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Table of Contents Summary
PART 1  CLASSICAL FIRST ORDER LOGIC WITH EQUALITY
      1.1  Pre-logic
      1.2  Conventions
      1.3  Propositional calculus
      1.4  Other axiomatizations of classical propositional calculus
      1.5  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
      1.6  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
      1.7  Predicate calculus with equality: Older axiomatization (1 rule, 14 schemes)
      1.8  Existential uniqueness
      1.9  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  MISCELLANEA
      14.1  Definitional Examples
      14.2  Natural deduction examples
      14.3  Humor
      14.4  (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 Frédéric Liné
      18.13  Mathbox for Jeff Hankins
      18.14  Mathbox for Jeff Madsen
      18.15  Mathbox for Rodolfo Medina
      18.16  Mathbox for Stefan O'Rear
      18.17  Mathbox for Steve Rodriguez
      18.18  Mathbox for Andrew Salmon
      18.19  Mathbox for Glauco Siliprandi
      18.20  Mathbox for Jarvin Udandy
      18.21  Mathbox for Alexander van der Vekens
      18.22  Mathbox for David A. Wheeler
      18.23  Mathbox for Alan Sare
      18.24  Mathbox for Jonathan Ben-Naim
      18.25  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  Conventions
      1.3  Propositional calculus
            1.3.1  Recursively define primitive wffs for propositional calculus   wn 5
            1.3.2  The axioms of propositional calculus   ax-1 7
            1.3.3  Logical implication   mp2b 11
            1.3.4  Logical negation   con4d 99
            1.3.5  Logical equivalence   wb 178
            1.3.6  Logical disjunction and conjunction   wo 359
            1.3.7  Miscellaneous theorems of propositional calculus   pm5.21nd 870
            1.3.8  Abbreviated conjunction and disjunction of three wff's   w3o 935
            1.3.9  Logical 'nand' (Sheffer stroke)   wnan 1289
            1.3.10  Logical 'xor'   wxo 1297
            1.3.11  True and false constants   wtru 1309
            1.3.12  Truth tables   truantru 1328
            1.3.13  Auxiliary theorems for Alan Sare's virtual deduction tool, part 1   ee22 1354
            1.3.14  Half-adders and full adders in propositional calculus   whad 1370
      1.4  Other axiomatizations of classical propositional calculus
            1.4.1  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1396
            1.4.2  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1414
            1.4.3  Derive Nicod's axiom from the standard axioms   nic-dfim 1425
            1.4.4  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1431
            1.4.5  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1450
            1.4.6  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1454
            1.4.7  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1469
            1.4.8  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1492
            1.4.9  Derive the Lukasiewicz axioms from the The Russell-Bernays Axioms   rb-bijust 1505
            1.4.10  Stoic logic indemonstrables (Chrysippus of Soli)   mpto1 1524
      1.5  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
            1.5.1  Universal quantifier; define "exists" and "not free"   wal 1528
            1.5.2  Rule scheme ax-gen (Generalization)   ax-gen 1534
            1.5.3  Axiom scheme ax-5 (Quantified Implication)   ax-5 1545
            1.5.4  Axiom scheme ax-17 (Distinctness) - first use of $d   ax-17 1604
            1.5.5  Equality predicate; define substitution   cv 1623
            1.5.6  Axiom scheme ax-9 (Existence)   ax-9 1637
            1.5.7  Axiom scheme ax-8 (Equality)   ax-8 1645
            1.5.8  Membership predicate   wcel 1685
            1.5.9  Axiom schemes ax-13 (Left Membership Equality)   ax-13 1687
            1.5.10  Axiom schemes ax-14 (Right Membership Equality)   ax-14 1689
            1.5.11  Logical redundancy of ax-6 , ax-7 , ax-11 , ax-12   ax9dgen 1691
      1.6  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.6.1  Axiom scheme ax-6 (Quantified Negation)   ax-6 1704
            1.6.2  Axiom scheme ax-7 (Quantifier Commutation)   ax-7 1709
            1.6.3  Axiom scheme ax-11 (Substitution)   ax-11 1716
            1.6.4  Axiom scheme ax-12 (Quantified Equality)   ax-12 1868
      1.7  Predicate calculus with equality: Older axiomatization (1 rule, 14 schemes)
            1.7.1  Obsolete schemes ax-5o ax-4 ax-6o ax-9o ax-10o ax-10 ax-11o ax-12o ax-15 ax-16   ax-4 2078
            1.7.2  Rederive new axioms from old: theorems ax5 , ax6 , ax9from9o , ax11 , ax12   ax5 2088
            1.7.3  Legacy theorems using obsolete axioms   ax17o 2099
      1.8  Existential uniqueness
      1.9  Other axiomatizations related to classical predicate calculus
            1.9.1  Predicate calculus with all distinct variables   ax-7d 2236
            1.9.2  Aristotelian logic: Assertic syllogisms   barbara 2242
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 2266
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2271
            2.1.3  Class form not-free predicate   wnfc 2408
            2.1.4  Negated equality and membership   wne 2448
            2.1.5  Restricted quantification   wral 2545
            2.1.6  The universal class   cvv 2790
            2.1.7  Conditional equality (experimental)   wcdeq 2976
            2.1.8  Russell's Paradox   ru 2992
            2.1.9  Proper substitution of classes for sets   wsbc 2993
            2.1.10  Proper substitution of classes for sets into classes   csb 3083
            2.1.11  Define basic set operations and relations   cdif 3151
            2.1.12  Subclasses and subsets   df-ss 3168
            2.1.13  The difference, union, and intersection of two classes   difeq1 3289
            2.1.14  The empty set   c0 3457
            2.1.15  "Weak deduction theorem" for set theory   cif 3567
            2.1.16  Power classes   cpw 3627
            2.1.17  Unordered and ordered pairs   csn 3642
            2.1.18  The union of a class   cuni 3829
            2.1.19  The intersection of a class   cint 3864
            2.1.20  Indexed union and intersection   ciun 3907
            2.1.21  Disjointness   wdisj 3995
            2.1.22  Binary relations   wbr 4025
            2.1.23  Ordered-pair class abstractions (class builders)   copab 4078
            2.1.24  Transitive classes   wtr 4115
      2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 4133
            2.2.2  Derive the Axiom of Separation   axsep 4142
            2.2.3  Derive the Null Set Axiom   zfnuleu 4148
            2.2.4  Theorems requiring subset and intersection existence   nalset 4153
