4  Basic number theory
PART 14  GUIDES AND MISCELLANEA
      14.1  Guides (conventions, explanations, and examples)
      14.2  Humor
      14.3  (Future - to be reviewed and classified)
PART 15  ADDITIONAL MATERIAL ON GROUPS, RINGS, AND FIELDS (DEPRECATED)
      15.1  Additional material on group theory
      15.2  Additional material on rings and fields
PART 16  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      16.1  Complex vector spaces
      16.2  Normed complex vector spaces
      16.3  Operators on complex vector spaces
      16.4  Inner product (pre-Hilbert) spaces
      16.5  Complex Banach spaces
      16.6  Complex Hilbert spaces
PART 17  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      17.1  Axiomatization of complex pre-Hilbert spaces
      17.2  Inner product and norms
      17.3  Cauchy sequences and completeness axiom
      17.4  Subspaces and projections
      17.5  Properties of Hilbert subspaces
      17.6  Operators on Hilbert spaces
      17.7  States on an Hilbert lattice and Godowski's equation
      17.8  Cover relation, atoms, exchange axiom, and modular symmetry
PART 18  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      18.1  Mathboxes for user contributions
      18.2  Mathbox for Stefan Allan
      18.3  Mathbox for Thierry Arnoux
      18.4  Mathbox for Mario Carneiro
      18.5  Mathbox for Paul Chapman
      18.6  Mathbox for Drahflow
      18.7  Mathbox for Scott Fenton
      18.8  Mathbox for Anthony Hart
      18.9  Mathbox for Chen-Pang He
      18.10  Mathbox for Jeff Hoffman
      18.11  Mathbox for Wolf Lammen
      18.12  Mathbox for Brendan Leahy
      18.13  Mathbox for Frédéric Liné
      18.14  Mathbox for Jeff Hankins
      18.15  Mathbox for Jeff Madsen
      18.16  Mathbox for Rodolfo Medina
      18.17  Mathbox for Stefan O'Rear
      18.18  Mathbox for Steve Rodriguez
      18.19  Mathbox for Andrew Salmon
      18.20  Mathbox for Glauco Siliprandi
      18.21  Mathbox for Saveliy Skresanov
      18.22  Mathbox for Jarvin Udandy
      18.23  Mathbox for Alexander van der Vekens
      18.24  Mathbox for David A. Wheeler
      18.25  Mathbox for Alan Sare
      18.26  Mathbox for Jonathan Ben-Naim
      18.27  Mathbox for Norm Megill

Detailed Table of Contents
PART 1  CLASSICAL FIRST ORDER LOGIC WITH EQUALITY
      1.1  Pre-logic
            1.1.1  Inferences for assisting proof development   dummylink 1
      1.2  Propositional calculus
            1.2.1  Recursively define primitive wffs for propositional calculus   wn 3
            1.2.2  The axioms of propositional calculus   ax-1 5
            1.2.3  Logical implication   mp2b 9
            1.2.4  Logical negation   con4d 97
            1.2.5  Logical equivalence   wb 176
            1.2.6  Logical disjunction and conjunction   wo 357
            1.2.7  Miscellaneous theorems of propositional calculus   pm5.21nd 868
            1.2.8  Abbreviated conjunction and disjunction of three wff's   w3o 933
            1.2.9  Logical 'nand' (Sheffer stroke)   wnan 1287
            1.2.10  Logical 'xor'   wxo 1295
            1.2.11  True and false constants   wtru 1307
            1.2.12  Truth tables   truantru 1326
            1.2.13  Auxiliary theorems for Alan Sare's virtual deduction tool, part 1   ee22 1352
            1.2.14  Half-adders and full adders in propositional calculus   whad 1368
      1.3  Other axiomatizations of classical propositional calculus
            1.3.1  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1394
            1.3.2  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1413
            1.3.3  Derive Nicod's axiom from the standard axioms   nic-dfim 1424
            1.3.4  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1430
            1.3.5  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1449
            1.3.6  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1453
            1.3.7  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1468
            1.3.8  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1491
            1.3.9  Derive the Lukasiewicz axioms from the The Russell-Bernays Axioms   rb-bijust 1504
            1.3.10  Stoic logic indemonstrables (Chrysippus of Soli)   mpto1 1523
      1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
            1.4.1  Universal quantifier; define "exists" and "not free"   wal 1527
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1533
            1.4.3  Axiom scheme ax-5 (Quantified Implication)   ax-5 1544
            1.4.4  Axiom scheme ax-17 (Distinctness) - first use of $d   ax-17 1603
            1.4.5  Equality predicate; define substitution   cv 1622
            1.4.6  Axiom scheme ax-9 (Existence)   ax-9 1635
            1.4.7  Axiom scheme ax-8 (Equality)   ax-8 1643
            1.4.8  Membership predicate   wcel 1684
            1.4.9  Axiom schemes ax-13 (Left Membership Equality)   ax-13 1686
            1.4.10  Axiom schemes ax-14 (Right Membership Equality)   ax-14 1688
            1.4.11  Logical redundancy of ax-6 , ax-7 , ax-11 , ax-12   ax9dgen 1690
      1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-6 (Quantified Negation)   ax-6 1703
            1.5.2  Axiom scheme ax-7 (Quantifier Commutation)   ax-7 1708
            1.5.3  Axiom scheme ax-11 (Substitution)   ax-11 1715
            1.5.4  Axiom scheme ax-12 (Quantified Equality)   ax-12 1866
      1.6  Predicate calculus with equality: Older axiomatization (1 rule, 14 schemes)
            1.6.1  Obsolete schemes ax-5o ax-4 ax-6o ax-9o ax-10o ax-10 ax-11o ax-12o ax-15 ax-16   ax-4 2074
            1.6.2  Rederive new axioms from old: theorems ax5 , ax6 , ax9from9o , ax11 , ax12   ax4 2084
            1.6.3  Legacy theorems using obsolete axioms   ax17o 2096
      1.7  Existential uniqueness
      1.8  Other axiomatizations related to classical predicate calculus
            1.8.1  Predicate calculus with all distinct variables   ax-7d 2234
            1.8.2  Aristotelian logic: Assertic syllogisms   barbara 2240
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 2264
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2269
            2.1.3  Class form not-free predicate   wnfc 2406
            2.1.4  Negated equality and membership   wne 2446
            2.1.5  Restricted quantification   wral 2543
            2.1.6  The universal class   cvv 2788
            2.1.7  Conditional equality (experimental)   wcdeq 2974
            2.1.8  Russell's Paradox   ru 2990
            2.1.9  Proper substitution of classes for sets   wsbc 2991
            2.1.10  Proper substitution of classes for sets into classes   csb 3081
            2.1.11  Define basic set operations and relations   cdif 3149
            2.1.12  Subclasses and subsets   df-ss 3166
            2.1.13  The difference, union, and intersection of two classes   difeq1 3287
            2.1.14  The empty set   c0 3455
            2.1.15  "Weak deduction theorem" for set theory   cif 3565
            2.1.16  Power classes   cpw 3625
            2.1.17  Unordered and ordered pairs   csn 3640
            2.1.18  The union of a class   cuni 3827
            2.1.19  The intersection of a class   cint 3862
            2.1.20  Indexed union and intersection   ciun 3905
            2.1.21  Disjointness   wdisj 3993
            2.1.22  Binary relations   wbr 4023
            2.1.23  Ordered-pair class abstractions (class builders)   copab 4076
            2.1.24  Transitive classes   wtr 4113
      2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 4131
            2.2.2  Derive the Axiom of Separation   axsep 4140
