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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 a 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 1530
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1536
            1.4.3  Axiom scheme ax-5 (Quantified Implication)   ax-5 1547
            1.4.4  Axiom scheme ax-17 (Distinctness) - first use of $d   ax-17 1606
            1.4.5  Equality predicate; define substitution   cv 1631
            1.4.6  Axiom scheme ax-9 (Existence)   ax-9 1644
            1.4.7  Axiom scheme ax-8 (Equality)   ax-8 1661
            1.4.8  Membership predicate   wcel 1696
            1.4.9  Axiom schemes ax-13 (Left Membership Equality)   ax-13 1698
            1.4.10  Axiom schemes ax-14 (Right Membership Equality)   ax-14 1700
            1.4.11  Logical redundancy of ax-6 , ax-7 , ax-11 , ax-12   ax9dgen 1702
      1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-6 (Quantified Negation)   ax-6 1715
            1.5.2  Axiom scheme ax-7 (Quantifier Commutation)   ax-7 1720
            1.5.3  Axiom scheme ax-11 (Substitution)   ax-11 1727
            1.5.4  Axiom scheme ax-12 (Quantified Equality)   ax-12 1878
      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 2087
            1.6.2  Rederive new axioms from old: ax5 , ax6 , ax9from9o , ax11 , ax12from12o   ax4 2097
            1.6.3  Legacy theorems using obsolete axioms   ax17o 2109
      1.7  Existential uniqueness
      1.8  Other axiomatizations related to classical predicate calculus
            1.8.1  Predicate calculus with all distinct variables   ax-7d 2247
            1.8.2  Aristotelian logic: Assertic syllogisms   barbara 2253
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 2277
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2282
            2.1.3  Class form not-free predicate   wnfc 2419
            2.1.4  Negated equality and membership   wne 2459
            2.1.5  Restricted quantification   wral 2556
            2.1.6  The universal class   cvv 2801
            2.1.7  Conditional equality (experimental)   wcdeq 2987
            2.1.8  Russell's Paradox   ru 3003
            2.1.9  Proper substitution of classes for sets   wsbc 3004
            2.1.10  Proper substitution of classes for sets into classes   csb 3094
            2.1.11  Define basic set operations and relations   cdif 3162
            2.1.12  Subclasses and subsets   df-ss 3179
            2.1.13  The difference, union, and intersection of two classes   difeq1 3300
            2.1.14  The empty set   c0 3468
            2.1.15  "Weak deduction theorem" for set theory   cif 3578
            2.1.16  Power classes   cpw 3638
            2.1.17  Unordered and ordered pairs   csn 3653
            2.1.18  The union of a class   cuni 3843
            2.1.19  The intersection of a class   cint 3878
            2.1.20  Indexed union and intersection   ciun 3921
            2.1.21  Disjointness   wdisj 4009
            2.1.22  Binary relations   wbr 4039
            2.1.23  Ordered-pair class abstractions (class builders)   copab 4092
            2.1.24  Transitive classes   wtr 4129
      2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 4147
            2.2.2  Derive the Axiom of Separation   axsep 4156
            2.2.3  Derive the Null Set Axiom   zfnuleu 4162
            2.2.4  Theorems requiring subset and intersection existence   nalset 4167
            2.2.5  Theorems requiring empty set existence   class2set 4194
      2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4204
            2.3.2  Derive the Axiom of Pairing   zfpair 4228
            2.3.3  Ordered pair theorem   opnz 4258
            2.3.4  Ordered-pair class abstractions (cont.)   opabid 4287
            2.3.5  Power class of union and intersection   pwin 4313
            2.3.6  Epsilon and identity relations   cep 4319
            2.3.7  Partial and complete ordering   wpo 4328
            2.3.8  Founded and well-ordering relations   wfr 4365
            2.3.9  Ordinals   word 4407
      2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4528
            2.4.2  Ordinals (continued)   ordon 4590
            2.4.3  Transfinite induction   tfi 4660
            2.4.4  The natural numbers (i.e. finite ordinals)   com 4672
            2.4.5  Peano's postulates   peano1 4691
            2.4.6  Finite induction (for finite ordinals)   find 4697
            2.4.7  Relations   cxp 4703
            2.4.8  Definite description binder (inverted iota)   cio 5233
            2.4.9  Functions   wfun 5265
            2.4.10  Operations   co 5874
            2.4.11  "Maps to" notation   elmpt2cl 6077
            2.4.12  Function operation   cof 6092
            2.4.13  First and second members of an ordered pair   c1st 6136
            2.4.14  Function transposition   ctpos 6249
            2.4.15  Curry and uncurry   ccur 6288
            2.4.16  Proper subset relation   crpss 6292
            2.4.17  Iota properties   fvopab5 6305
            2.4.18  Cantor's Theorem   canth 6310
            2.4.19  Undefined values and restricted iota (description binder)   cund 6312
            2.4.20  Functions on ordinals; strictly monotone ordinal functions   iunon 6371
            2.4.21  "Strong" transfinite recursion   crecs 6403
            2.4.22  Recursive definition generator   crdg 6438
            2.4.23  Finite recursion   frfnom 6463
            2.4.24  Abian's "most fundamental" fixed point theorem   abianfplem 6486
            2.4.25  Ordinal arithmetic   c1o 6488
            2.4.26  Natural number arithmetic   nna0 6618
            2.4.27  Equivalence relations and classes   wer 6673
            2.4.28  The mapping operation   cmap 6788
            2.4.29  Infinite Cartesian products   cixp 6833
            2.4.30  Equinumerosity   cen 6876
            2.4.31  Schroeder-Bernstein Theorem   sbthlem1 6987
            2.4.32  Equinumerosity (cont.)   xpf1o 7039
            2.4.33  Pigeonhole Principle   phplem1 7056
            2.4.34  Finite sets   onomeneq 7066
            2.4.35  Finite intersections   cfi 7180
            2.4.36  Hall's marriage theorem   marypha1lem 7202
            2.4.37  Supremum   csup 7209
            2.4.38  Ordinal isomorphism, Hartog's theorem   coi 7240
            2.4.39  Hartogs function, order types, weak dominance   char 7286
      2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 7322
            2.5.2  Axiom of Infinity equivalents   inf0 7338
      2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 7355
            2.6.2  Existence of omega (the set of natural numbers)   omex 7360
            2.6.3  Cantor normal form   ccnf 7378
            2.6.4  Transitive closure   trcl 7426
            2.6.5  Rank   cr1 7450
            2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 7571
            2.6.7  Cardinal numbers   ccrd 7584
            2.6.8  Axiom of Choice equivalents   wac 7758
            2.6.9  Cardinal number arithmetic   ccda 7809
            2.6.10  The Ackermann bijection   ackbij2lem1 7861
            2.6.11  Cofinality (without Axiom of Choice)   cflem 7888