            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  Functions   wfun 5216
            2.4.9  Operations   co 5820
            2.4.10  "Maps to" notation   elmpt2cl 6023
            2.4.11  Function operation   cof 6038
            2.4.12  First and second members of an ordered pair   c1st 6082
            2.4.13  Function transposition   ctpos 6195
            2.4.14  Curry and uncurry   ccur 6234
            2.4.15  Proper subset relation   crpss 6238
            2.4.16  Definite description binder (inverted iota)   cio 6251
            2.4.17  Cantor's Theorem   canth 6288
            2.4.18  Undefined values and restricted iota (description binder)   cund 6290
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 6351
            2.4.20  "Strong" transfinite recursion   crecs 6383
            2.4.21  Recursive definition generator   crdg 6418
            2.4.22  Finite recursion   frfnom 6443
            2.4.23  Abian's "most fundamental" fixed point theorem   abianfplem 6466
            2.4.24  Ordinal arithmetic   c1o 6468
            2.4.25  Natural number arithmetic   nna0 6598
            2.4.26  Equivalence relations and classes   wer 6653
            2.4.27  The mapping operation   cmap 6768
            2.4.28  Infinite Cartesian products   cixp 6813
            2.4.29  Equinumerosity   cen 6856
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 6967
            2.4.31  Equinumerosity (cont.)   xpf1o 7019
            2.4.32  Pigeonhole Principle   phplem1 7036
            2.4.33  Finite sets   onomeneq 7046
            2.4.34  Finite intersections   cfi 7160
            2.4.35  Hall's marriage theorem   marypha1lem 7182
            2.4.36  Supremum   csup 7189
            2.4.37  Ordinal isomorphism, Hartog's theorem   coi 7220
            2.4.38  Hartogs function, order types, weak dominance   char 7266
      2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 7302
            2.5.2  Axiom of Infinity equivalents   inf0 7318
      2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 7335
            2.6.2  Existence of omega (the set of natural numbers)   omex 7340
            2.6.3  Cantor normal form   ccnf 7358
            2.6.4  Transitive closure   trcl 7406
            2.6.5  Rank   cr1 7430
            2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 7551
            2.6.7  Cardinal numbers   ccrd 7564
            2.6.8  Axiom of Choice equivalents   wac 7738
            2.6.9  Cardinal number arithmetic   ccda 7789
            2.6.10  The Ackermann bijection   ackbij2lem1 7841
            2.6.11  Cofinality (without Axiom of Choice)   cflem 7868
            2.6.12  Eight inequivalent definitions of finite set   sornom 7899
            2.6.13  Hereditarily size-limited sets without Choice   itunifval 8038
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 8081
            3.2.2  AC equivalents: well ordering, Zorn's lemma   numthcor 8117
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 8164
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 8192
            3.2.5  Cofinality using Axiom of Choice   alephreg 8200
      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 8300
            4.1.2  Weak universes   cwun 8318
            4.1.3  Tarski's classes   ctsk 8366
            4.1.4  Grothendieck's universes   cgru 8408
      4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 8441
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 8444
            4.2.3  Tarski map function   ctskm 8455
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 8462
            5.1.2  Final derivation of real and complex number postulates   axaddf 8763
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 8789
      5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 8814
            5.2.2  Infinity and the extended real number system   cpnf 8860
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 8890
            5.2.4  Ordering on reals   lttr 8895
            5.2.5  Initial properties of the complex numbers   mul12 8974
      5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 9021
            5.3.2  Subtraction   cmin 9033
            5.3.3  Multiplication   muladd 9208
            5.3.4  Ordering on reals (cont.)   gt0ne0 9235
            5.3.5  Reciprocals   ixi 9393
            5.3.6  Division   cdiv 9419
            5.3.7  Ordering on reals (cont.)   elimgt0 9588
            5.3.8  Completeness Axiom and Suprema   fimaxre 9697
            5.3.9  Imaginary and complex number properties   inelr 9732
            5.3.10  Function operation analogue theorems   ofsubeq0 9739
      5.4  Integer sets
            5.4.1  Natural numbers (as a subset of complex numbers)   cn 9742
            5.4.2  Principle of mathematical induction   nnind 9760
            5.4.3  Decimal representation of numbers   c2 9791
            5.4.4  Some properties of specific numbers   0p1e1 9835
            5.4.5  The Archimedean property   nnunb 9957
            5.4.6  Nonnegative integers (as a subset of complex numbers)   cn0 9961
            5.4.7  Integers (as a subset of complex numbers)   cz 10020
            5.4.8  Decimal arithmetic   cdc 10120
            5.4.9  Upper partititions of integers   cuz 10226
            5.4.10  Well-ordering principle for bounded-below sets of integers   uzwo3 10307
            5.4.11  Rational numbers (as a subset of complex numbers)   cq 10312
            5.4.12  Existence of the set of complex numbers   rpnnen1lem1 10338
      5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 10350
            5.5.2  Infinity and the extended real number system (cont.)   cxne 10445
            5.5.3  Supremum on the extended reals   xrsupexmnf 10618
            5.5.4  Real number intervals   cioo 10651
            5.5.5  Finite intervals of integers   cfz 10777
            5.5.6  Half-open integer ranges   cfzo 10865
      5.6  Elementary integer functions
            5.6.1  The floor (greatest integer) function   cfl 10919
            5.6.2  The modulo (remainder) operation   cmo 10968
            5.6.3  The infinite sequence builder "seq"   om2uz0i 11005
            5.6.4  Integer powers   cexp 11099
            5.6.5  Ordered pair theorem for nonnegative integers   nn0le2msqi 11277
            5.6.6  Factorial function   cfa 11283
            5.6.7  The binomial coefficient operation   cbc 11310
            5.6.8  The ` # ` (finite set size) function   chash 11332
            5.6.9  Words over a set   cword 11398
            5.6.10  Longer string literals   cs2 11486
      5.7  Elementary real and complex functions
            5.7.1  The "shift" operation   cshi 11556
            5.7.2  Real and imaginary parts; conjugate   ccj 11576