            2.2.3  Derive the Null Set Axiom   zfnuleu 4146
            2.2.4  Theorems requiring subset and intersection existence   nalset 4151
            2.2.5  Theorems requiring empty set existence   class2set 4178
      2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4188
            2.3.2  Derive the Axiom of Pairing   zfpair 4212
            2.3.3  Ordered pair theorem   opnz 4242
            2.3.4  Ordered-pair class abstractions (cont.)   opabid 4271
            2.3.5  Power class of union and intersection   pwin 4297
            2.3.6  Epsilon and identity relations   cep 4303
            2.3.7  Partial and complete ordering   wpo 4312
            2.3.8  Founded and well-ordering relations   wfr 4349
            2.3.9  Ordinals   word 4391
      2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4512
            2.4.2  Ordinals (continued)   ordon 4574
            2.4.3  Transfinite induction   tfi 4644
            2.4.4  The natural numbers (i.e. finite ordinals)   com 4656
            2.4.5  Peano's postulates   peano1 4675
            2.4.6  Finite induction (for finite ordinals)   find 4681
            2.4.7  Relations   cxp 4687
            2.4.8  Definite description binder (inverted iota)   cio 5217
            2.4.9  Functions   wfun 5249
            2.4.10  Operations   co 5858
            2.4.11  "Maps to" notation   elmpt2cl 6061
            2.4.12  Function operation   cof 6076
            2.4.13  First and second members of an ordered pair   c1st 6120
            2.4.14  Function transposition   ctpos 6233
            2.4.15  Curry and uncurry   ccur 6272
            2.4.16  Proper subset relation   crpss 6276
            2.4.17  Iota properties   fvopab5 6289
            2.4.18  Cantor's Theorem   canth 6294
            2.4.19  Undefined values and restricted iota (description binder)   cund 6296
            2.4.20  Functions on ordinals; strictly monotone ordinal functions   iunon 6355
            2.4.21  "Strong" transfinite recursion   crecs 6387
            2.4.22  Recursive definition generator   crdg 6422
            2.4.23  Finite recursion   frfnom 6447
            2.4.24  Abian's "most fundamental" fixed point theorem   abianfplem 6470
            2.4.25  Ordinal arithmetic   c1o 6472
            2.4.26  Natural number arithmetic   nna0 6602
            2.4.27  Equivalence relations and classes   wer 6657
            2.4.28  The mapping operation   cmap 6772
            2.4.29  Infinite Cartesian products   cixp 6817
            2.4.30  Equinumerosity   cen 6860
            2.4.31  Schroeder-Bernstein Theorem   sbthlem1 6971
            2.4.32  Equinumerosity (cont.)   xpf1o 7023
            2.4.33  Pigeonhole Principle   phplem1 7040
            2.4.34  Finite sets   onomeneq 7050
            2.4.35  Finite intersections   cfi 7164
            2.4.36  Hall's marriage theorem   marypha1lem 7186
            2.4.37  Supremum   csup 7193
            2.4.38  Ordinal isomorphism, Hartog's theorem   coi 7224
            2.4.39  Hartogs function, order types, weak dominance   char 7270
      2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 7306
            2.5.2  Axiom of Infinity equivalents   inf0 7322
      2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 7339
            2.6.2  Existence of omega (the set of natural numbers)   omex 7344
            2.6.3  Cantor normal form   ccnf 7362
            2.6.4  Transitive closure   trcl 7410
            2.6.5  Rank   cr1 7434
            2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 7555
            2.6.7  Cardinal numbers   ccrd 7568
            2.6.8  Axiom of Choice equivalents   wac 7742
            2.6.9  Cardinal number arithmetic   ccda 7793
            2.6.10  The Ackermann bijection   ackbij2lem1 7845
            2.6.11  Cofinality (without Axiom of Choice)   cflem 7872
            2.6.12  Eight inequivalent definitions of finite set   sornom 7903
            2.6.13  Hereditarily size-limited sets without Choice   itunifval 8042
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
            3.2.1  Introduce the Axiom of Choice   ax-ac 8085
            3.2.2  AC equivalents: well ordering, Zorn's lemma   numthcor 8121
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 8168
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 8196
            3.2.5  Cofinality using Axiom of Choice   alephreg 8204
      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
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 8304
            4.1.2  Weak universes   cwun 8322
            4.1.3  Tarski's classes   ctsk 8370
            4.1.4  Grothendieck's universes   cgru 8412
      4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 8445
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 8448
            4.2.3  Tarski map function   ctskm 8459
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 8466
            5.1.2  Final derivation of real and complex number postulates   axaddf 8767
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 8793
      5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 8818
            5.2.2  Infinity and the extended real number system   cpnf 8864
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 8894
            5.2.4  Ordering on reals   lttr 8899
            5.2.5  Initial properties of the complex numbers   mul12 8978
      5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 9025
            5.3.2  Subtraction   cmin 9037
            5.3.3  Multiplication   muladd 9212
            5.3.4  Ordering on reals (cont.)   gt0ne0 9239
            5.3.5  Reciprocals   ixi 9397
            5.3.6  Division   cdiv 9423
            5.3.7  Ordering on reals (cont.)   elimgt0 9592
            5.3.8  Completeness Axiom and Suprema   fimaxre 9701
            5.3.9  Imaginary and complex number properties   inelr 9736
            5.3.10  Function operation analogue theorems   ofsubeq0 9743
      5.4  Integer sets
            5.4.1  Natural numbers (as a subset of complex numbers)   cn 9746
            5.4.2  Principle of mathematical induction   nnind 9764
            5.4.3  Decimal representation of numbers   c2 9795
            5.4.4  Some properties of specific numbers   0p1e1 9839
            5.4.5  The Archimedean property   nnunb 9961
            5.4.6  Nonnegative integers (as a subset of complex numbers)   cn0 9965
            5.4.7  Integers (as a subset of complex numbers)   cz 10024
            5.4.8  Decimal arithmetic   cdc 10124
            5.4.9  Upper partititions of integers   cuz 10230
            5.4.10  Well-ordering principle for bounded-below sets of integers   uzwo3 10311
            5.4.11  Rational numbers (as a subset of complex numbers)   cq 10316
            5.4.12  Existence of the set of complex numbers   rpnnen1lem1 10342
      5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 10354
            5.5.2  Infinity and the extended real number system (cont.)   cxne 10449
            5.5.3  Supremum on the extended reals   xrsupexmnf 10623
            5.5.4  Real number intervals   cioo 10656
            5.5.5  Finite intervals of integers   cfz 10782
            5.5.6  Half-open integer ranges   cfzo 10870
      5.6  Elementary integer functions
            5.6.1  The floor (greatest integer) function   cfl 10924
            5.6.2  The modulo (remainder) operation   cmo 10973
            5.6.3  The infinite sequence builder "seq"   om2uz0i 11010
            5.6.4  Integer powers   cexp 11104
            5.6.5  Ordered pair theorem for nonnegative integers   nn0le2msqi 11282