            2.6.12  Eight inequivalent definitions of finite set   sornom 7919
            2.6.13  Hereditarily size-limited sets without Choice   itunifval 8058
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 8101
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 8137
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 8184
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 8212
            3.2.5  Cofinality using Axiom of Choice   alephreg 8220
      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 8320
            4.1.2  Weak universes   cwun 8338
            4.1.3  Tarski's classes   ctsk 8386
            4.1.4  Grothendieck's universes   cgru 8428
      4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 8461
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 8464
            4.2.3  Tarski map function   ctskm 8475
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 8482
            5.1.2  Final derivation of real and complex number postulates   axaddf 8783
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 8809
      5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 8834
            5.2.2  Infinity and the extended real number system   cpnf 8880
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 8910
            5.2.4  Ordering on reals   lttr 8915
            5.2.5  Initial properties of the complex numbers   mul12 8994
      5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 9041
            5.3.2  Subtraction   cmin 9053
            5.3.3  Multiplication   muladd 9228
            5.3.4  Ordering on reals (cont.)   gt0ne0 9255
            5.3.5  Reciprocals   ixi 9413
            5.3.6  Division   cdiv 9439
            5.3.7  Ordering on reals (cont.)   elimgt0 9608
            5.3.8  Completeness Axiom and Suprema   fimaxre 9717
            5.3.9  Imaginary and complex number properties   inelr 9752
            5.3.10  Function operation analogue theorems   ofsubeq0 9759
      5.4  Integer sets
            5.4.1  Natural numbers (as a subset of complex numbers)   cn 9762
            5.4.2  Principle of mathematical induction   nnind 9780
            5.4.3  Decimal representation of numbers   c2 9811
            5.4.4  Some properties of specific numbers   0p1e1 9855
            5.4.5  The Archimedean property   nnunb 9977
            5.4.6  Nonnegative integers (as a subset of complex numbers)   cn0 9981
            5.4.7  Integers (as a subset of complex numbers)   cz 10040
            5.4.8  Decimal arithmetic   cdc 10140
            5.4.9  Upper partititions of integers   cuz 10246
            5.4.10  Well-ordering principle for bounded-below sets of integers   uzwo3 10327
            5.4.11  Rational numbers (as a subset of complex numbers)   cq 10332
            5.4.12  Existence of the set of complex numbers   rpnnen1lem1 10358
      5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 10370
            5.5.2  Infinity and the extended real number system (cont.)   cxne 10465
            5.5.3  Supremum on the extended reals   xrsupexmnf 10639
            5.5.4  Real number intervals   cioo 10672
            5.5.5  Finite intervals of integers   cfz 10798
            5.5.6  Half-open integer ranges   cfzo 10886
      5.6  Elementary integer functions
            5.6.1  The floor (greatest integer) function   cfl 10940
            5.6.2  The modulo (remainder) operation   cmo 10989
            5.6.3  The infinite sequence builder "seq"   om2uz0i 11026
            5.6.4  Integer powers   cexp 11120
            5.6.5  Ordered pair theorem for nonnegative integers   nn0le2msqi 11298
            5.6.6  Factorial function   cfa 11304
            5.6.7  The binomial coefficient operation   cbc 11331
            5.6.8  The ` # ` (finite set size) function   chash 11353
            5.6.9  Words over a set   cword 11419
            5.6.10  Longer string literals   cs2 11507
      5.7  Elementary real and complex functions
            5.7.1  The "shift" operation   cshi 11577
            5.7.2  Real and imaginary parts; conjugate   ccj 11597
            5.7.3  Square root; absolute value   csqr 11734
      5.8  Elementary limits and convergence
            5.8.1  Superior limit (lim sup)   clsp 11960
            5.8.2  Limits   cli 11974
            5.8.3  Finite and infinite sums   csu 12174
            5.8.4  The binomial theorem   binomlem 12303
            5.8.5  The inclusion/exclusion principle   incexclem 12311
            5.8.6  Infinite sums (cont.)   isumshft 12314
            5.8.7  Miscellaneous converging and diverging sequences   divrcnv 12327
            5.8.8  Arithmetic series   arisum 12334
            5.8.9  Geometric series   expcnv 12338
            5.8.10  Ratio test for infinite series convergence   cvgrat 12355
            5.8.11  Mertens' theorem   mertenslem1 12356
      5.9  Elementary trigonometry
            5.9.1  The exponential, sine, and cosine functions   ce 12359
            5.9.2  _e is irrational   eirrlem 12498
      5.10  Cardinality of real and complex number subsets
            5.10.1  Countability of integers and rationals   xpnnen 12503
            5.10.2  The reals are uncountable   rpnnen2lem1 12509
PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqr2irrlem 12542
            6.1.2  Some Number sets are chains of proper subsets   nthruc 12545
            6.1.3  The divides relation   cdivides 12547
            6.1.4  The division algorithm   divalglem0 12608
            6.1.5  Bit sequences   cbits 12626
            6.1.6  The greatest common divisor operator   cgcd 12701
            6.1.7  Bézout's identity   bezoutlem1 12733
            6.1.8  Algorithms   nn0seqcvgd 12756
            6.1.9  Euclid's Algorithm   eucalgval2 12767
      6.2  Elementary prime number theory
            6.2.1  Elementary properties   cprime 12774
            6.2.2  Properties of the canonical representation of a rational   cnumer 12820
            6.2.3  Euler's theorem   codz 12847
            6.2.4  Pythagorean Triples   coprimeprodsq 12878
            6.2.5  The prime count function   cpc 12905
            6.2.6  Pocklington's theorem   prmpwdvds 12967
            6.2.7  Infinite primes theorem   unbenlem 12971
            6.2.8  Sum of prime reciprocals   prmreclem1 12979
            6.2.9  Fundamental theorem of arithmetic   1arithlem1 12986
            6.2.10  Lagrange's four-square theorem   cgz 12992
            6.2.11  Van der Waerden's theorem   cvdwa 13028
            6.2.12  Ramsey's theorem   cram 13062
            6.2.13  Decimal arithmetic (cont.)   dec2dvds 13094
            6.2.14  Specific prime numbers   4nprm 13122
            6.2.15  Very large primes   1259lem1 13145
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
            7.1.1  Basic definitions   cstr 13160
            7.1.2  Slot definitions   cplusg 13224
            7.1.3  Definition of the structure product   crest 13341
            7.1.4  Definition of the structure quotient   cordt 13414
      7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 13524
            7.2.2  Independent sets in a Moore system   mrisval 13548