            5.7.3  Square root; absolute value   csqr 11713
      5.8  Elementary limits and convergence
            5.8.1  Superior limit (lim sup)   clsp 11939
            5.8.2  Limits   cli 11953
            5.8.3  Finite and infinite sums   csu 12153
            5.8.4  The binomial theorem   binomlem 12282
            5.8.5  The inclusion/exclusion principle   incexclem 12290
            5.8.6  Infinite sums (cont.)   isumshft 12293
            5.8.7  Miscellaneous converging and diverging sequences   divrcnv 12306
            5.8.8  Arithmetic series   arisum 12313
            5.8.9  Geometric series   expcnv 12317
            5.8.10  Ratio test for infinite series convergence   cvgrat 12334
            5.8.11  Mertens' theorem   mertenslem1 12335
      5.9  Elementary trigonometry
            5.9.1  The exponential, sine, and cosine functions   ce 12338
            5.9.2  _e is irrational   eirrlem 12477
      5.10  Cardinality of real and complex number subsets
            5.10.1  Countability of integers and rationals   xpnnen 12482
            5.10.2  The reals are uncountable   rpnnen2lem1 12488
PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqr2irrlem 12521
            6.1.2  Some Number sets are chains of proper subsets   nthruc 12524
            6.1.3  The divides relation   cdivides 12526
            6.1.4  The division algorithm   divalglem0 12587
            6.1.5  Bit sequences   cbits 12605
            6.1.6  The greatest common divisor operator   cgcd 12680
            6.1.7  Bézout's identity   bezoutlem1 12712
            6.1.8  Algorithms   nn0seqcvgd 12735
            6.1.9  Euclid's Algorithm   eucalgval2 12746
      6.2  Elementary prime number theory
            6.2.1  Elementary properties   cprime 12753
            6.2.2  Properties of the canonical representation of a rational   cnumer 12799
            6.2.3  Euler's theorem   codz 12826
            6.2.4  Pythagorean Triples   coprimeprodsq 12857
            6.2.5  The prime count function   cpc 12884
            6.2.6  Pocklington's theorem   prmpwdvds 12946
            6.2.7  Infinite primes theorem   unbenlem 12950
            6.2.8  Sum of prime reciprocals   prmreclem1 12958
            6.2.9  Fundamental theorem of arithmetic   1arithlem1 12965
            6.2.10  Lagrange's four-square theorem   cgz 12971
            6.2.11  Van der Waerden's theorem   cvdwa 13007
            6.2.12  Ramsey's theorem   cram 13041
            6.2.13  Decimal arithmetic (cont.)   dec2dvds 13073
            6.2.14  Specific prime numbers   4nprm 13101
            6.2.15  Very large primes   1259lem1 13124
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
            7.1.1  Basic definitions   cstr 13139
            7.1.2  Slot definitions   cplusg 13203
            7.1.3  Definition of the structure product   crest 13320
            7.1.4  Definition of the structure quotient   cordt 13393
      7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 13503
            7.2.2  Independent sets in a Moore system   mrisval 13527
            7.2.3  Algebraic closure systems   isacs 13548
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
            8.1.1  Categories   ccat 13561
            8.1.2  Opposite category   coppc 13609
            8.1.3  Monomorphisms and epimorphisms   cmon 13626
            8.1.4  Sections, inverses, isomorphisms   csect 13642
            8.1.5  Subcategories   cssc 13679
            8.1.6  Functors   cfunc 13723
            8.1.7  Full & faithful functors   cful 13771
            8.1.8  Natural transformations and the functor category   cnat 13810
      8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 13880
      8.3  Examples of categories
            8.3.1  The category of sets   csetc 13902
            8.3.2  The category of categories   ccatc 13921
      8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 13937
            8.4.2  Functor evaluation   cevlf 13978
            8.4.3  Hom functor   chof 14017
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 14069
            9.2.2  Lattices   clat 14146
            9.2.3  The dual of an ordered set   codu 14227
            9.2.4  Subset order structures   cipo 14249
            9.2.5  Distributive lattices   latmass 14286
            9.2.6  Posets and lattices as relations   cps 14296
            9.2.7  Directed sets, nets   cdir 14345
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
            10.1.1  Definition and basic properties   cmnd 14356
            10.1.2  Monoid homomorphisms and submonoids   cmhm 14408
            10.1.3  Ordered group sum operation   gsumvallem1 14443
            10.1.4  Free monoids   cfrmd 14464
      10.2  Groups
            10.2.1  Definition and basic properties   df-grp 14484
            10.2.2  Subgroups and Quotient groups   csubg 14610
            10.2.3  Elementary theory of group homomorphisms   cghm 14675
            10.2.4  Isomorphisms of groups   cgim 14716
            10.2.5  Group actions   cga 14738
            10.2.6  Symmetry groups and Cayley's Theorem   csymg 14764
            10.2.7  Centralizers and centers   ccntz 14786
            10.2.8  The opposite group   coppg 14813
            10.2.9  p-Groups and Sylow groups; Sylow's theorems   cod 14835
            10.2.10  Direct products   clsm 14940
            10.2.11  Free groups   cefg 15010
      10.3  Abelian groups
            10.3.1  Definition and basic properties   ccmn 15084
            10.3.2  Cyclic groups   ccyg 15159
            10.3.3  Group sum operation   gsumval3a 15184
            10.3.4  Internal direct products   cdprd 15226
            10.3.5  The Fundamental Theorem of Abelian Groups   ablfacrplem 15295
      10.4  Rings
            10.4.1  Multiplicative Group   cmgp 15320
            10.4.2  Definition and basic properties   crg 15332
            10.4.3  Opposite ring   coppr 15399
            10.4.4  Divisibility   cdsr 15415
            10.4.5  Ring homomorphisms   crh 15489
      10.5  Division rings and Fields
            10.5.1  Definition and basic properties   cdr 15507
            10.5.2  Subrings of a ring   csubrg 15536
            10.5.3  Absolute value (abstract algebra)   cabv 15576
            10.5.4  Star rings   cstf 15603
      10.6  Left Modules
            10.6.1  Definition and basic properties   clmod 15622
            10.6.2  Subspaces and spans in a left module   clss 15684
            10.6.3  Homomorphisms and isomorphisms of left modules   clmhm 15771
            10.6.4  Subspace sum; bases for a left module   clbs 15822
      10.7  Vector Spaces
            10.7.1  Definition and basic properties   clvec 15850
      10.8  Ideals
            10.8.1  The subring algebra; ideals   csra 15916