            5.6.6  Factorial function   cfa 11288
            5.6.7  The binomial coefficient operation   cbc 11315
            5.6.8  The ` # ` (finite set size) function   chash 11337
            5.6.9  Words over a set   cword 11403
            5.6.10  Longer string literals   cs2 11491
      5.7  Elementary real and complex functions
            5.7.1  The "shift" operation   cshi 11561
            5.7.2  Real and imaginary parts; conjugate   ccj 11581
            5.7.3  Square root; absolute value   csqr 11718
      5.8  Elementary limits and convergence
            5.8.1  Superior limit (lim sup)   clsp 11944
            5.8.2  Limits   cli 11958
            5.8.3  Finite and infinite sums   csu 12158
            5.8.4  The binomial theorem   binomlem 12287
            5.8.5  The inclusion/exclusion principle   incexclem 12295
            5.8.6  Infinite sums (cont.)   isumshft 12298
            5.8.7  Miscellaneous converging and diverging sequences   divrcnv 12311
            5.8.8  Arithmetic series   arisum 12318
            5.8.9  Geometric series   expcnv 12322
            5.8.10  Ratio test for infinite series convergence   cvgrat 12339
            5.8.11  Mertens' theorem   mertenslem1 12340
      5.9  Elementary trigonometry
            5.9.1  The exponential, sine, and cosine functions   ce 12343
            5.9.2  _e is irrational   eirrlem 12482
      5.10  Cardinality of real and complex number subsets
            5.10.1  Countability of integers and rationals   xpnnen 12487
            5.10.2  The reals are uncountable   rpnnen2lem1 12493
PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqr2irrlem 12526
            6.1.2  Some Number sets are chains of proper subsets   nthruc 12529
            6.1.3  The divides relation   cdivides 12531
            6.1.4  The division algorithm   divalglem0 12592
            6.1.5  Bit sequences   cbits 12610
            6.1.6  The greatest common divisor operator   cgcd 12685
            6.1.7  Bézout's identity   bezoutlem1 12717
            6.1.8  Algorithms   nn0seqcvgd 12740
            6.1.9  Euclid's Algorithm   eucalgval2 12751
      6.2  Elementary prime number theory
            6.2.1  Elementary properties   cprime 12758
            6.2.2  Properties of the canonical representation of a rational   cnumer 12804
            6.2.3  Euler's theorem   codz 12831
            6.2.4  Pythagorean Triples   coprimeprodsq 12862
            6.2.5  The prime count function   cpc 12889
            6.2.6  Pocklington's theorem   prmpwdvds 12951
            6.2.7  Infinite primes theorem   unbenlem 12955
            6.2.8  Sum of prime reciprocals   prmreclem1 12963
            6.2.9  Fundamental theorem of arithmetic   1arithlem1 12970
            6.2.10  Lagrange's four-square theorem   cgz 12976
            6.2.11  Van der Waerden's theorem   cvdwa 13012
            6.2.12  Ramsey's theorem   cram 13046
            6.2.13  Decimal arithmetic (cont.)   dec2dvds 13078
            6.2.14  Specific prime numbers   4nprm 13106
            6.2.15  Very large primes   1259lem1 13129
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
            7.1.1  Basic definitions   cstr 13144
            7.1.2  Slot definitions   cplusg 13208
            7.1.3  Definition of the structure product   crest 13325
            7.1.4  Definition of the structure quotient   cordt 13398
      7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 13508
            7.2.2  Independent sets in a Moore system   mrisval 13532
            7.2.3  Algebraic closure systems   isacs 13553
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
            8.1.1  Categories   ccat 13566
            8.1.2  Opposite category   coppc 13614
            8.1.3  Monomorphisms and epimorphisms   cmon 13631
            8.1.4  Sections, inverses, isomorphisms   csect 13647
            8.1.5  Subcategories   cssc 13684
            8.1.6  Functors   cfunc 13728
            8.1.7  Full & faithful functors   cful 13776
            8.1.8  Natural transformations and the functor category   cnat 13815
      8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 13885
      8.3  Examples of categories
            8.3.1  The category of sets   csetc 13907
            8.3.2  The category of categories   ccatc 13926
      8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 13942
            8.4.2  Functor evaluation   cevlf 13983
            8.4.3  Hom functor   chof 14022
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 14074
            9.2.2  Lattices   clat 14151
            9.2.3  The dual of an ordered set   codu 14232
            9.2.4  Subset order structures   cipo 14254
            9.2.5  Distributive lattices   latmass 14291
            9.2.6  Posets and lattices as relations   cps 14301
            9.2.7  Directed sets, nets   cdir 14350
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
            10.1.1  Definition and basic properties   cmnd 14361
            10.1.2  Monoid homomorphisms and submonoids   cmhm 14413
            10.1.3  Ordered group sum operation   gsumvallem1 14448
            10.1.4  Free monoids   cfrmd 14469
      10.2  Groups
            10.2.1  Definition and basic properties   df-grp 14489
            10.2.2  Subgroups and Quotient groups   csubg 14615
            10.2.3  Elementary theory of group homomorphisms   cghm 14680
            10.2.4  Isomorphisms of groups   cgim 14721
            10.2.5  Group actions   cga 14743
            10.2.6  Symmetry groups and Cayley's Theorem   csymg 14769
            10.2.7  Centralizers and centers   ccntz 14791
            10.2.8  The opposite group   coppg 14818
            10.2.9  p-Groups and Sylow groups; Sylow's theorems   cod 14840
            10.2.10  Direct products   clsm 14945
            10.2.11  Free groups   cefg 15015
      10.3  Abelian groups
            10.3.1  Definition and basic properties   ccmn 15089
            10.3.2  Cyclic groups   ccyg 15164
            10.3.3  Group sum operation   gsumval3a 15189
            10.3.4  Internal direct products   cdprd 15231
            10.3.5  The Fundamental Theorem of Abelian Groups   ablfacrplem 15300
      10.4  Rings
            10.4.1  Multiplicative Group   cmgp 15325
            10.4.2  Definition and basic properties   crg 15337
            10.4.3  Opposite ring   coppr 15404
            10.4.4  Divisibility   cdsr 15420
            10.4.5  Ring homomorphisms   crh 15494
      10.5  Division rings and fields
            10.5.1  Definition and basic properties   cdr 15512
            10.5.2  Subrings of a ring   csubrg 15541
            10.5.3  Absolute value (abstract algebra)   cabv 15581
            10.5.4  Star rings   cstf 15608
      10.6  Left modules
            10.6.1  Definition and basic properties   clmod 15627
            10.6.2  Subspaces and spans in a left module   clss 15689
            10.6.3  Homomorphisms and isomorphisms of left modules   clmhm 15776
            10.6.4  Subspace sum; bases for a left module   clbs 15827
      10.7  Vector spaces
            10.7.1  Definition and basic properties   clvec 15855
      10.8  Ideals
            10.8.1  The subring algebra; ideals   csra 15921
            10.8.2  Two-sided ideals and quotient rings   c2idl 15983
            10.8.3  Principal ideal rings. Divisibility in the integers   clpidl 15993
            10.8.4  Nonzero rings   cnzr 16009
            10.8.5  Left regular elements. More kinds of rings   crlreg 16020
      10.9  Associative algebras
            10.9.1  Definition and basic properties   casa 16050