            7.2.3  Algebraic closure systems   isacs 13569
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
            8.1.1  Categories   ccat 13582
            8.1.2  Opposite category   coppc 13630
            8.1.3  Monomorphisms and epimorphisms   cmon 13647
            8.1.4  Sections, inverses, isomorphisms   csect 13663
            8.1.5  Subcategories   cssc 13700
            8.1.6  Functors   cfunc 13744
            8.1.7  Full & faithful functors   cful 13792
            8.1.8  Natural transformations and the functor category   cnat 13831
      8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 13901
      8.3  Examples of categories
            8.3.1  The category of sets   csetc 13923
            8.3.2  The category of categories   ccatc 13942
      8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 13958
            8.4.2  Functor evaluation   cevlf 13999
            8.4.3  Hom functor   chof 14038
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 14090
            9.2.2  Lattices   clat 14167
            9.2.3  The dual of an ordered set   codu 14248
            9.2.4  Subset order structures   cipo 14270
            9.2.5  Distributive lattices   latmass 14307
            9.2.6  Posets and lattices as relations   cps 14317
            9.2.7  Directed sets, nets   cdir 14366
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
            10.1.1  Definition and basic properties   cmnd 14377
            10.1.2  Monoid homomorphisms and submonoids   cmhm 14429
            10.1.3  Ordered group sum operation   gsumvallem1 14464
            10.1.4  Free monoids   cfrmd 14485
      10.2  Groups
            10.2.1  Definition and basic properties   df-grp 14505
            10.2.2  Subgroups and Quotient groups   csubg 14631
            10.2.3  Elementary theory of group homomorphisms   cghm 14696
            10.2.4  Isomorphisms of groups   cgim 14737
            10.2.5  Group actions   cga 14759
            10.2.6  Symmetry groups and Cayley's Theorem   csymg 14785
            10.2.7  Centralizers and centers   ccntz 14807
            10.2.8  The opposite group   coppg 14834
            10.2.9  p-Groups and Sylow groups; Sylow's theorems   cod 14856
            10.2.10  Direct products   clsm 14961
            10.2.11  Free groups   cefg 15031
      10.3  Abelian groups
            10.3.1  Definition and basic properties   ccmn 15105
            10.3.2  Cyclic groups   ccyg 15180
            10.3.3  Group sum operation   gsumval3a 15205
            10.3.4  Internal direct products   cdprd 15247
            10.3.5  The Fundamental Theorem of Abelian Groups   ablfacrplem 15316
      10.4  Rings
            10.4.1  Multiplicative Group   cmgp 15341
            10.4.2  Definition and basic properties   crg 15353
            10.4.3  Opposite ring   coppr 15420
            10.4.4  Divisibility   cdsr 15436
            10.4.5  Ring homomorphisms   crh 15510
      10.5  Division rings and fields
            10.5.1  Definition and basic properties   cdr 15528
            10.5.2  Subrings of a ring   csubrg 15557
            10.5.3  Absolute value (abstract algebra)   cabv 15597
            10.5.4  Star rings   cstf 15624
      10.6  Left modules
            10.6.1  Definition and basic properties   clmod 15643
            10.6.2  Subspaces and spans in a left module   clss 15705
            10.6.3  Homomorphisms and isomorphisms of left modules   clmhm 15792
            10.6.4  Subspace sum; bases for a left module   clbs 15843
      10.7  Vector spaces
            10.7.1  Definition and basic properties   clvec 15871
      10.8  Ideals
            10.8.1  The subring algebra; ideals   csra 15937
            10.8.2  Two-sided ideals and quotient rings   c2idl 15999
            10.8.3  Principal ideal rings. Divisibility in the integers   clpidl 16009
            10.8.4  Nonzero rings   cnzr 16025
            10.8.5  Left regular elements. More kinds of rings   crlreg 16036
      10.9  Associative algebras
            10.9.1  Definition and basic properties   casa 16066
      10.10  Abstract multivariate polynomials
            10.10.1  Definition and basic properties   cmps 16103
            10.10.2  Polynomial evaluation   evlslem4 16261
            10.10.3  Univariate polynomials   cps1 16266
      10.11  The complex numbers as an extensible structure
            10.11.1  Definition and basic properties   cxmt 16385
            10.11.2  Algebraic constructions based on the complexes   czrh 16467
      10.12  Hilbert spaces
            10.12.1  Definition and basic properties   cphl 16544
            10.12.2  Orthocomplements and closed subspaces   cocv 16576
            10.12.3  Orthogonal projection and orthonormal bases   cpj 16616
PART 11  BASIC TOPOLOGY
      11.1  Topology
            11.1.1  Topological spaces   ctop 16647
            11.1.2  TopBases for topologies   isbasisg 16701
            11.1.3  Examples of topologies   distop 16749
            11.1.4  Closure and interior   ccld 16769
            11.1.5  Neighborhoods   cnei 16850
            11.1.6  Limit points and perfect sets   clp 16882
            11.1.7  Subspace topologies   restrcl 16904
            11.1.8  Order topology   ordtbaslem 16934
            11.1.9  Limits and continuity in topological spaces   ccn 16970
            11.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 17050
            11.1.11  Compactness   ccmp 17129
            11.1.12  Connectedness   ccon 17153
            11.1.13  First- and second-countability   c1stc 17179
            11.1.14  Local topological properties   clly 17206
            11.1.15  Compactly generated spaces   ckgen 17244
            11.1.16  Product topologies   ctx 17271
            11.1.17  Continuous function-builders   cnmptid 17371
            11.1.18  Quotient maps and quotient topology   ckq 17400
            11.1.19  Homeomorphisms   chmeo 17460
      11.2  Filters and filter bases
            11.2.1  Filter bases   cfbas 17534
            11.2.2  Filters   cfil 17556
            11.2.3  Ultrafilters   cufil 17610
            11.2.4  Filter limits   cfm 17644
            11.2.5  Topological groups   ctmd 17769
            11.2.6  Infinite group sum on topological groups   ctsu 17824
            11.2.7  Topological rings, fields, vector spaces   ctrg 17854
      11.3  Metric spaces
            11.3.1  Basic metric space properties   cxme 17898
            11.3.2  Metric space balls   blfval 17963
            11.3.3  Open sets of a metric space   mopnval 18000
            11.3.4  Continuity in metric spaces   metcnp3 18102
            11.3.5  Examples of metric spaces   dscmet 18111
            11.3.6  Normed algebraic structures   cnm 18115
            11.3.7  Normed space homomorphisms (bounded linear operators)   cnmo 18230
            11.3.8  Topology on the reals   qtopbaslem 18283
            11.3.9  Topological definitions using the reals   cii 18395
            11.3.10  Path homotopy   chtpy 18481
            11.3.11  The fundamental group   cpco 18514
      11.4  Complex metric vector spaces
            11.4.1  Complex left modules   cclm 18576