            10.8.2  Two-sided ideals and quotient rings   c2idl 15978
            10.8.3  Principal ideal rings. Divisibility in the integers   clpidl 15988
            10.8.4  Nonzero rings   cnzr 16004
            10.8.5  Left regular elements. More kinds of ring   crlreg 16015
      10.9  Associative algebras
            10.9.1  Definition and basic properties   casa 16045
      10.10  Abstract Multivariate Polynomials
            10.10.1  Definition and basic properties   cmps 16082
            10.10.2  Polynomial evaluation   evlslem4 16240
            10.10.3  Univariate Polynomials   cps1 16245
      10.11  The complex numbers as an extensible structure
            10.11.1  Definition and basic properties   cxmt 16364
            10.11.2  Algebraic constructions based on the complexes   czrh 16446
      10.12  Hilbert spaces
            10.12.1  Definition and basic properties   cphl 16523
            10.12.2  Orthocomplements and closed subspaces   cocv 16555
            10.12.3  Orthogonal projection and orthonormal bases   cpj 16595
PART 11  BASIC TOPOLOGY
      11.1  Topology
            11.1.1  Topological spaces   ctop 16626
            11.1.2  TopBases for topologies   isbasisg 16680
            11.1.3  Examples of topologies   distop 16728
            11.1.4  Closure and interior   ccld 16748
            11.1.5  Neighborhoods   cnei 16829
            11.1.6  Limit points and perfect sets   clp 16861
            11.1.7  Subspace topologies   restrcl 16883
            11.1.8  Order topology   ordtbaslem 16913
            11.1.9  Limits and Continuity in topological spaces   ccn 16949
            11.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 17029
            11.1.11  Compactness   ccmp 17108
            11.1.12  Connectedness   ccon 17132
            11.1.13  First- and second-countability   c1stc 17158
            11.1.14  Local topological properties   clly 17185
            11.1.15  Compactly generated spaces   ckgen 17223
            11.1.16  Product topologies   ctx 17250
            11.1.17  Continuous function-builders   cnmptid 17350
            11.1.18  Quotient maps and quotient topology   ckq 17379
            11.1.19  Homeomorphisms   chmeo 17439
      11.2  Filters and filter bases
            11.2.1  Filter Bases   cfbas 17513
            11.2.2  Filters   cfil 17535
            11.2.3  Ultrafilters   cufil 17589
            11.2.4  Filter limits   cfm 17623
            11.2.5  Topological groups   ctmd 17748
            11.2.6  Infinite group sum on topological groups   ctsu 17803
            11.2.7  Topological rings, fields, vector spaces   ctrg 17833
      11.3  Metric spaces
            11.3.1  Basic metric space properties   cxme 17877
            11.3.2  Metric space balls   blfval 17942
            11.3.3  Open sets of a metric space   mopnval 17979
            11.3.4  Continuity in metric spaces   metcnp3 18081
            11.3.5  Examples of metric spaces   dscmet 18090
            11.3.6  Normed algebraic structures   cnm 18094
            11.3.7  Normed space homomorphisms (bounded linear operators)   cnmo 18209
            11.3.8  Topology on the Reals   qtopbaslem 18262
            11.3.9  Topological definitions using the reals   cii 18374
            11.3.10  Path homotopy   chtpy 18460
            11.3.11  The fundamental group   cpco 18493
      11.4  Complex metric vector spaces
            11.4.1  Complex left modules   cclm 18555
            11.4.2  Complex pre-Hilbert space   ccph 18597
            11.4.3  Convergence and completeness   ccfil 18673
            11.4.4  Baire's Category Theorem   bcthlem1 18741
            11.4.5  Banach spaces and complex Hilbert spaces   ccms 18749
            11.4.6  Minimizing Vector Theorem   minveclem1 18783
            11.4.7  Projection theorem   pjthlem1 18796
PART 12  BASIC REAL AND COMPLEX ANALYSIS
      12.1  Continuity
            12.1.1  Intermediate value theorem   pmltpclem1 18803
      12.2  Integrals
            12.2.1  Lebesgue measure   covol 18817
            12.2.2  Lebesgue integration   cmbf 18964
      12.3  Derivatives
            12.3.1  Real and Complex Differentiation   climc 19207
PART 13  BASIC REAL AND COMPLEX FUNCTIONS
      13.1  Polynomials
            13.1.1  Abstract polynomials, continued   evlslem6 19392
            13.1.2  Polynomial degrees   cmdg 19434
            13.1.3  The division algorithm for univariate polynomials   cmn1 19506
            13.1.4  Elementary properties of complex polynomials   cply 19561
            13.1.5  The Division algorithm for polynomials   cquot 19665
            13.1.6  Algebraic numbers   caa 19689
            13.1.7  Liouville's approximation theorem   aalioulem1 19707
      13.2  Sequences and series
            13.2.1  Taylor polynomials and Taylor's theorem   ctayl 19727
            13.2.2  Uniform convergence   culm 19750
            13.2.3  Power series   pserval 19781
      13.3  Basic trigonometry
            13.3.1  The exponential, sine, and cosine functions (cont.)   efcn 19814
            13.3.2  Properties of pi = 3.14159...   pilem1 19822
            13.3.3  Mapping of the exponential function   efgh 19898
            13.3.4  The natural logarithm on complex numbers   clog 19907
            13.3.5  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 20094
            13.3.6  Solutions of quadratic, cubic, and quartic equations   quad2 20130
            13.3.7  Inverse trigonometric functions   casin 20153
            13.3.8  The Birthday Problem   log2ublem1 20237
            13.3.9  Areas in R^2   carea 20245
            13.3.10  More miscellaneous converging sequences   rlimcnp 20255
            13.3.11  Inequality of arithmetic and geometric means   cvxcl 20274
            13.3.12  Euler-Mascheroni constant   cem 20281
      13.4  Basic number theory
            13.4.1  Wilson's theorem   wilthlem1 20301
            13.4.2  The Fundamental Theorem of Algebra   ftalem1 20305
            13.4.3  The Basel problem (ζ(2) = π2/6)   basellem1 20313
            13.4.4  Number-theoretical functions   ccht 20323
            13.4.5  Perfect Number Theorem   mersenne 20461
            13.4.6  Characters of Z/nZ   cdchr 20466
            13.4.7  Bertrand's postulate   bcctr 20509
            13.4.8  Legendre symbol   clgs 20528
            13.4.9  Quadratic Reciprocity   lgseisenlem1 20583
            13.4.10  All primes 4n+1 are the sum of two squares   2sqlem1 20597
            13.4.11  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 20613
            13.4.12  The Prime Number Theorem   mudivsum 20674
            13.4.13  Ostrowski's theorem   abvcxp 20759
PART 14  MISCELLANEA
      14.1  Definitional Examples
      14.2  Natural deduction examples