      10.10  Abstract multivariate polynomials
            10.10.1  Definition and basic properties   cmps 16087
            10.10.2  Polynomial evaluation   evlslem4 16245
            10.10.3  Univariate polynomials   cps1 16250
      10.11  The complex numbers as an extensible structure
            10.11.1  Definition and basic properties   cxmt 16369
            10.11.2  Algebraic constructions based on the complexes   czrh 16451
      10.12  Hilbert spaces
            10.12.1  Definition and basic properties   cphl 16528
            10.12.2  Orthocomplements and closed subspaces   cocv 16560
            10.12.3  Orthogonal projection and orthonormal bases   cpj 16600
PART 11  BASIC TOPOLOGY
      11.1  Topology
            11.1.1  Topological spaces   ctop 16631
            11.1.2  TopBases for topologies   isbasisg 16685
            11.1.3  Examples of topologies   distop 16733
            11.1.4  Closure and interior   ccld 16753
            11.1.5  Neighborhoods   cnei 16834
            11.1.6  Limit points and perfect sets   clp 16866
            11.1.7  Subspace topologies   restrcl 16888
            11.1.8  Order topology   ordtbaslem 16918
            11.1.9  Limits and continuity in topological spaces   ccn 16954
            11.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 17034
            11.1.11  Compactness   ccmp 17113
            11.1.12  Connectedness   ccon 17137
            11.1.13  First- and second-countability   c1stc 17163
            11.1.14  Local topological properties   clly 17190
            11.1.15  Compactly generated spaces   ckgen 17228
            11.1.16  Product topologies   ctx 17255
            11.1.17  Continuous function-builders   cnmptid 17355
            11.1.18  Quotient maps and quotient topology   ckq 17384
            11.1.19  Homeomorphisms   chmeo 17444
      11.2  Filters and filter bases
            11.2.1  Filter bases   cfbas 17518
            11.2.2  Filters   cfil 17540
            11.2.3  Ultrafilters   cufil 17594
            11.2.4  Filter limits   cfm 17628
            11.2.5  Topological groups   ctmd 17753
            11.2.6  Infinite group sum on topological groups   ctsu 17808
            11.2.7  Topological rings, fields, vector spaces   ctrg 17838
      11.3  Metric spaces
            11.3.1  Basic metric space properties   cxme 17882
            11.3.2  Metric space balls   blfval 17947
            11.3.3  Open sets of a metric space   mopnval 17984
            11.3.4  Continuity in metric spaces   metcnp3 18086
            11.3.5  Examples of metric spaces   dscmet 18095
            11.3.6  Normed algebraic structures   cnm 18099
            11.3.7  Normed space homomorphisms (bounded linear operators)   cnmo 18214
            11.3.8  Topology on the reals   qtopbaslem 18267
            11.3.9  Topological definitions using the reals   cii 18379
            11.3.10  Path homotopy   chtpy 18465
            11.3.11  The fundamental group   cpco 18498
      11.4  Complex metric vector spaces
            11.4.1  Complex left modules   cclm 18560
            11.4.2  Complex pre-Hilbert space   ccph 18602
            11.4.3  Convergence and completeness   ccfil 18678
            11.4.4  Baire's Category Theorem   bcthlem1 18746
            11.4.5  Banach spaces and complex Hilbert spaces   ccms 18754
            11.4.6  Minimizing Vector Theorem   minveclem1 18788
            11.4.7  Projection Theorem   pjthlem1 18801
PART 12  BASIC REAL AND COMPLEX ANALYSIS
      12.1  Continuity
            12.1.1  Intermediate value theorem   pmltpclem1 18808
      12.2  Integrals
            12.2.1  Lebesgue measure   covol 18822
            12.2.2  Lebesgue integration   cmbf 18969
      12.3  Derivatives
            12.3.1  Real and complex differentiation   climc 19212
PART 13  BASIC REAL AND COMPLEX FUNCTIONS
      13.1  Polynomials
            13.1.1  Abstract polynomials, continued   evlslem6 19397
            13.1.2  Polynomial degrees   cmdg 19439
            13.1.3  The division algorithm for univariate polynomials   cmn1 19511
            13.1.4  Elementary properties of complex polynomials   cply 19566
            13.1.5  The division algorithm for polynomials   cquot 19670
            13.1.6  Algebraic numbers   caa 19694
            13.1.7  Liouville's approximation theorem   aalioulem1 19712
      13.2  Sequences and series
            13.2.1  Taylor polynomials and Taylor's theorem   ctayl 19732
            13.2.2  Uniform convergence   culm 19755
            13.2.3  Power series   pserval 19786
      13.3  Basic trigonometry
            13.3.1  The exponential, sine, and cosine functions (cont.)   efcn 19819
            13.3.2  Properties of pi = 3.14159...   pilem1 19827
            13.3.3  Mapping of the exponential function   efgh 19903
            13.3.4  The natural logarithm on complex numbers   clog 19912
            13.3.5  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 20099
            13.3.6  Solutions of quadratic, cubic, and quartic equations   quad2 20135
            13.3.7  Inverse trigonometric functions   casin 20158
            13.3.8  The Birthday Problem   log2ublem1 20242
            13.3.9  Areas in R^2   carea 20250
            13.3.10  More miscellaneous converging sequences   rlimcnp 20260
            13.3.11  Inequality of arithmetic and geometric means   cvxcl 20279
            13.3.12  Euler-Mascheroni constant   cem 20286
      13.4  Basic number theory
            13.4.1  Wilson's theorem   wilthlem1 20306
            13.4.2  The Fundamental Theorem of Algebra   ftalem1 20310
            13.4.3  The Basel problem (ζ(2) = π2/6)   basellem1 20318
            13.4.4  Number-theoretical functions   ccht 20328
            13.4.5  Perfect Number Theorem   mersenne 20466
            13.4.6  Characters of Z/nZ   cdchr 20471
            13.4.7  Bertrand's postulate   bcctr 20514
            13.4.8  Legendre symbol   clgs 20533
            13.4.9  Quadratic reciprocity   lgseisenlem1 20588
            13.4.10  All primes 4n+1 are the sum of two squares   2sqlem1 20602
            13.4.11  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 20618
            13.4.12  The Prime Number Theorem   mudivsum 20679
            13.4.13  Ostrowski's theorem   abvcxp 20764
PART 14  GUIDES AND MISCELLANEA
      14.1  Guides (conventions, explanations, and examples)
            14.1.1  Conventions   conventions 20789
            14.1.2  Natural deduction   natded 20790
            14.1.3  Natural deduction examples   ex-natded5.2 20791
            14.1.4  Definitional examples   ex-or 20808
      14.2  Humor
            14.2.1  April Fool's theorem   avril1 20836
      14.3  (Future - to be reviewed and classified)
            14.3.1  Planar incidence geometry   cplig 20842
            14.3.2  Algebra preliminaries   crpm 20847
            14.3.3  Transitive closure   ctcl 20849
PART 15  ADDITIONAL MATERIAL ON GROUPS, RINGS, AND FIELDS (DEPRECATED)
      15.1  Additional material on group theory
            15.1.1  Definitions and basic properties for groups   cgr 20853
            15.1.2  Definition and basic properties of Abelian groups   cablo 20948
            15.1.3  Subgroups   csubgo 20968
            15.1.4  Operation properties   cass 20979
            15.1.5  Group-like structures   cmagm 20985
            15.1.6  Examples of Abelian groups   ablosn 21014
            15.1.7  Group homomorphism and isomorphism   cghom 21024
      15.2  Additional material on rings and fields
            15.2.1  Definition and basic properties   crngo 21042