            11.4.2  Complex pre-Hilbert space   ccph 18618
            11.4.3  Convergence and completeness   ccfil 18694
            11.4.4  Baire's Category Theorem   bcthlem1 18762
            11.4.5  Banach spaces and complex Hilbert spaces   ccms 18770
            11.4.6  Minimizing Vector Theorem   minveclem1 18804
            11.4.7  Projection Theorem   pjthlem1 18817
PART 12  BASIC REAL AND COMPLEX ANALYSIS
      12.1  Continuity
            12.1.1  Intermediate value theorem   pmltpclem1 18824
      12.2  Integrals
            12.2.1  Lebesgue measure   covol 18838
            12.2.2  Lebesgue integration   cmbf 18985
      12.3  Derivatives
            12.3.1  Real and complex differentiation   climc 19228
PART 13  BASIC REAL AND COMPLEX FUNCTIONS
      13.1  Polynomials
            13.1.1  Abstract polynomials, continued   evlslem6 19413
            13.1.2  Polynomial degrees   cmdg 19455
            13.1.3  The division algorithm for univariate polynomials   cmn1 19527
            13.1.4  Elementary properties of complex polynomials   cply 19582
            13.1.5  The division algorithm for polynomials   cquot 19686
            13.1.6  Algebraic numbers   caa 19710
            13.1.7  Liouville's approximation theorem   aalioulem1 19728
      13.2  Sequences and series
            13.2.1  Taylor polynomials and Taylor's theorem   ctayl 19748
            13.2.2  Uniform convergence   culm 19771
            13.2.3  Power series   pserval 19802
      13.3  Basic trigonometry
            13.3.1  The exponential, sine, and cosine functions (cont.)   efcn 19835
            13.3.2  Properties of pi = 3.14159...   pilem1 19843
            13.3.3  Mapping of the exponential function   efgh 19919
            13.3.4  The natural logarithm on complex numbers   clog 19928
            13.3.5  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 20115
            13.3.6  Solutions of quadratic, cubic, and quartic equations   quad2 20151
            13.3.7  Inverse trigonometric functions   casin 20174
            13.3.8  The Birthday Problem   log2ublem1 20258
            13.3.9  Areas in R^2   carea 20266
            13.3.10  More miscellaneous converging sequences   rlimcnp 20276
            13.3.11  Inequality of arithmetic and geometric means   cvxcl 20295
            13.3.12  Euler-Mascheroni constant   cem 20302
      13.4  Basic number theory
            13.4.1  Wilson's theorem   wilthlem1 20322
            13.4.2  The Fundamental Theorem of Algebra   ftalem1 20326
            13.4.3  The Basel problem (ζ(2) = π2/6)   basellem1 20334
            13.4.4  Number-theoretical functions   ccht 20344
            13.4.5  Perfect Number Theorem   mersenne 20482
            13.4.6  Characters of Z/nZ   cdchr 20487
            13.4.7  Bertrand's postulate   bcctr 20530
            13.4.8  Legendre symbol   clgs 20549
            13.4.9  Quadratic reciprocity   lgseisenlem1 20604
            13.4.10  All primes 4n+1 are the sum of two squares   2sqlem1 20618
            13.4.11  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 20634
            13.4.12  The Prime Number Theorem   mudivsum 20695
            13.4.13  Ostrowski's theorem   abvcxp 20780
PART 14  GUIDES AND MISCELLANEA
      14.1  Guides (conventions, explanations, and examples)
            14.1.1  Conventions   conventions 20805
            14.1.2  Natural deduction   natded 20806
            14.1.3  Natural deduction examples   ex-natded5.2 20807
            14.1.4  Definitional examples   ex-or 20824
      14.2  Humor
            14.2.1  April Fool's theorem   avril1 20852
      14.3  (Future - to be reviewed and classified)
            14.3.1  Planar incidence geometry   cplig 20858
            14.3.2  Algebra preliminaries   crpm 20863
            14.3.3  Transitive closure   ctcl 20865
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 20869
            15.1.2  Definition and basic properties of Abelian groups   cablo 20964
            15.1.3  Subgroups   csubgo 20984
            15.1.4  Operation properties   cass 20995
            15.1.5  Group-like structures   cmagm 21001
            15.1.6  Examples of Abelian groups   ablosn 21030
            15.1.7  Group homomorphism and isomorphism   cghom 21040
      15.2  Additional material on rings and fields
            15.2.1  Definition and basic properties   crngo 21058
            15.2.2  Examples of rings   cnrngo 21086
            15.2.3  Division Rings   cdrng 21088
            15.2.4  Star Fields   csfld 21091
            15.2.5  Fields and Rings   ccm2 21093
PART 16  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      16.1  Complex vector spaces
            16.1.1  Definition and basic properties   cvc 21117
            16.1.2  Examples of complex vector spaces   cncvc 21155
      16.2  Normed complex vector spaces
            16.2.1  Definition and basic properties   cnv 21156
            16.2.2  Examples of normed complex vector spaces   cnnv 21261
            16.2.3  Induced metric of a normed complex vector space   imsval 21270
            16.2.4  Inner product   cdip 21289
            16.2.5  Subspaces   css 21313
      16.3  Operators on complex vector spaces
            16.3.1  Definitions and basic properties   clno 21334
      16.4  Inner product (pre-Hilbert) spaces
            16.4.1  Definition and basic properties   ccphlo 21406
            16.4.2  Examples of pre-Hilbert spaces   cncph 21413
            16.4.3  Properties of pre-Hilbert spaces   isph 21416
      16.5  Complex Banach spaces
            16.5.1  Definition and basic properties   ccbn 21457
            16.5.2  Examples of complex Banach spaces   cnbn 21464
            16.5.3  Uniform Boundedness Theorem   ubthlem1 21465
            16.5.4  Minimizing Vector Theorem   minvecolem1 21469
      16.6  Complex Hilbert spaces
            16.6.1  Definition and basic properties   chlo 21480
            16.6.2  Standard axioms for a complex Hilbert space   hlex 21493
            16.6.3  Examples of complex Hilbert spaces   cnchl 21511
            16.6.4  Subspaces   ssphl 21512
            16.6.5  Hellinger-Toeplitz Theorem   htthlem 21513
PART 17  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      17.1  Axiomatization of complex pre-Hilbert spaces
            17.1.1  Basic Hilbert space definitions   chil 21515
            17.1.2  Preliminary ZFC lemmas   df-hnorm 21564
            17.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 21577
            17.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 21595
            17.1.5  Vector operations   hvmulex 21607
            17.1.6  Inner product postulates for a Hilbert space   ax-hfi 21674
      17.2  Inner product and norms
            17.2.1  Inner product   his5 21681
            17.2.2  Norms   dfhnorm2 21717
            17.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 21755
            17.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 21774
      17.3  Cauchy sequences and completeness axiom
            17.3.1  Cauchy sequences and limits   hcau 21779
            17.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 21789