      14.3  Humor
            14.3.1  April Fool's theorem   avril1 20829
      14.4  (Future - to be reviewed and classified)
            14.4.1  Planar incidence geometry   cplig 20835
            14.4.2  Algebra preliminaries   crpm 20840
            14.4.3  Transitive closure   ctcl 20842
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 20846
            15.1.2  Definition and basic properties of Abelian groups   cablo 20941
            15.1.3  Subgroups   csubgo 20961
            15.1.4  Operation properties   cass 20972
            15.1.5  Group-like structures   cmagm 20978
            15.1.6  Examples of Abelian groups   ablosn 21007
            15.1.7  Group homomorphism and isomorphism   cghom 21017
      15.2  Additional material on Rings and Fields
            15.2.1  Definition and basic properties   crngo 21035
            15.2.2  Examples of rings   cnrngo 21063
            15.2.3  Division Rings   cdrng 21065
            15.2.4  Star Fields   csfld 21068
            15.2.5  Fields and Rings   ccm2 21070
PART 16  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      16.1  Complex vector spaces
            16.1.1  Definition and basic properties   cvc 21094
            16.1.2  Examples of complex vector spaces   cncvc 21132
      16.2  Normed complex vector spaces
            16.2.1  Definition and basic properties   cnv 21133
            16.2.2  Examples of normed complex vector spaces   cnnv 21238
            16.2.3  Induced metric of a normed complex vector space   imsval 21247
            16.2.4  Inner product   cdip 21266
            16.2.5  Subspaces   css 21290
      16.3  Operators on complex vector spaces
            16.3.1  Definitions and basic properties   clno 21311
      16.4  Inner product (pre-Hilbert) spaces
            16.4.1  Definition and basic properties   ccphlo 21383
            16.4.2  Examples of pre-Hilbert spaces   cncph 21390
            16.4.3  Properties of pre-Hilbert spaces   isph 21393
      16.5  Complex Banach spaces
            16.5.1  Definition and basic properties   ccbn 21434
            16.5.2  Examples of complex Banach spaces   cnbn 21441
            16.5.3  Uniform Boundedness Theorem   ubthlem1 21442
            16.5.4  Minimizing Vector Theorem   minvecolem1 21446
      16.6  Complex Hilbert spaces
            16.6.1  Definition and basic properties   chlo 21457
            16.6.2  Standard axioms for a complex Hilbert space   hlex 21470
            16.6.3  Examples of complex Hilbert spaces   cnchl 21488
            16.6.4  Subspaces   ssphl 21489
            16.6.5  Hellinger-Toeplitz Theorem   htthlem 21490
PART 17  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      17.1  Axiomatization of complex pre-Hilbert spaces
            17.1.1  Basic Hilbert space definitions   chil 21492
            17.1.2  Preliminary ZFC lemmas   df-hnorm 21541
            17.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 21554
            17.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 21572
            17.1.5  Vector operations   hvmulex 21584
            17.1.6  Inner product postulates for a Hilbert space   ax-hfi 21651
      17.2  Inner product and norms
            17.2.1  Inner product   his5 21658
            17.2.2  Norms   dfhnorm2 21694
            17.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 21732
            17.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 21751
      17.3  Cauchy sequences and completeness axiom
            17.3.1  Cauchy sequences and limits   hcau 21756
            17.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 21766
            17.3.3  Completeness postulate for a Hilbert space   ax-hcompl 21774
            17.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 21775
      17.4  Subspaces and projections
            17.4.1  Subspaces   df-sh 21779
            17.4.2  Closed subspaces   df-ch 21794
            17.4.3  Orthocomplements   df-oc 21824
            17.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 21880
            17.4.5  Projection theorem   pjhthlem1 21963
            17.4.6  Projectors   df-pjh 21967
      17.5  Properties of Hilbert subspaces
            17.5.1  Orthomodular law   omlsilem 21974
            17.5.2  Projectors (cont.)   pjhtheu2 21988
            17.5.3  Hilbert lattice operations   sh0le 22012
            17.5.4  Span (cont.) and one-dimensional subspaces   spansn0 22113
            17.5.5  Commutes relation for Hilbert lattice elements   df-cm 22155
            17.5.6  Foulis-Holland theorem   fh1 22190
            17.5.7  Quantum Logic Explorer axioms   qlax1i 22199
            17.5.8  Orthogonal subspaces   chscllem1 22209
            17.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 22226
            17.5.10  Projectors (cont.)   pjorthi 22241
            17.5.11  Mayet's equation E_3   mayete3i 22300
      17.6  Operators on Hilbert spaces
            17.6.1  Operator sum, difference, and scalar multiplication   df-hosum 22303
            17.6.2  Zero and identity operators   df-h0op 22321
            17.6.3  Operations on Hilbert space operators   hoaddcl 22331
            17.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 22412
            17.6.5  Linear and continuous functionals and norms   df-nmfn 22418
            17.6.6  Adjoint   df-adjh 22422
            17.6.7  Dirac bra-ket notation   df-bra 22423
            17.6.8  Positive operators   df-leop 22425
            17.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 22426
            17.6.10  Theorems about operators and functionals   nmopval 22429
            17.6.11  Riesz lemma   riesz3i 22635
            17.6.12  Adjoints (cont.)   cnlnadjlem1 22640
            17.6.13  Quantum computation error bound theorem   unierri 22677
            17.6.14  Dirac bra-ket notation (cont.)   branmfn 22678
            17.6.15  Positive operators (cont.)   leopg 22695
            17.6.16  Projectors as operators   pjhmopi 22719
      17.7  States on an Hilbert lattice and Godowski's equation
            17.7.1  States on a Hilbert lattice   df-st 22784
            17.7.2  Godowski's equation   golem1 22844
      17.8  Cover relation, atoms, exchange axiom, and modular symmetry
            17.8.1  Covers relation; modular pairs   df-cv 22852
            17.8.2  Atoms   df-at 22911
            17.8.3  Superposition principle   superpos 22927
            17.8.4  Atoms, exchange and covering properties, atomicity   chcv1 22928
            17.8.5  Irreducibility   chirredlem1 22963
            17.8.6  Atoms (cont.)   atcvat3i 22969
            17.8.7  Modular symmetry   mdsymlem1 22976
PART 18  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      18.1  Mathboxes for user contributions