            15.2.2  Examples of rings   cnrngo 21070
            15.2.3  Division Rings   cdrng 21072
            15.2.4  Star Fields   csfld 21075
            15.2.5  Fields and Rings   ccm2 21077
PART 16  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      16.1  Complex vector spaces
            16.1.1  Definition and basic properties   cvc 21101
            16.1.2  Examples of complex vector spaces   cncvc 21139
      16.2  Normed complex vector spaces
            16.2.1  Definition and basic properties   cnv 21140
            16.2.2  Examples of normed complex vector spaces   cnnv 21245
            16.2.3  Induced metric of a normed complex vector space   imsval 21254
            16.2.4  Inner product   cdip 21273
            16.2.5  Subspaces   css 21297
      16.3  Operators on complex vector spaces
            16.3.1  Definitions and basic properties   clno 21318
      16.4  Inner product (pre-Hilbert) spaces
            16.4.1  Definition and basic properties   ccphlo 21390
            16.4.2  Examples of pre-Hilbert spaces   cncph 21397
            16.4.3  Properties of pre-Hilbert spaces   isph 21400
      16.5  Complex Banach spaces
            16.5.1  Definition and basic properties   ccbn 21441
            16.5.2  Examples of complex Banach spaces   cnbn 21448
            16.5.3  Uniform Boundedness Theorem   ubthlem1 21449
            16.5.4  Minimizing Vector Theorem   minvecolem1 21453
      16.6  Complex Hilbert spaces
            16.6.1  Definition and basic properties   chlo 21464
            16.6.2  Standard axioms for a complex Hilbert space   hlex 21477
            16.6.3  Examples of complex Hilbert spaces   cnchl 21495
            16.6.4  Subspaces   ssphl 21496
            16.6.5  Hellinger-Toeplitz Theorem   htthlem 21497
PART 17  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      17.1  Axiomatization of complex pre-Hilbert spaces
            17.1.1  Basic Hilbert space definitions   chil 21499
            17.1.2  Preliminary ZFC lemmas   df-hnorm 21548
            17.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 21561
            17.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 21579
            17.1.5  Vector operations   hvmulex 21591
            17.1.6  Inner product postulates for a Hilbert space   ax-hfi 21658
      17.2  Inner product and norms
            17.2.1  Inner product   his5 21665
            17.2.2  Norms   dfhnorm2 21701
            17.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 21739
            17.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 21758
      17.3  Cauchy sequences and completeness axiom
            17.3.1  Cauchy sequences and limits   hcau 21763
            17.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 21773
            17.3.3  Completeness postulate for a Hilbert space   ax-hcompl 21781
            17.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 21782
      17.4  Subspaces and projections
            17.4.1  Subspaces   df-sh 21786
            17.4.2  Closed subspaces   df-ch 21801
            17.4.3  Orthocomplements   df-oc 21831
            17.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 21887
            17.4.5  Projection theorem   pjhthlem1 21970
            17.4.6  Projectors   df-pjh 21974
      17.5  Properties of Hilbert subspaces
            17.5.1  Orthomodular law   omlsilem 21981
            17.5.2  Projectors (cont.)   pjhtheu2 21995
            17.5.3  Hilbert lattice operations   sh0le 22019
            17.5.4  Span (cont.) and one-dimensional subspaces   spansn0 22120
            17.5.5  Commutes relation for Hilbert lattice elements   df-cm 22162
            17.5.6  Foulis-Holland theorem   fh1 22197
            17.5.7  Quantum Logic Explorer axioms   qlax1i 22206
            17.5.8  Orthogonal subspaces   chscllem1 22216
            17.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 22233
            17.5.10  Projectors (cont.)   pjorthi 22248
            17.5.11  Mayet's equation E_3   mayete3i 22307
      17.6  Operators on Hilbert spaces
            17.6.1  Operator sum, difference, and scalar multiplication   df-hosum 22310
            17.6.2  Zero and identity operators   df-h0op 22328
            17.6.3  Operations on Hilbert space operators   hoaddcl 22338
            17.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 22419
            17.6.5  Linear and continuous functionals and norms   df-nmfn 22425
            17.6.6  Adjoint   df-adjh 22429
            17.6.7  Dirac bra-ket notation   df-bra 22430
            17.6.8  Positive operators   df-leop 22432
            17.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 22433
            17.6.10  Theorems about operators and functionals   nmopval 22436
            17.6.11  Riesz lemma   riesz3i 22642
            17.6.12  Adjoints (cont.)   cnlnadjlem1 22647
            17.6.13  Quantum computation error bound theorem   unierri 22684
            17.6.14  Dirac bra-ket notation (cont.)   branmfn 22685
            17.6.15  Positive operators (cont.)   leopg 22702
            17.6.16  Projectors as operators   pjhmopi 22726
      17.7  States on an Hilbert lattice and Godowski's equation
            17.7.1  States on a Hilbert lattice   df-st 22791
            17.7.2  Godowski's equation   golem1 22851
      17.8  Cover relation, atoms, exchange axiom, and modular symmetry
            17.8.1  Covers relation; modular pairs   df-cv 22859
            17.8.2  Atoms   df-at 22918
            17.8.3  Superposition principle   superpos 22934
            17.8.4  Atoms, exchange and covering properties, atomicity   chcv1 22935
            17.8.5  Irreducibility   chirredlem1 22970
            17.8.6  Atoms (cont.)   atcvat3i 22976
            17.8.7  Modular symmetry   mdsymlem1 22983
PART 18  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      18.1  Mathboxes for user contributions
            18.1.1  Mathbox guidelines   mathbox 23022
      18.2  Mathbox for Stefan Allan
      18.3  Mathbox for Thierry Arnoux
            18.3.1  Bertrand's Ballot Problem   ballotlemoex 23044
            18.3.2  Division in the extended real number system   cxdiv 23100
            18.3.3  Propositional Calculus - misc additions   bisimpd 23120
            18.3.4  Subclass relations - misc additions   ssrd 23125
            18.3.5  Restricted Quantification - misc additions   abeq2f 23129
            18.3.6  Substitution (without distinct variables) - misc additions   sbcss12g 23141
            18.3.7  Existential Uniqueness - misc additions   mo5f 23143
            18.3.8  Conditional operator - misc additions   ifbieq12d2 23149
            18.3.9  Indexed union - misc additions   iuneq12daf 23154
            18.3.10  Miscellaneous   ceqsexv2d 23162
            18.3.11  Functions and relations - misc additions   xpdisjres 23197
            18.3.12  First and second members of an ordered pair - misc additions   df1stres 23243
            18.3.13  Supremum - misc additions   supssd 23248
            18.3.14  Ordering on reals - misc additions   lt2addrd 23249
            18.3.15  Extended reals - misc additions   xrlelttric 23250
            18.3.16  Real number intervals - misc additions   icossicc 23258
            18.3.17  Finite intervals of integers - misc additions   fzssnn 23276
            18.3.18  Half-open integer ranges - misc additions   fzossnn 23278
            18.3.19  Closed unit   unitsscn 23280
            18.3.20  Topology of ` ( RR X. RR ) `   tpr2tp 23287
            18.3.21  Order topology - misc. additions   cnvordtrestixx 23297