            17.3.3  Completeness postulate for a Hilbert space   ax-hcompl 21797
            17.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 21798
      17.4  Subspaces and projections
            17.4.1  Subspaces   df-sh 21802
            17.4.2  Closed subspaces   df-ch 21817
            17.4.3  Orthocomplements   df-oc 21847
            17.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 21903
            17.4.5  Projection theorem   pjhthlem1 21986
            17.4.6  Projectors   df-pjh 21990
      17.5  Properties of Hilbert subspaces
            17.5.1  Orthomodular law   omlsilem 21997
            17.5.2  Projectors (cont.)   pjhtheu2 22011
            17.5.3  Hilbert lattice operations   sh0le 22035
            17.5.4  Span (cont.) and one-dimensional subspaces   spansn0 22136
            17.5.5  Commutes relation for Hilbert lattice elements   df-cm 22178
            17.5.6  Foulis-Holland theorem   fh1 22213
            17.5.7  Quantum Logic Explorer axioms   qlax1i 22222
            17.5.8  Orthogonal subspaces   chscllem1 22232
            17.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 22249
            17.5.10  Projectors (cont.)   pjorthi 22264
            17.5.11  Mayet's equation E_3   mayete3i 22323
      17.6  Operators on Hilbert spaces
            17.6.1  Operator sum, difference, and scalar multiplication   df-hosum 22326
            17.6.2  Zero and identity operators   df-h0op 22344
            17.6.3  Operations on Hilbert space operators   hoaddcl 22354
            17.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 22435
            17.6.5  Linear and continuous functionals and norms   df-nmfn 22441
            17.6.6  Adjoint   df-adjh 22445
            17.6.7  Dirac bra-ket notation   df-bra 22446
            17.6.8  Positive operators   df-leop 22448
            17.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 22449
            17.6.10  Theorems about operators and functionals   nmopval 22452
            17.6.11  Riesz lemma   riesz3i 22658
            17.6.12  Adjoints (cont.)   cnlnadjlem1 22663
            17.6.13  Quantum computation error bound theorem   unierri 22700
            17.6.14  Dirac bra-ket notation (cont.)   branmfn 22701
            17.6.15  Positive operators (cont.)   leopg 22718
            17.6.16  Projectors as operators   pjhmopi 22742
      17.7  States on a Hilbert lattice and Godowski's equation
            17.7.1  States on a Hilbert lattice   df-st 22807
            17.7.2  Godowski's equation   golem1 22867
      17.8  Cover relation, atoms, exchange axiom, and modular symmetry
            17.8.1  Covers relation; modular pairs   df-cv 22875
            17.8.2  Atoms   df-at 22934
            17.8.3  Superposition principle   superpos 22950
            17.8.4  Atoms, exchange and covering properties, atomicity   chcv1 22951
            17.8.5  Irreducibility   chirredlem1 22986
            17.8.6  Atoms (cont.)   atcvat3i 22992
            17.8.7  Modular symmetry   mdsymlem1 22999
PART 18  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      18.1  Mathboxes for user contributions
            18.1.1  Mathbox guidelines   mathbox 23038
      18.2  Mathbox for Stefan Allan
      18.3  Mathbox for Thierry Arnoux
            18.3.1  Bertrand's Ballot Problem   ballotlemoex 23060
            18.3.2  Division in the extended real number system   cxdiv 23116
            18.3.3  Propositional Calculus - misc additions   bisimpd 23136
            18.3.4  Subclass relations - misc additions   ssrd 23141
            18.3.5  Restricted Quantification - misc additions   abeq2f 23145
            18.3.6  Substitution (without distinct variables) - misc additions   sbcss12g 23157
            18.3.7  Existential Uniqueness - misc additions   mo5f 23159
            18.3.8  Conditional operator - misc additions   ifbieq12d2 23165
            18.3.9  Indexed union - misc additions   iuneq12daf 23170
            18.3.10  Miscellaneous   ceqsexv2d 23178
            18.3.11  Functions and relations - misc additions   xpdisjres 23212
            18.3.12  First and second members of an ordered pair - misc additions   df1stres 23258
            18.3.13  Supremum - misc additions   supssd 23263
            18.3.14  Ordering on reals - misc additions   lt2addrd 23264
            18.3.15  Extended reals - misc additions   xrlelttric 23265
            18.3.16  Real number intervals - misc additions   icossicc 23273
            18.3.17  Finite intervals of integers - misc additions   fzssnn 23291
            18.3.18  Half-open integer ranges - misc additions   fzossnn 23293
            18.3.19  Closed unit   unitsscn 23295
            18.3.20  Topology of ` ( RR X. RR ) `   tpr2tp 23302
            18.3.21  Order topology - misc. additions   cnvordtrestixx 23312
            18.3.22  Continuity in topological spaces - misc. additions   ressplusf 23313
            18.3.23  Extended reals Structure - misc additions   xaddeq0 23319
            18.3.24  The extended non-negative real numbers monoid   xrge0base 23325
            18.3.25  Countable Sets   nnct 23350
            18.3.26  Disjointness - misc additions   cbvdisjf 23365
            18.3.27  Limits - misc additions   lmlim 23386
            18.3.28  Finitely supported group sums - misc additions   gsumsn2 23393
            18.3.29  Logarithm laws generalized to an arbitrary base - logb   clogb 23405
            18.3.30  Extended sum   cesum 23425
            18.3.31  Mixed Function/Constant operation   cofc 23471
            18.3.32  Sigma-Algebra   csiga 23483
            18.3.33  Generated Sigma-Algebra   csigagen 23514
            18.3.34  The Borel Algebra on real numbers   cbrsiga 23527
            18.3.35  Product Sigma-Algebra   csx 23534
            18.3.36  Measures   cmeas 23541
            18.3.37  Measurable functions   cmbfm 23570
            18.3.38  Borel Algebra on ` ( RR X. RR ) `   br2base 23589
            18.3.39  Integration with respect to a Measure   cibfm 23598
            18.3.40  Indicator Functions   cind 23609
            18.3.41  Probability Theory   cprb 23625
            18.3.42  Conditional Probabilities   ccprob 23649
            18.3.43  Real Valued Random Variables   crrv 23658
            18.3.44  Preimage set mapping operator   corvc 23671
            18.3.45  Distribution Functions   orvcelval 23684
            18.3.46  Cumulative Distribution Functions   orvclteel 23688
            18.3.47  Probabilities - example   coinfliplem 23694
      18.4  Mathbox for Mario Carneiro
            18.4.1  Miscellaneous stuff   quartfull 23701
            18.4.2  Zeta function   czeta 23702
            18.4.3  Gamma function   clgam 23705
            18.4.4  Derangements and the Subfactorial   deranglem 23712
            18.4.5  The Erdős-Szekeres theorem   erdszelem1 23737
            18.4.6  The Kuratowski closure-complement theorem   kur14lem1 23752
            18.4.7  Retracts and sections   cretr 23763
            18.4.8  Path-connected and simply connected spaces   cpcon 23765
            18.4.9  Covering maps   ccvm 23801
            18.4.10  Undirected multigraphs   cumg 23875