            18.1.1  Mathbox guidelines   mathbox 23015
      18.2  Mathbox for Stefan Allan
      18.3  Mathbox for Thierry Arnoux
            18.3.1  Bertrand's Ballot Problem   ballotlemoex 23038
      18.4  Mathbox for Mario Carneiro
            18.4.1  Miscellaneous stuff   quartfull 23091
            18.4.2  Zeta function   czeta 23092
            18.4.3  Gamma function   clgam 23095
            18.4.4  Derangements and the Subfactorial   deranglem 23102
            18.4.5  The Erdős-Szekeres theorem   erdszelem1 23127
            18.4.6  The Kuratowski closure-complement theorem   kur14lem1 23142
            18.4.7  Retracts and sections   cretr 23153
            18.4.8  Path-connected and simply connected spaces   cpcon 23155
            18.4.9  Covering maps   ccvm 23191
            18.4.10  Undirected multigraphs   cumg 23265
            18.4.11  Normal numbers   snmlff 23317
            18.4.12  Godel-sets of formulas   cgoe 23321
            18.4.13  Models of ZF   cgze 23349
            18.4.14  Splitting fields   citr 23363
            18.4.15  p-adic number fields   czr 23379
      18.5  Mathbox for Paul Chapman
            18.5.1  Group homomorphism and isomorphism   ghomgrpilem1 23397
            18.5.2  Real and complex numbers (cont.)   climuzcnv 23409
            18.5.3  Miscellaneous theorems   elfzm12 23413
      18.6  Mathbox for Drahflow
      18.7  Mathbox for Scott Fenton
            18.7.1  ZFC Axioms in primitive form   axextprim 23452
            18.7.2  Untangled classes   untelirr 23459
            18.7.3  Extra propositional calculus theorems   3orel1 23466
            18.7.4  Misc. Useful Theorems   nepss 23477
            18.7.5  Properties of reals and complexes   sqdivzi 23483
            18.7.6  Greatest common divisor and divisibility   pdivsq 23506
            18.7.7  Properties of relationships   brtp 23510
            18.7.8  Properties of functions and mappings   funpsstri 23523
            18.7.9  Epsilon induction   setinds 23536
            18.7.10  Ordinal numbers   elpotr 23539
            18.7.11  Defined equality axioms   axextdfeq 23556
            18.7.12  Hypothesis builders   hbntg 23564
            18.7.13  The Predecessor Class   cpred 23569
            18.7.14  (Trans)finite Recursion Theorems   tfisg 23606
            18.7.15  Well-founded induction   tz6.26 23607
            18.7.16  Transitive closure under a relationship   ctrpred 23622
            18.7.17  Founded Induction   frmin 23644
            18.7.18  Ordering Ordinal Sequences   orderseqlem 23654
            18.7.19  Well-founded recursion   wfr3g 23657
            18.7.20  Transfinite recursion via Well-founded recursion   tfrALTlem 23678
            18.7.21  Founded Recursion   frr3g 23682
            18.7.22  Surreal Numbers   csur 23696
            18.7.23  Surreal Numbers: Ordering   axsltsolem1 23723
            18.7.24  Surreal Numbers: Birthday Function   axbday 23730
            18.7.25  Surreal Numbers: Density   axdenselem1 23737
            18.7.26  Surreal Numbers: Full-Eta Property   axfelem1 23748
            18.7.27  Symmetric difference   csymdif 23770
            18.7.28  Quantifier-free definitions   ctxp 23782
            18.7.29  Alternate ordered pairs   caltop 23898
            18.7.30  Tarskian geometry   cee 23924
            18.7.31  Tarski's axioms for geometry   axdimuniq 23949
            18.7.32  Congruence properties   cofs 24013
            18.7.33  Betweenness properties   btwntriv2 24043
            18.7.34  Segment Transportation   ctransport 24060
            18.7.35  Properties relating betweenness and congruence   cifs 24066
            18.7.36  Connectivity of betweenness   btwnconn1lem1 24118
            18.7.37  Segment less than or equal to   csegle 24137
            18.7.38  Outside of relationship   coutsideof 24150
            18.7.39  Lines and Rays   cline2 24165
            18.7.40  Bernoulli polynomials and sums of k-th powers   cbp 24189
            18.7.41  Rank theorems   rankung 24204
            18.7.42  Hereditarily Finite Sets   chf 24210
      18.8  Mathbox for Anthony Hart
            18.8.1  Propositional Calculus   tb-ax1 24225
            18.8.2  Predicate Calculus   quantriv 24247
            18.8.3  Misc. Single Axiom Systems   meran1 24258
            18.8.4  Connective Symmetry   negsym1 24264
      18.9  Mathbox for Chen-Pang He
            18.9.1  Ordinal topology   ontopbas 24275
      18.10  Mathbox for Jeff Hoffman
            18.10.1  Inferences for finite induction on generic function values   fveleq 24298
            18.10.2  gdc.mm   nnssi2 24302
      18.11  Mathbox for Wolf Lammen
      18.12  Mathbox for Frédéric Liné
            18.12.1  Theorems from other workspaces   tpssg 24331
            18.12.2  Propositional and predicate calculus   neleq12d 24332
            18.12.3  Linear temporal logic   wbox 24369
            18.12.4  Operations   ssoprab2g 24431
            18.12.5  General Set Theory   uninqs 24438
            18.12.6  The "maps to" notation   cmpfun 24542
            18.12.7  Cartesian Products   cpro 24550
            18.12.8  Operations on subsets and functions   ccst 24572
            18.12.9  Arithmetic   3timesi 24578
            18.12.10  Lattice (algebraic definition)   clatalg 24581
            18.12.11  Currying and Partial Mappings   ccur1 24594
            18.12.12  Order theory (Extensible Structure Builder)   corhom 24607
            18.12.13  Order theory   cpresetrel 24615
            18.12.14  Finite composites ( i. e. finite sums, products ... )   cprd 24698
            18.12.15  Operation properties   ccm1 24731
            18.12.16  Groups and related structures   ridlideq 24735
            18.12.17  Free structures   csubsmg 24783
            18.12.18  Translations   trdom2 24791
            18.12.19  Fields and Rings   com2i 24816
            18.12.20  Ideals   cidln 24843
            18.12.21  Generic modules and vector spaces (New Structure builder)   cact 24847
            18.12.22  Generic modules and vector spaces   cvec 24849
            18.12.23  Real vector spaces   cvr 24889
            18.12.24  Matrices   cmmat 24893
            18.12.25  Affine spaces   craffsp 24899
            18.12.26  Intervals of reals and extended reals   bsi 24901
            18.12.27  Topology   topnem 24912
            18.12.28  Continuous functions   cnrsfin 24925
            18.12.29  Homeomorphisms   dmhmph 24933
            18.12.30  Initial and final topologies   intopcoaconlem3b 24938
            18.12.31  Filters   efilcp 24952
            18.12.32  Limits   plimfil 24958
            18.12.33  Uniform spaces   cunifsp 24985