            18.3.22  Continuity in topological spaces - misc. additions   ressplusf 23298
            18.3.23  Extended reals Structure - misc additions   xaddeq0 23304
            18.3.24  The extended non-negative real numbers monoid   xrge0base 23310
            18.3.25  Countable Sets   nnct 23335
            18.3.26  Disjointness - misc additions   cbvdisjf 23350
            18.3.27  Limits - misc additions   lmlim 23371
            18.3.28  Finitely supported group sums - misc additions   gsumsn2 23378
            18.3.29  Logarithm laws generalized to an arbitrary base - logb   clogb 23390
            18.3.30  Extended sum   cesum 23410
            18.3.31  Mixed Function/Constant operation   cofc 23456
            18.3.32  Sigma-Algebra   csiga 23468
            18.3.33  Generated Sigma-Algebra   csigagen 23499
            18.3.34  The Borel Algebra on real numbers   cbrsiga 23512
            18.3.35  Product Sigma-Algebra   csx 23519
            18.3.36  Measures   cmeas 23526
            18.3.37  Measurable functions   cmbfm 23555
            18.3.38  Borel Algebra on ` ( RR X. RR ) `   br2base 23574
            18.3.39  Integration with respect to a Measure   cibfm 23583
            18.3.40  Indicator Functions   cind 23594
            18.3.41  Probability Theory   cprb 23610
            18.3.42  Conditional Probabilities   ccprob 23634
            18.3.43  Real Valued Random Variables   crrv 23643
            18.3.44  Preimage set mapping operator   corvc 23656
            18.3.45  Distribution Functions   orvcelval 23669
            18.3.46  Cumulative Distribution Functions   orvclteel 23673
            18.3.47  Probabilities - example   coinfliplem 23679
      18.4  Mathbox for Mario Carneiro
            18.4.1  Miscellaneous stuff   quartfull 23686
            18.4.2  Zeta function   czeta 23687
            18.4.3  Gamma function   clgam 23690
            18.4.4  Derangements and the Subfactorial   deranglem 23697
            18.4.5  The Erdős-Szekeres theorem   erdszelem1 23722
            18.4.6  The Kuratowski closure-complement theorem   kur14lem1 23737
            18.4.7  Retracts and sections   cretr 23748
            18.4.8  Path-connected and simply connected spaces   cpcon 23750
            18.4.9  Covering maps   ccvm 23786
            18.4.10  Undirected multigraphs   cumg 23860
            18.4.11  Normal numbers   snmlff 23912
            18.4.12  Godel-sets of formulas   cgoe 23916
            18.4.13  Models of ZF   cgze 23944
            18.4.14  Splitting fields   citr 23958
            18.4.15  p-adic number fields   czr 23974
      18.5  Mathbox for Paul Chapman
            18.5.1  Group homomorphism and isomorphism   ghomgrpilem1 23992
            18.5.2  Real and complex numbers (cont.)   climuzcnv 24004
            18.5.3  Miscellaneous theorems   elfzm12 24008
      18.6  Mathbox for Drahflow
      18.7  Mathbox for Scott Fenton
            18.7.1  ZFC Axioms in primitive form   axextprim 24047
            18.7.2  Untangled classes   untelirr 24054
            18.7.3  Extra propositional calculus theorems   3orel1 24061
            18.7.4  Misc. Useful Theorems   nepss 24072
            18.7.5  Properties of reals and complexes   sqdivzi 24079
            18.7.6  Greatest common divisor and divisibility   pdivsq 24102
            18.7.7  Properties of relationships   brtp 24106
            18.7.8  Properties of functions and mappings   funpsstri 24121
            18.7.9  Epsilon induction   setinds 24134
            18.7.10  Ordinal numbers   elpotr 24137
            18.7.11  Defined equality axioms   axextdfeq 24154
            18.7.12  Hypothesis builders   hbntg 24162
            18.7.13  The Predecessor Class   cpred 24167
            18.7.14  (Trans)finite Recursion Theorems   tfisg 24204
            18.7.15  Well-founded induction   tz6.26 24205
            18.7.16  Transitive closure under a relationship   ctrpred 24220
            18.7.17  Founded Induction   frmin 24242
            18.7.18  Ordering Ordinal Sequences   orderseqlem 24252
            18.7.19  Well-founded recursion   wfr3g 24255
            18.7.20  Transfinite recursion via Well-founded recursion   tfrALTlem 24276
            18.7.21  Founded Recursion   frr3g 24280
            18.7.22  Surreal Numbers   csur 24294
            18.7.23  Surreal Numbers: Ordering   sltsolem1 24322
            18.7.24  Surreal Numbers: Birthday Function   bdayfo 24329
            18.7.25  Surreal Numbers: Density   fvnobday 24336
            18.7.26  Surreal Numbers: Density   nodenselem3 24337
            18.7.27  Surreal Numbers: Upper and Lower Bounds   nobndlem1 24346
            18.7.28  Surreal Numbers: Full-Eta Property   nofulllem1 24356
            18.7.29  Symmetric difference   csymdif 24361
            18.7.30  Quantifier-free definitions   ctxp 24373
            18.7.31  Alternate ordered pairs   caltop 24490
            18.7.32  Tarskian geometry   cee 24516
            18.7.33  Tarski's axioms for geometry   axdimuniq 24541
            18.7.34  Congruence properties   cofs 24605
            18.7.35  Betweenness properties   btwntriv2 24635
            18.7.36  Segment Transportation   ctransport 24652
            18.7.37  Properties relating betweenness and congruence   cifs 24658
            18.7.38  Connectivity of betweenness   btwnconn1lem1 24710
            18.7.39  Segment less than or equal to   csegle 24729
            18.7.40  Outside of relationship   coutsideof 24742
            18.7.41  Lines and Rays   cline2 24757
            18.7.42  Bernoulli polynomials and sums of k-th powers   cbp 24781
            18.7.43  Rank theorems   rankung 24796
            18.7.44  Hereditarily Finite Sets   chf 24802
      18.8  Mathbox for Anthony Hart
            18.8.1  Propositional Calculus   tb-ax1 24817
            18.8.2  Predicate Calculus   quantriv 24839
            18.8.3  Misc. Single Axiom Systems   meran1 24850
            18.8.4  Connective Symmetry   negsym1 24856
      18.9  Mathbox for Chen-Pang He
            18.9.1  Ordinal topology   ontopbas 24867
      18.10  Mathbox for Jeff Hoffman
            18.10.1  Inferences for finite induction on generic function values   fveleq 24890
            18.10.2  gdc.mm   nnssi2 24894
      18.11  Mathbox for Wolf Lammen
      18.12  Mathbox for Brendan Leahy
      18.13  Mathbox for Frédéric Liné
            18.13.1  Theorems from other workspaces   tpssg 24932
            18.13.2  Propositional and predicate calculus   neleq12d 24933
            18.13.3  Linear temporal logic   wbox 24970
            18.13.4  Operations   ssoprab2g 25032
            18.13.5  General Set Theory   uninqs 25039
            18.13.6  The "maps to" notation   cmpfunOLD 25142
            18.13.7  Cartesian Products   cpro 25150
            18.13.8  Operations on subsets and functions   ccst 25172
            18.13.9  Arithmetic   3timesi 25178
            18.13.10  Lattice (algebraic definition)   clatalg 25181
            18.13.11  Currying and Partial Mappings   ccur1 25194
            18.13.12  Order theory (Extensible Structure Builder)   corhom 25207
            18.13.13  Order theory   cpresetrel 25215
            18.13.14  Finite composites ( i. e. finite sums, products ... )   cprd 25298
            18.13.15  Operation properties   ccm1 25331
            18.13.16  Groups and related structures   ridlideq 25335
            18.13.17  Free structures   csubsmg 25383
            18.13.18  Translations   trdom2 25391
            18.13.19  Fields and Rings   com2i 25416
            18.13.20  Ideals   cidln 25443