            18.4.11  Normal numbers   snmlff 23927
            18.4.12  Godel-sets of formulas   cgoe 23931
            18.4.13  Models of ZF   cgze 23959
            18.4.14  Splitting fields   citr 23973
            18.4.15  p-adic number fields   czr 23989
      18.5  Mathbox for Paul Chapman
            18.5.1  Group homomorphism and isomorphism   ghomgrpilem1 24007
            18.5.2  Real and complex numbers (cont.)   climuzcnv 24019
            18.5.3  Miscellaneous theorems   elfzm12 24023
      18.6  Mathbox for Drahflow
      18.7  Mathbox for Scott Fenton
            18.7.1  ZFC Axioms in primitive form   axextprim 24062
            18.7.2  Untangled classes   untelirr 24069
            18.7.3  Extra propositional calculus theorems   3orel1 24076
            18.7.4  Misc. Useful Theorems   nepss 24087
            18.7.5  Properties of reals and complexes   sqdivzi 24094
            18.7.6  Complex products   cprod 24127
            18.7.7  Greatest common divisor and divisibility   pdivsq 24172
            18.7.8  Properties of relationships   brtp 24176
            18.7.9  Properties of functions and mappings   funpsstri 24191
            18.7.10  Epsilon induction   setinds 24204
            18.7.11  Ordinal numbers   elpotr 24207
            18.7.12  Defined equality axioms   axextdfeq 24224
            18.7.13  Hypothesis builders   hbntg 24232
            18.7.14  The Predecessor Class   cpred 24237
            18.7.15  (Trans)finite Recursion Theorems   tfisg 24274
            18.7.16  Well-founded induction   tz6.26 24275
            18.7.17  Transitive closure under a relationship   ctrpred 24290
            18.7.18  Founded Induction   frmin 24312
            18.7.19  Ordering Ordinal Sequences   orderseqlem 24322
            18.7.20  Well-founded recursion   wfr3g 24325
            18.7.21  Transfinite recursion via Well-founded recursion   tfrALTlem 24346
            18.7.22  Founded Recursion   frr3g 24350
            18.7.23  Surreal Numbers   csur 24364
            18.7.24  Surreal Numbers: Ordering   sltsolem1 24392
            18.7.25  Surreal Numbers: Birthday Function   bdayfo 24399
            18.7.26  Surreal Numbers: Density   fvnobday 24406
            18.7.27  Surreal Numbers: Density   nodenselem3 24407
            18.7.28  Surreal Numbers: Upper and Lower Bounds   nobndlem1 24416
            18.7.29  Surreal Numbers: Full-Eta Property   nofulllem1 24426
            18.7.30  Symmetric difference   csymdif 24431
            18.7.31  Quantifier-free definitions   ctxp 24443
            18.7.32  Alternate ordered pairs   caltop 24561
            18.7.33  Tarskian geometry   cee 24587
            18.7.34  Tarski's axioms for geometry   axdimuniq 24612
            18.7.35  Congruence properties   cofs 24676
            18.7.36  Betweenness properties   btwntriv2 24706
            18.7.37  Segment Transportation   ctransport 24723
            18.7.38  Properties relating betweenness and congruence   cifs 24729
            18.7.39  Connectivity of betweenness   btwnconn1lem1 24781
            18.7.40  Segment less than or equal to   csegle 24800
            18.7.41  Outside of relationship   coutsideof 24813
            18.7.42  Lines and Rays   cline2 24828
            18.7.43  Bernoulli polynomials and sums of k-th powers   cbp 24852
            18.7.44  Rank theorems   rankung 24867
            18.7.45  Hereditarily Finite Sets   chf 24873
      18.8  Mathbox for Anthony Hart
            18.8.1  Propositional Calculus   tb-ax1 24888
            18.8.2  Predicate Calculus   quantriv 24910
            18.8.3  Misc. Single Axiom Systems   meran1 24921
            18.8.4  Connective Symmetry   negsym1 24927
      18.9  Mathbox for Chen-Pang He
            18.9.1  Ordinal topology   ontopbas 24938
      18.10  Mathbox for Jeff Hoffman
            18.10.1  Inferences for finite induction on generic function values   fveleq 24961
            18.10.2  gdc.mm   nnssi2 24965
      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 25034
            18.13.2  Propositional and predicate calculus   neleq12d 25035
            18.13.3  Linear temporal logic   wbox 25072
            18.13.4  Operations   ssoprab2g 25134
            18.13.5  General Set Theory   uninqs 25141
            18.13.6  The "maps to" notation   cmpfunOLD 25244
            18.13.7  Cartesian Products   cpro 25252
            18.13.8  Operations on subsets and functions   ccst 25274
            18.13.9  Arithmetic   3timesi 25280
            18.13.10  Lattice (algebraic definition)   clatalg 25283
            18.13.11  Currying and Partial Mappings   ccur1 25296
            18.13.12  Order theory (Extensible Structure Builder)   corhom 25309
            18.13.13  Order theory   cpresetrel 25317
            18.13.14  Finite composites ( i.e. finite sums, products ... )   cgprd 25400
            18.13.15  Operation properties   ccm1 25433
            18.13.16  Groups and related structures   ridlideq 25437
            18.13.17  Free structures   csubsmg 25485
            18.13.18  Translations   trdom2 25493
            18.13.19  Fields and Rings   com2i 25518
            18.13.20  Ideals   cidln 25545
            18.13.21  Generic modules and vector spaces (New Structure builder)   cact 25549
            18.13.22  Generic modules and vector spaces   cvec 25551
            18.13.23  Real vector spaces   cvr 25591
            18.13.24  Matrices   cmmat 25595
            18.13.25  Affine spaces   craffsp 25601
            18.13.26  Intervals of reals and extended reals   bsi 25603
            18.13.27  Topology   topnem 25614
            18.13.28  Continuous functions   cnrsfin 25627
            18.13.29  Homeomorphisms   dmhmph 25635
            18.13.30  Initial and final topologies   intopcoaconlem3b 25640
            18.13.31  Filters   efilcp 25654
            18.13.32  Limits   plimfil 25660
            18.13.33  Uniform spaces   cunifsp 25687
            18.13.34  Separated spaces: T0, T1, T2 (Hausdorff) ...   hst1 25689
            18.13.35  Compactness   indcomp 25691
            18.13.36  Connectedness   singempcon 25695
            18.13.37  Topological fields   ctopfld 25699
            18.13.38  Standard topology on RR   intrn 25701
            18.13.39  Standard topology of intervals of RR   stoi 25703
            18.13.40  Cantor's set   cntrset 25704
            18.13.41  Pre-calculus and Cartesian geometry   dmse1 25705
            18.13.42  Extended Real numbers   nolimf 25721
            18.13.43  ( RR ^ N ) and ( CC ^ N )   cplcv 25746
            18.13.44  Calculus   cintvl 25798
            18.13.45  Directed multi graphs   cmgra 25810
            18.13.46  Category and deductive system underlying "structure"   calg 25813
            18.13.47  Deductive systems   cded 25836
            18.13.48  Categories   ccatOLD 25854
            18.13.49  Homsets   chomOLD 25887
            18.13.50  Monomorphisms, Epimorphisms, Isomorphisms   cepiOLD 25905
            18.13.51  Functors   cfuncOLD 25933
            18.13.52  Subcategories   csubcat 25945