            18.12.34  Separated spaces: T0, T1, T2 (Hausdorff) ...   hst1 24987
            18.12.35  Compactness   indcomp 24989
            18.12.36  Connectedness   singempcon 24993
            18.12.37  Topological fields   ctopfld 24997
            18.12.38  Standard topology on RR   intrn 24999
            18.12.39  Standard topology of intervals of RR   stoi 25001
            18.12.40  Cantor's set   cntrset 25002
            18.12.41  Pre-calculus and Cartesian geometry   dmse1 25003
            18.12.42  Extended Real numbers   nolimf 25019
            18.12.43  ( RR ^ N ) and ( CC ^ N )   cplcv 25044
            18.12.44  Calculus   cintvl 25096
            18.12.45  Directed multi graphs   cmgra 25108
            18.12.46  Category and deductive system underlying "structure"   calg 25111
            18.12.47  Deductive systems   cded 25134
            18.12.48  Categories   ccatOLD 25152
            18.12.49  Homsets   chomOLD 25185
            18.12.50  Monomorphisms, Epimorphisms, Isomorphisms   cepiOLD 25203
            18.12.51  Functors   cfuncOLD 25231
            18.12.52  Subcategories   csubcat 25243
            18.12.53  Terminal and initial objects   ciobj 25260
            18.12.54  Sources and sinks   csrce 25265
            18.12.55  Limits and co-limits   clmct 25274
            18.12.56  Product and sum of two objects   cprodo 25277
            18.12.57  Tarski's classes   ctar 25281
            18.12.58  Category Set   ccmrcase 25310
            18.12.59  Grammars, Logics, Machines and Automata   ckln 25380
            18.12.60  Words   cwrd 25381
            18.12.61  Planar geometry   cpoints 25456
      18.13  Mathbox for Jeff Hankins
            18.13.1  Miscellany   a1i13 25600
            18.13.2  Basic topological facts   topbnd 25642
            18.13.3  Topology of the real numbers   reconnOLD 25655
            18.13.4  Refinements   cfne 25659
            18.13.5  Neighborhood bases determine topologies   neibastop1 25708
            18.13.6  Lattice structure of topologies   topmtcl 25712
            18.13.7  Filter bases   fgmin 25719
            18.13.8  Directed sets, nets   tailfval 25721
      18.14  Mathbox for Jeff Madsen
            18.14.1  Logic and set theory   anim12da 25732
            18.14.2  Real and complex numbers; integers   fimaxreOLD 25830
            18.14.3  Sequences and sums   sdclem2 25852
            18.14.4  Topology   unopnOLD 25864
            18.14.5  Metric spaces   metf1o 25869
            18.14.6  Continuous maps and homeomorphisms   constcncf 25878
            18.14.7  Product topologies   txtopiOLD 25886
            18.14.8  Boundedness   ctotbnd 25890
            18.14.9  Isometries   cismty 25922
            18.14.10  Heine-Borel Theorem   heibor1lem 25933
            18.14.11  Banach Fixed Point Theorem   bfplem1 25946
            18.14.12  Euclidean space   crrn 25949
            18.14.13  Intervals (continued)   ismrer1 25962
            18.14.14  Groups and related structures   exidcl 25966
            18.14.15  Rings   rngonegcl 25976
            18.14.16  Ring homomorphisms   crnghom 25991
            18.14.17  Commutative rings   ccring 26020
            18.14.18  Ideals   cidl 26032
            18.14.19  Prime rings and integral domains   cprrng 26071
            18.14.20  Ideal generators   cigen 26084
      18.15  Mathbox for Rodolfo Medina
            18.15.1  Partitions   prtlem60 26103
      18.16  Mathbox for Stefan O'Rear
            18.16.1  Additional elementary logic and set theory   nelss 26151
            18.16.2  Additional theory of functions   fninfp 26154
            18.16.3  Extensions beyond function theory   gsumvsmul 26164
            18.16.4  Additional topology   elrfi 26169
            18.16.5  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 26173
            18.16.6  Algebraic closure systems   cnacs 26177
            18.16.7  Miscellanea 1. Map utilities   constmap 26188
            18.16.8  Miscellanea for polynomials   ofmpteq 26197
            18.16.9  Multivariate polynomials over the integers   cmzpcl 26199
            18.16.10  Miscellanea for Diophantine sets 1   coeq0 26231
            18.16.11  Diophantine sets 1: definitions   cdioph 26234
            18.16.12  Diophantine sets 2 miscellanea   ellz1 26246
            18.16.13  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 26252
            18.16.14  Diophantine sets 3: construction   diophrex 26255
            18.16.15  Diophantine sets 4 miscellanea   2sbcrex 26264
            18.16.16  Diophantine sets 4: Quantification   rexrabdioph 26275
            18.16.17  Diophantine sets 5: Arithmetic sets   rabdiophlem1 26282
            18.16.18  Diophantine sets 6 miscellanea   fz1ssnn 26292
            18.16.19  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 26294
            18.16.20  Pigeonhole Principle and cardinality helpers   fphpd 26299
            18.16.21  A non-closed set of reals is infinite   rencldnfilem 26303
            18.16.22  Miscellanea for Lagrange's theorem   icodiamlt 26305
            18.16.23  Lagrange's rational approximation theorem   irrapxlem1 26307
            18.16.24  Pell equations 1: A nontrivial solution always exists   pellexlem1 26314
            18.16.25  Pell equations 2: Algebraic number theory of the solution set   csquarenn 26321
            18.16.26  Pell equations 3: characterizing fundamental solution   infmrgelbi 26363
            18.16.27  Logarithm laws generalized to an arbitrary base   reglogcl 26375
            18.16.28  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 26383
            18.16.29  X and Y sequences 1: Definition and recurrence laws   crmx 26385
            18.16.30  Ordering and induction lemmas for the integers   monotuz 26426
            18.16.31  X and Y sequences 2: Order properties   rmxypos 26434
            18.16.32  Congruential equations   congtr 26452
            18.16.33  Alternating congruential equations   acongid 26462
            18.16.34  Additional theorems on integer divisibility   bezoutr 26472
            18.16.35  X and Y sequences 3: Divisibility properties   jm2.18 26481
            18.16.36  X and Y sequences 4: Diophantine representability of Y   jm2.27a 26498
            18.16.37  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 26508
            18.16.38  Uncategorized stuff not associated with a major project   setindtr 26517
            18.16.39  More equivalents of the Axiom of Choice   axac10 26526
            18.16.40  Finitely generated left modules   clfig 26565
            18.16.41  Noetherian left modules I   clnm 26573
            18.16.42  Addenda for structure powers   pwssplit0 26587
            18.16.43  Direct sum of left modules   cdsmm 26597