            18.13.21  Generic modules and vector spaces (New Structure builder)   cact 25447
            18.13.22  Generic modules and vector spaces   cvec 25449
            18.13.23  Real vector spaces   cvr 25489
            18.13.24  Matrices   cmmat 25493
            18.13.25  Affine spaces   craffsp 25499
            18.13.26  Intervals of reals and extended reals   bsi 25501
            18.13.27  Topology   topnem 25512
            18.13.28  Continuous functions   cnrsfin 25525
            18.13.29  Homeomorphisms   dmhmph 25533
            18.13.30  Initial and final topologies   intopcoaconlem3b 25538
            18.13.31  Filters   efilcp 25552
            18.13.32  Limits   plimfil 25558
            18.13.33  Uniform spaces   cunifsp 25585
            18.13.34  Separated spaces: T0, T1, T2 (Hausdorff) ...   hst1 25587
            18.13.35  Compactness   indcomp 25589
            18.13.36  Connectedness   singempcon 25593
            18.13.37  Topological fields   ctopfld 25597
            18.13.38  Standard topology on RR   intrn 25599
            18.13.39  Standard topology of intervals of RR   stoi 25601
            18.13.40  Cantor's set   cntrset 25602
            18.13.41  Pre-calculus and Cartesian geometry   dmse1 25603
            18.13.42  Extended Real numbers   nolimf 25619
            18.13.43  ( RR ^ N ) and ( CC ^ N )   cplcv 25644
            18.13.44  Calculus   cintvl 25696
            18.13.45  Directed multi graphs   cmgra 25708
            18.13.46  Category and deductive system underlying "structure"   calg 25711
            18.13.47  Deductive systems   cded 25734
            18.13.48  Categories   ccatOLD 25752
            18.13.49  Homsets   chomOLD 25785
            18.13.50  Monomorphisms, Epimorphisms, Isomorphisms   cepiOLD 25803
            18.13.51  Functors   cfuncOLD 25831
            18.13.52  Subcategories   csubcat 25843
            18.13.53  Terminal and initial objects   ciobj 25860
            18.13.54  Sources and sinks   csrce 25865
            18.13.55  Limits and co-limits   clmct 25874
            18.13.56  Product and sum of two objects   cprodo 25877
            18.13.57  Tarski's classes   ctar 25881
            18.13.58  Category Set   ccmrcase 25910
            18.13.59  Grammars, Logics, Machines and Automata   ckln 25980
            18.13.60  Words   cwrd 25981
            18.13.61  Planar geometry   cpoints 26056
      18.14  Mathbox for Jeff Hankins
            18.14.1  Miscellany   a1i13 26200
            18.14.2  Basic topological facts   topbnd 26242
            18.14.3  Topology of the real numbers   reconnOLD 26255
            18.14.4  Refinements   cfne 26259
            18.14.5  Neighborhood bases determine topologies   neibastop1 26308
            18.14.6  Lattice structure of topologies   topmtcl 26312
            18.14.7  Filter bases   fgmin 26319
            18.14.8  Directed sets, nets   tailfval 26321
      18.15  Mathbox for Jeff Madsen
            18.15.1  Logic and set theory   anim12da 26332
            18.15.2  Real and complex numbers; integers   fimaxreOLD 26430
            18.15.3  Sequences and sums   sdclem2 26452
            18.15.4  Topology   unopnOLD 26464
            18.15.5  Metric spaces   metf1o 26469
            18.15.6  Continuous maps and homeomorphisms   constcncf 26478
            18.15.7  Product topologies   txtopiOLD 26486
            18.15.8  Boundedness   ctotbnd 26490
            18.15.9  Isometries   cismty 26522
            18.15.10  Heine-Borel Theorem   heibor1lem 26533
            18.15.11  Banach Fixed Point Theorem   bfplem1 26546
            18.15.12  Euclidean space   crrn 26549
            18.15.13  Intervals (continued)   ismrer1 26562
            18.15.14  Groups and related structures   exidcl 26566
            18.15.15  Rings   rngonegcl 26576
            18.15.16  Ring homomorphisms   crnghom 26591
            18.15.17  Commutative rings   ccring 26620
            18.15.18  Ideals   cidl 26632
            18.15.19  Prime rings and integral domains   cprrng 26671
            18.15.20  Ideal generators   cigen 26684
      18.16  Mathbox for Rodolfo Medina
            18.16.1  Partitions   prtlem60 26703
      18.17  Mathbox for Stefan O'Rear
            18.17.1  Additional elementary logic and set theory   nelss 26751
            18.17.2  Additional theory of functions   fninfp 26754
            18.17.3  Extensions beyond function theory   gsumvsmul 26764
            18.17.4  Additional topology   elrfi 26769
            18.17.5  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 26773
            18.17.6  Algebraic closure systems   cnacs 26777
            18.17.7  Miscellanea 1. Map utilities   constmap 26788
            18.17.8  Miscellanea for polynomials   ofmpteq 26797
            18.17.9  Multivariate polynomials over the integers   cmzpcl 26799
            18.17.10  Miscellanea for Diophantine sets 1   coeq0 26831
            18.17.11  Diophantine sets 1: definitions   cdioph 26834
            18.17.12  Diophantine sets 2 miscellanea   ellz1 26846
            18.17.13  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 26852
            18.17.14  Diophantine sets 3: construction   diophrex 26855
            18.17.15  Diophantine sets 4 miscellanea   2sbcrex 26864
            18.17.16  Diophantine sets 4: Quantification   rexrabdioph 26875
            18.17.17  Diophantine sets 5: Arithmetic sets   rabdiophlem1 26882
            18.17.18  Diophantine sets 6 miscellanea   fz1ssnn 26892
            18.17.19  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 26894
            18.17.20  Pigeonhole Principle and cardinality helpers   fphpd 26899
            18.17.21  A non-closed set of reals is infinite   rencldnfilem 26903
            18.17.22  Miscellanea for Lagrange's theorem   icodiamlt 26905
            18.17.23  Lagrange's rational approximation theorem   irrapxlem1 26907
            18.17.24  Pell equations 1: A nontrivial solution always exists   pellexlem1 26914
            18.17.25  Pell equations 2: Algebraic number theory of the solution set   csquarenn 26921
            18.17.26  Pell equations 3: characterizing fundamental solution   infmrgelbi 26963
            18.17.27  Logarithm laws generalized to an arbitrary base   reglogcl 26975
            18.17.28  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 26983
            18.17.29  X and Y sequences 1: Definition and recurrence laws   crmx 26985
            18.17.30  Ordering and induction lemmas for the integers   monotuz 27026
            18.17.31  X and Y sequences 2: Order properties   rmxypos 27034
            18.17.32  Congruential equations   congtr 27052
            18.17.33  Alternating congruential equations   acongid 27062
            18.17.34  Additional theorems on integer divisibility   bezoutr 27072
            18.17.35  X and Y sequences 3: Divisibility properties   jm2.18 27081
            18.17.36  X and Y sequences 4: Diophantine representability of Y   jm2.27a 27098
            18.17.37  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 27108
            18.17.38  Uncategorized stuff not associated with a major project   setindtr 27117
            18.17.39  More equivalents of the Axiom of Choice   axac10 27126
            18.17.40  Finitely generated left modules   clfig 27165
            18.17.41  Noetherian left modules I   clnm 27173
            18.17.42  Addenda for structure powers   pwssplit0 27187
            18.17.43  Direct sum of left modules   cdsmm 27197
            18.17.44  Free modules   cfrlm 27212