            18.13.53  Terminal and initial objects   ciobj 25962
            18.13.54  Sources and sinks   csrce 25967
            18.13.55  Limits and co-limits   clmct 25976
            18.13.56  Product and sum of two objects   cprodo 25979
            18.13.57  Tarski's classes   ctar 25983
            18.13.58  Category Set   ccmrcase 26012
            18.13.59  Grammars, Logics, Machines and Automata   ckln 26082
            18.13.60  Words   cwrd 26083
            18.13.61  Planar geometry   cpoints 26158
      18.14  Mathbox for Jeff Hankins
            18.14.1  Miscellany   a1i13 26302
            18.14.2  Basic topological facts   topbnd 26344
            18.14.3  Topology of the real numbers   reconnOLD 26357
            18.14.4  Refinements   cfne 26361
            18.14.5  Neighborhood bases determine topologies   neibastop1 26410
            18.14.6  Lattice structure of topologies   topmtcl 26414
            18.14.7  Filter bases   fgmin 26421
            18.14.8  Directed sets, nets   tailfval 26423
      18.15  Mathbox for Jeff Madsen
            18.15.1  Logic and set theory   anim12da 26434
            18.15.2  Real and complex numbers; integers   fimaxreOLD 26532
            18.15.3  Sequences and sums   sdclem2 26554
            18.15.4  Topology   unopnOLD 26566
            18.15.5  Metric spaces   metf1o 26571
            18.15.6  Continuous maps and homeomorphisms   constcncf 26580
            18.15.7  Product topologies   txtopiOLD 26588
            18.15.8  Boundedness   ctotbnd 26592
            18.15.9  Isometries   cismty 26624
            18.15.10  Heine-Borel Theorem   heibor1lem 26635
            18.15.11  Banach Fixed Point Theorem   bfplem1 26648
            18.15.12  Euclidean space   crrn 26651
            18.15.13  Intervals (continued)   ismrer1 26664
            18.15.14  Groups and related structures   exidcl 26668
            18.15.15  Rings   rngonegcl 26678
            18.15.16  Ring homomorphisms   crnghom 26693
            18.15.17  Commutative rings   ccring 26722
            18.15.18  Ideals   cidl 26734
            18.15.19  Prime rings and integral domains   cprrng 26773
            18.15.20  Ideal generators   cigen 26786
      18.16  Mathbox for Rodolfo Medina
            18.16.1  Partitions   prtlem60 26805
      18.17  Mathbox for Stefan O'Rear
            18.17.1  Additional elementary logic and set theory   nelss 26853
            18.17.2  Additional theory of functions   fninfp 26856
            18.17.3  Extensions beyond function theory   gsumvsmul 26866
            18.17.4  Additional topology   elrfi 26871
            18.17.5  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 26875
            18.17.6  Algebraic closure systems   cnacs 26879
            18.17.7  Miscellanea 1. Map utilities   constmap 26890
            18.17.8  Miscellanea for polynomials   ofmpteq 26899
            18.17.9  Multivariate polynomials over the integers   cmzpcl 26901
            18.17.10  Miscellanea for Diophantine sets 1   coeq0 26933
            18.17.11  Diophantine sets 1: definitions   cdioph 26936
            18.17.12  Diophantine sets 2 miscellanea   ellz1 26948
            18.17.13  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 26954
            18.17.14  Diophantine sets 3: construction   diophrex 26957
            18.17.15  Diophantine sets 4 miscellanea   2sbcrex 26966
            18.17.16  Diophantine sets 4: Quantification   rexrabdioph 26977
            18.17.17  Diophantine sets 5: Arithmetic sets   rabdiophlem1 26984
            18.17.18  Diophantine sets 6 miscellanea   fz1ssnn 26994
            18.17.19  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 26996
            18.17.20  Pigeonhole Principle and cardinality helpers   fphpd 27001
            18.17.21  A non-closed set of reals is infinite   rencldnfilem 27005
            18.17.22  Miscellanea for Lagrange's theorem   icodiamlt 27007
            18.17.23  Lagrange's rational approximation theorem   irrapxlem1 27009
            18.17.24  Pell equations 1: A nontrivial solution always exists   pellexlem1 27016
            18.17.25  Pell equations 2: Algebraic number theory of the solution set   csquarenn 27023
            18.17.26  Pell equations 3: characterizing fundamental solution   infmrgelbi 27065
            18.17.27  Logarithm laws generalized to an arbitrary base   reglogcl 27077
            18.17.28  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 27085
            18.17.29  X and Y sequences 1: Definition and recurrence laws   crmx 27087
            18.17.30  Ordering and induction lemmas for the integers   monotuz 27128
            18.17.31  X and Y sequences 2: Order properties   rmxypos 27136
            18.17.32  Congruential equations   congtr 27154
            18.17.33  Alternating congruential equations   acongid 27164
            18.17.34  Additional theorems on integer divisibility   bezoutr 27174
            18.17.35  X and Y sequences 3: Divisibility properties   jm2.18 27183
            18.17.36  X and Y sequences 4: Diophantine representability of Y   jm2.27a 27200
            18.17.37  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 27210
            18.17.38  Uncategorized stuff not associated with a major project   setindtr 27219
            18.17.39  More equivalents of the Axiom of Choice   axac10 27228
            18.17.40  Finitely generated left modules   clfig 27267
            18.17.41  Noetherian left modules I   clnm 27275
            18.17.42  Addenda for structure powers   pwssplit0 27289
            18.17.43  Direct sum of left modules   cdsmm 27299
            18.17.44  Free modules   cfrlm 27314
            18.17.45  Every set admits a group structure iff choice   unxpwdom3 27358
            18.17.46  Independent sets and families   clindf 27376
            18.17.47  Characterization of free modules   lmimlbs 27408
            18.17.48  Noetherian rings and left modules II   clnr 27415
            18.17.49  Hilbert's Basis Theorem   cldgis 27427
            18.17.50  Additional material on polynomials [DEPRECATED]   cmnc 27437
            18.17.51  Degree and minimal polynomial of algebraic numbers   cdgraa 27447
            18.17.52  Algebraic integers I   citgo 27464
            18.17.53  Finite cardinality [SO]   en1uniel 27482
            18.17.54  Words in monoids and ordered group sum   issubmd 27485
            18.17.55  Transpositions in the symmetric group   cpmtr 27486
            18.17.56  The sign of a permutation   cpsgn 27516
            18.17.57  The matrix algebra   cmmul 27541
            18.17.58  The determinant   cmdat 27585
            18.17.59  Endomorphism algebra   cmend 27591
            18.17.60  Subfields   csdrg 27605
            18.17.61  Cyclic groups and order   idomrootle 27613
            18.17.62  Cyclotomic polynomials   ccytp 27623
            18.17.63  Miscellaneous topology   fgraphopab 27631
      18.18  Mathbox for Steve Rodriguez
            18.18.1  Miscellanea   iso0 27638
            18.18.2  Function operations   caofcan 27642
            18.18.3  Calculus   lhe4.4ex1a 27648
      18.19  Mathbox for Andrew Salmon