            18.16.44  Free modules   cfrlm 26612
            18.16.45  Every set admits a group structure iff choice   unxpwdom3 26656
            18.16.46  Independent sets and families   clindf 26674
            18.16.47  Characterization of free modules   lmimlbs 26706
            18.16.48  Noetherian rings and left modules II   clnr 26713
            18.16.49  Hilbert's Basis Theorem   cldgis 26725
            18.16.50  Additional material on polynomials [DEPRECATED]   cmnc 26735
            18.16.51  Degree and minimal polynomial of algebraic numbers   cdgraa 26745
            18.16.52  Algebraic integers I   citgo 26762
            18.16.53  Finite cardinality [SO]   en1uniel 26780
            18.16.54  Words in monoids and ordered group sum   issubmd 26783
            18.16.55  Transpositions in the symmetric group   cpmtr 26784
            18.16.56  The sign of a permutation   cpsgn 26814
            18.16.57  The matrix algebra   cmmul 26839
            18.16.58  The determinant   cmdat 26883
            18.16.59  Endomorphism algebra   cmend 26889
            18.16.60  Subfields   csdrg 26903
            18.16.61  Cyclic groups and order   idomrootle 26911
            18.16.62  Cyclotomic polynomials   ccytp 26921
            18.16.63  Miscellaneous topology   fgraphopab 26929
      18.17  Mathbox for Steve Rodriguez
            18.17.1  Miscellanea   iso0 26936
            18.17.2  Function operations   caofcan 26940
            18.17.3  Calculus   lhe4.4ex1a 26946
      18.18  Mathbox for Andrew Salmon
            18.18.1  Principia Mathematica * 10   pm10.12 26953
            18.18.2  Principia Mathematica * 11   2alanimi 26967
            18.18.3  Predicate Calculus   sbeqal1 26997
            18.18.4  Principia Mathematica * 13 and * 14   pm13.13a 27007
            18.18.5  Set Theory   elnev 27038
            18.18.6  Arithmetic   addcomgi 27061
            18.18.7  Geometry   cplusr 27062
      18.19  Mathbox for Glauco Siliprandi
            18.19.1  Miscellanea   ssrexf 27084
            18.19.2  Finite multiplication of numbers and finite multiplication of functions   fmul01 27110
            18.19.3  Limits   clim1fr1 27127
            18.19.4  Derivatives   dvsinexp 27140
            18.19.5  Integrals   ioovolcl 27142
            18.19.6  Stone Weierstrass theorem - real version   stoweidlem1 27150
            18.19.7  Wallis' product for π   wallispilem1 27214
            18.19.8  Stirling's approximation formula for ` n ` factorial   stirlinglem1 27223
      18.20  Mathbox for Jarvin Udandy
      18.21  Mathbox for Alexander van der Vekens
            18.21.1  1.1 Restricted quantification (extension)   r19.32 27325
            18.21.2  1.2 The empty set (extension)   raaan2 27333
            18.21.3  1.3 Restricted uniqueness and "at most one" quantification   rmoimi 27334
            18.21.4  1.4 Analogs to 1.6.6 Existential uniqueness (double quantification)   2reurex 27339
            18.21.5  2. === Alternative definitions of function's and operation's values ===   wdfat 27351
            18.21.6  2.1a Restricted quantification (extension)   ralbinrald 27357
            18.21.7  2.1b The universal class (extension)   nvelim 27358
            18.21.8  2.2 Relations (extension)   sbcrel 27359
            18.21.9  2.3 Functions (extension)   sbcfun 27365
            18.21.10  2.4 Predicate "defined at"   dfateq12d 27372
            18.21.11  2.5 Alternative definition of the value of a function   dfafv2 27375
            18.21.12  2.6 Alternative definition of the value of an operation   aoveq123d 27418
      18.22  Mathbox for David A. Wheeler
            18.22.1  Natural deduction   19.8ad 27448
            18.22.2  Greater than, greater than or equal to.   cge-real 27451
            18.22.3  Hyperbolic trig functions   csinh 27461
            18.22.4  Reciprocal trig functions (sec, csc, cot)   csec 27472
            18.22.5  Identities for "if"   ifnmfalse 27494
            18.22.6  Not-member-of   AnelBC 27495
            18.22.7  Decimal point   cdp2 27496
            18.22.8  Signum (sgn or sign) function   csgn 27504
            18.22.9  Ceiling function   ccei 27514
            18.22.10  Logarithm laws generalized to an arbitrary base - logb   clogb 27518
            18.22.11  Logarithm laws generalized to an arbitrary base - log_   clog_ 27529
            18.22.12  Miscellaneous   5m4e1 27531
      18.23  Mathbox for Alan Sare
            18.23.1  Conventional Metamath proofs, some derived from VD proofs   iidn3 27534
            18.23.2  What is Virtual Deduction?   wvd1 27609
            18.23.3  Virtual Deduction Theorems   df-vd1 27610
            18.23.4  Theorems proved using virtual deduction   trsspwALT 27861
            18.23.5  Theorems proved using virtual deduction with mmj2 assistance   simplbi2VD 27891
            18.23.6  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 27958
            18.23.7  Theorems proved using conjunction-form virtual deduction   elpwgdedVD 27962
            18.23.8  Theorems with VD proofs in conventional notation derived from VD proofs   suctrALT3 27969
            18.23.9  Theorems with a proof in conventional notation automatically derived   notnot2ALT2 27972
      18.24  Mathbox for Jonathan Ben-Naim
            18.24.1  First order logic and set theory   bnj170 27991
            18.24.2  Well founded induction and recursion   bnj110 28158
            18.24.3  The existence of a minimal element in certain classes   bnj69 28308
            18.24.4  Well-founded induction   bnj1204 28310
            18.24.5  Well-founded recursion, part 1 of 3   bnj60 28360
            18.24.6  Well-founded recursion, part 2 of 3   bnj1500 28366
            18.24.7  Well-founded recursion, part 3 of 3   bnj1522 28370
      18.25  Mathbox for Norm Megill
            18.25.1  Obsolete experiments to study ax-12o   ax12-2 28371
            18.25.2  Miscellanea   cnaddcom 28429
            18.25.3  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 28432
            18.25.4  Functionals and kernels of a left vector space (or module)   clfn 28515
            18.25.5  Opposite rings and dual vector spaces   cld 28581
            18.25.6  Ortholattices and orthomodular lattices   cops 28630
            18.25.7  Atomic lattices with covering property   ccvr 28720
            18.25.8  Hilbert lattices   chlt 28808
            18.25.9  Projective geometries based on Hilbert lattices   clln 28948
            18.25.10  Construction of a vector space from a Hilbert lattice   cdlema1N 29248
            18.25.11  Construction of involution and inner product from a Hilbert lattice   clpoN 30938

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