            18.17.45  Every set admits a group structure iff choice   unxpwdom3 27256
            18.17.46  Independent sets and families   clindf 27274
            18.17.47  Characterization of free modules   lmimlbs 27306
            18.17.48  Noetherian rings and left modules II   clnr 27313
            18.17.49  Hilbert's Basis Theorem   cldgis 27325
            18.17.50  Additional material on polynomials [DEPRECATED]   cmnc 27335
            18.17.51  Degree and minimal polynomial of algebraic numbers   cdgraa 27345
            18.17.52  Algebraic integers I   citgo 27362
            18.17.53  Finite cardinality [SO]   en1uniel 27380
            18.17.54  Words in monoids and ordered group sum   issubmd 27383
            18.17.55  Transpositions in the symmetric group   cpmtr 27384
            18.17.56  The sign of a permutation   cpsgn 27414
            18.17.57  The matrix algebra   cmmul 27439
            18.17.58  The determinant   cmdat 27483
            18.17.59  Endomorphism algebra   cmend 27489
            18.17.60  Subfields   csdrg 27503
            18.17.61  Cyclic groups and order   idomrootle 27511
            18.17.62  Cyclotomic polynomials   ccytp 27521
            18.17.63  Miscellaneous topology   fgraphopab 27529
      18.18  Mathbox for Steve Rodriguez
            18.18.1  Miscellanea   iso0 27536
            18.18.2  Function operations   caofcan 27540
            18.18.3  Calculus   lhe4.4ex1a 27546
      18.19  Mathbox for Andrew Salmon
            18.19.1  Principia Mathematica * 10   pm10.12 27553
            18.19.2  Principia Mathematica * 11   2alanimi 27567
            18.19.3  Predicate Calculus   sbeqal1 27597
            18.19.4  Principia Mathematica * 13 and * 14   pm13.13a 27607
            18.19.5  Set Theory   elnev 27638
            18.19.6  Arithmetic   addcomgi 27661
            18.19.7  Geometry   cplusr 27662
      18.20  Mathbox for Glauco Siliprandi
            18.20.1  Miscellanea   ssrexf 27684
            18.20.2  Finite multiplication of numbers and finite multiplication of functions   fmul01 27710
            18.20.3  Limits   clim1fr1 27727
            18.20.4  Derivatives   dvsinexp 27740
            18.20.5  Integrals   ioovolcl 27742
            18.20.6  Stone Weierstrass theorem - real version   stoweidlem1 27750
            18.20.7  Wallis' product for π   wallispilem1 27814
            18.20.8  Stirling's approximation formula for ` n ` factorial   stirlinglem1 27823
      18.21  Mathbox for Saveliy Skresanov
            18.21.1  Ceva's theorem   sigarval 27840
      18.22  Mathbox for Jarvin Udandy
      18.23  Mathbox for Alexander van der Vekens
            18.23.1  Double restricted existential uniqueness   r19.32 27945
                  18.23.1.1  Restricted quantification (extension)   r19.32 27945
                  18.23.1.2  The empty set (extension)   raaan2 27953
                  18.23.1.3  Restricted uniqueness and "at most one" quantification   rmoimi 27954
                  18.23.1.4  Analogs to Existential uniqueness (double quantification)   2reurex 27959
            18.23.2  Alternative definitions of function's and operation's values   wdfat 27971
                  18.23.2.1  Restricted quantification (extension)   ralbinrald 27977
                  18.23.2.2  The universal class (extension)   nvelim 27978
                  18.23.2.3  Relations (extension)   sbcrel 27979
                  18.23.2.4  Functions (extension)   sbcfun 27985
                  18.23.2.5  Predicate "defined at"   dfateq12d 27992
                  18.23.2.6  Alternative definition of the value of a function   dfafv2 27995
                  18.23.2.7  Alternative definition of the value of an operation   aoveq123d 28038
            18.23.3  Graph theory   difprsneq 28068
                  18.23.3.1  Unordered and ordered pairs (extension)   difprsneq 28068
                  18.23.3.2  Functions (extension)   f1oprg 28075
                  18.23.3.3  Operations (Extension)   nssdmovg 28077
                  18.23.3.4  "Maps to" notation (Extension)   mpt2xopn0yelv 28079
                  18.23.3.5  The ` # ` (finite set size) function (extension)   elprchashprn2 28088
                  18.23.3.6  Longer string literals (extension)   s2prop 28089
                  18.23.3.7  Undirected simple graphs   cuslg 28094
                  18.23.3.8  Undirected simple graphs (examples)   usgra1v 28126
                  18.23.3.9  Neighbors, complete graphs and universal vertices   cnbgra 28134
                  18.23.3.10  Friendship graphs   cfrgra 28169
      18.24  Mathbox for David A. Wheeler
            18.24.1  Natural deduction   19.8ad 28187
            18.24.2  Greater than, greater than or equal to.   cge-real 28190
            18.24.3  Hyperbolic trig functions   csinh 28200
            18.24.4  Reciprocal trig functions (sec, csc, cot)   csec 28211
            18.24.5  Identities for "if"   ifnmfalse 28233
            18.24.6  Not-member-of   AnelBC 28234
            18.24.7  Decimal point   cdp2 28235
            18.24.8  Signum (sgn or sign) function   csgn 28243
            18.24.9  Ceiling function   ccei 28253
            18.24.10  Logarithm laws generalized to an arbitrary base - log_   clog_ 28257
            18.24.11  Miscellaneous   5m4e1 28259
      18.25  Mathbox for Alan Sare
            18.25.1  Conventional Metamath proofs, some derived from VD proofs   iidn3 28262
            18.25.2  What is Virtual Deduction?   wvd1 28337
            18.25.3  Virtual Deduction Theorems   df-vd1 28338
            18.25.4  Theorems proved using virtual deduction   trsspwALT 28592
            18.25.5  Theorems proved using virtual deduction with mmj2 assistance   simplbi2VD 28622
            18.25.6  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 28689
            18.25.7  Theorems proved using conjunction-form virtual deduction   elpwgdedVD 28693
            18.25.8  Theorems with VD proofs in conventional notation derived from VD proofs   suctrALT3 28700
            18.25.9  Theorems with a proof in conventional notation automatically derived   notnot2ALT2 28703
      18.26  Mathbox for Jonathan Ben-Naim
            18.26.1  First order logic and set theory   bnj170 28723
            18.26.2  Well founded induction and recursion   bnj110 28890
            18.26.3  The existence of a minimal element in certain classes   bnj69 29040
            18.26.4  Well-founded induction   bnj1204 29042
            18.26.5  Well-founded recursion, part 1 of 3   bnj60 29092
            18.26.6  Well-founded recursion, part 2 of 3   bnj1500 29098
            18.26.7  Well-founded recursion, part 3 of 3   bnj1522 29102
      18.27  Mathbox for Norm Megill
            18.27.1  Obsolete experiments to study ax-12o   ax12-2 29103
            18.27.2  Miscellanea   cnaddcom 29161
            18.27.3  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 29164
            18.27.4  Functionals and kernels of a left vector space (or module)   clfn 29247
            18.27.5  Opposite rings and dual vector spaces   cld 29313
            18.27.6  Ortholattices and orthomodular lattices   cops 29362
            18.27.7  Atomic lattices with covering property   ccvr 29452
            18.27.8  Hilbert lattices   chlt 29540
            18.27.9  Projective geometries based on Hilbert lattices   clln 29680
            18.27.10  Construction of a vector space from a Hilbert lattice   cdlema1N 29980
            18.27.11  Construction of involution and inner product from a Hilbert lattice   clpoN 31670

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