            18.19.1  Principia Mathematica * 10   pm10.12 27655
            18.19.2  Principia Mathematica * 11   2alanimi 27669
            18.19.3  Predicate Calculus   sbeqal1 27699
            18.19.4  Principia Mathematica * 13 and * 14   pm13.13a 27709
            18.19.5  Set Theory   elnev 27740
            18.19.6  Arithmetic   addcomgi 27763
            18.19.7  Geometry   cplusr 27764
      18.20  Mathbox for Glauco Siliprandi
            18.20.1  Miscellanea   ssrexf 27786
            18.20.2  Finite multiplication of numbers and finite multiplication of functions   fmul01 27812
            18.20.3  Limits   clim1fr1 27829
            18.20.4  Derivatives   dvsinexp 27842
            18.20.5  Integrals   ioovolcl 27844
            18.20.6  Stone Weierstrass theorem - real version   stoweidlem1 27852
            18.20.7  Wallis' product for π   wallispilem1 27916
            18.20.8  Stirling's approximation formula for ` n ` factorial   stirlinglem1 27925
      18.21  Mathbox for Saveliy Skresanov
            18.21.1  Ceva's theorem   sigarval 27942
      18.22  Mathbox for Jarvin Udandy
      18.23  Mathbox for Alexander van der Vekens
            18.23.1  Double restricted existential uniqueness   r19.32 28047
                  18.23.1.1  Restricted quantification (extension)   r19.32 28047
                  18.23.1.2  The empty set (extension)   raaan2 28055
                  18.23.1.3  Restricted uniqueness and "at most one" quantification   rmoimi 28056
                  18.23.1.4  Analogs to Existential uniqueness (double quantification)   2reurex 28061
            18.23.2  Alternative definitions of function's and operation's values   wdfat 28073
                  18.23.2.1  Restricted quantification (extension)   ralbinrald 28079
                  18.23.2.2  The universal class (extension)   nvelim 28080
                  18.23.2.3  Introduce the Axiom of Power Sets (extension)   alneu 28081
                  18.23.2.4  Relations (extension)   sbcrel 28083
                  18.23.2.5  Functions (extension)   sbcfun 28089
                  18.23.2.6  Predicate "defined at"   dfateq12d 28096
                  18.23.2.7  Alternative definition of the value of a function   dfafv2 28099
                  18.23.2.8  Alternative definition of the value of an operation   aoveq123d 28145
            18.23.3  Graph theory   jaoi2 28175
                  18.23.3.1  Logical disjunction and conjunction (extension)   jaoi2 28175
                  18.23.3.2  Abbreviated conjunction and disjunction of three wff's (extension)   3bior1fd 28176
                  18.23.3.3  Unordered and ordered pairs (extension)   tppreq3 28180
                  18.23.3.4  Functions (extension)   f1oprg 28185
                  18.23.3.5  Operations (Extension)   nssdmovg 28193
                  18.23.3.6  "Maps to" notation (Extension)   mpt2xopn0yelv 28194
                  18.23.3.7  Half-open integer ranges (extension)   fzossrbm1 28208
                  18.23.3.8  The ` # ` (finite set size) function (extension)   elprchashprn2 28215
                  18.23.3.9  Words over a set (extension)   4fvwrd4 28219
                  18.23.3.10  Longer string literals (extension)   s2prop 28220
                  18.23.3.11  Undirected simple graphs   cuslg 28225
                  18.23.3.12  Undirected simple graphs (examples)   usgra1v 28259
                  18.23.3.13  Neighbors, complete graphs and universal vertices   cnbgra 28267
                  18.23.3.14  Paths and Cycles   cwalk 28309
                  18.23.3.15  Friendship graphs   cfrgra 28414
      18.24  Mathbox for David A. Wheeler
            18.24.1  Natural deduction   19.8ad 28440
            18.24.2  Greater than, greater than or equal to.   cge-real 28443
            18.24.3  Hyperbolic trig functions   csinh 28453
            18.24.4  Reciprocal trig functions (sec, csc, cot)   csec 28464
            18.24.5  Identities for "if"   ifnmfalse 28486
            18.24.6  Not-member-of   AnelBC 28487
            18.24.7  Decimal point   cdp2 28488
            18.24.8  Signum (sgn or sign) function   csgn 28496
            18.24.9  Ceiling function   ccei 28506
            18.24.10  Logarithms generalized to arbitrary base using ` logb `   ene0 28510
            18.24.11  Logarithm laws generalized to an arbitrary base - log_   clog_ 28513
            18.24.12  Miscellaneous   5m4e1 28515
      18.25  Mathbox for Alan Sare
            18.25.1  Supplementary "adant" deductions   ad4ant13 28518
            18.25.2  Supplementary unification deductions   biimp 28544
            18.25.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 28560
            18.25.4  What is Virtual Deduction?   wvd1 28635
            18.25.5  Virtual Deduction Theorems   df-vd1 28636
            18.25.6  Theorems proved using virtual deduction   trsspwALT 28907
            18.25.7  Theorems proved using virtual deduction with mmj2 assistance   simplbi2VD 28937
            18.25.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 29004
            18.25.9  Theorems proved using conjunction-form virtual deduction   elpwgdedVD 29008
            18.25.10  Theorems with VD proofs in conventional notation derived from VD proofs   suctrALT3 29015
            18.25.11  Theorems with a proof in conventional notation automatically derived   notnot2ALT2 29018
      18.26  Mathbox for Jonathan Ben-Naim
            18.26.1  First order logic and set theory   bnj170 29038
            18.26.2  Well founded induction and recursion   bnj110 29205
            18.26.3  The existence of a minimal element in certain classes   bnj69 29355
            18.26.4  Well-founded induction   bnj1204 29357
            18.26.5  Well-founded recursion, part 1 of 3   bnj60 29407
            18.26.6  Well-founded recursion, part 2 of 3   bnj1500 29413
            18.26.7  Well-founded recursion, part 3 of 3   bnj1522 29417
      18.27  Mathbox for Norm Megill
            18.27.1  Experiments to study ax-7 unbundling   ax-7v 29418
                  18.27.1.1  Theorems derived from ax-7v (suffixes NEW7 and AUX7)   ax-7v 29418
                  18.27.1.2  Theorems derived from ax-7 (suffix OLD7)   ax-7OLD7 29632
            18.27.2  Obsolete experiments to study ax-12o   ax12-2 29725
            18.27.3  Miscellanea   cnaddcom 29783
            18.27.4  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 29786
            18.27.5  Functionals and kernels of a left vector space (or module)   clfn 29869
            18.27.6  Opposite rings and dual vector spaces   cld 29935
            18.27.7  Ortholattices and orthomodular lattices   cops 29984
            18.27.8  Atomic lattices with covering property   ccvr 30074
            18.27.9  Hilbert lattices   chlt 30162
            18.27.10  Projective geometries based on Hilbert lattices   clln 30302
            18.27.11  Construction of a vector space from a Hilbert lattice   cdlema1N 30602
            18.27.12  Construction of involution and inner product from a Hilbert lattice   clpoN 32292

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