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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  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
      16.7  Conventions
PART 17  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      17.1  Axiomatization of complex pre-Hilbert spaces
      17.2  Inner product and norms
      17.3  Cauchy sequences and completeness axiom
      17.4  Subspaces and projections
      17.5  Properties of Hilbert subspaces
      17.6  Operators on Hilbert spaces
      17.7  States on an Hilbert lattice and Godowski's equation
      17.8  Cover relation, atoms, exchange axiom, and modular symmetry
PART 18  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      18.1  Mathboxes for user contributions
      18.2  Mathbox for Stefan Allan
      18.3  Mathbox for Thierry Arnoux
      18.4  Mathbox for Mario Carneiro
      18.5  Mathbox for Paul Chapman
      18.6  Mathbox for Drahflow
      18.7  Mathbox for Scott Fenton
      18.8  Mathbox for Anthony Hart
      18.9  Mathbox for Chen-Pang He
      18.10  Mathbox for Jeff Hoffman
      18.11  Mathbox for Wolf Lammen
      18.12  Mathbox for Brendan Leahy
      18.13  Mathbox for Frédéric Liné
      18.14  Mathbox for Jeff Hankins
      18.15  Mathbox for Jeff Madsen
      18.16  Mathbox for Rodolfo Medina
      18.17  Mathbox for Stefan O'Rear
      18.18  Mathbox for Steve Rodriguez
      18.19  Mathbox for Andrew Salmon
      18.20  Mathbox for Glauco Siliprandi
      18.21  Mathbox for Jarvin Udandy
      18.22  Mathbox for Alexander van der Vekens
      18.23  Mathbox for David A. Wheeler
      18.24  Mathbox for Alan Sare
      18.25  Mathbox for Jonathan Ben-Naim
      18.26  Mathbox for Norm Megill

Detailed Table of Contents
PART 1  CLASSICAL FIRST ORDER LOGIC WITH EQUALITY
      1.1  Pre-logic
            1.1.1  Inferences for assisting proof development   dummylink 1
      1.2  Propositional calculus
            1.2.1  Recursively define primitive wffs for propositional calculus   wn 3
            1.2.2  The axioms of propositional calculus   ax-1 5
            1.2.3  Logical implication   mp2b 9
            1.2.4  Logical negation   con4d 97
            1.2.5  Logical equivalence   wb 176
            1.2.6  Logical disjunction and conjunction   wo 357
            1.2.7  Miscellaneous theorems of propositional calculus   pm5.21nd 868
            1.2.8  Abbreviated conjunction and disjunction of three wff's   w3o 933
            1.2.9  Logical 'nand' (Sheffer stroke)   wnan 1287
            1.2.10  Logical 'xor'   wxo 1295
            1.2.11  True and false constants   wtru 1307
            1.2.12  Truth tables   truantru 1326
            1.2.13  Auxiliary theorems for Alan Sare's virtual deduction tool, part 1   ee22 1352
            1.2.14  Half-adders and full adders in propositional calculus   whad 1368
      1.3  Other axiomatizations of classical propositional calculus
            1.3.1  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1394
            1.3.2  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1413
            1.3.3  Derive Nicod's axiom from the standard axioms   nic-dfim 1424
            1.3.4  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1430
            1.3.5  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1449
            1.3.6  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1453
            1.3.7  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1468
            1.3.8  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1491
            1.3.9  Derive the Lukasiewicz axioms from the The Russell-Bernays Axioms   rb-bijust 1504
            1.3.10  Stoic logic indemonstrables (Chrysippus of Soli)   mpto1 1523
      1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
            1.4.1  Universal quantifier; define "exists" and "not free"   wal 1527
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1533
            1.4.3  Axiom scheme ax-5 (Quantified Implication)   ax-5 1544
            1.4.4  Axiom scheme ax-17 (Distinctness) - first use of $d   ax-17 1603
            1.4.5  Equality predicate; define substitution   cv 1622
            1.4.6  Axiom scheme ax-9 (Existence)   ax-9 1636
            1.4.7  Axiom scheme ax-8 (Equality)   ax-8 1644
            1.4.8  Membership predicate   wcel 1685
            1.4.9  Axiom schemes ax-13 (Left Membership Equality)   ax-13 1687
            1.4.10  Axiom schemes ax-14 (Right Membership Equality)   ax-14 1689
            1.4.11  Logical redundancy of ax-6 , ax-7 , ax-11 , ax-12   ax9dgen 1691
      1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-6 (Quantified Negation)   ax-6 1704
            1.5.2  Axiom scheme ax-7 (Quantifier Commutation)   ax-7 1709
            1.5.3  Axiom scheme ax-11 (Substitution)   ax-11 1716
            1.5.4  Axiom scheme ax-12 (Quantified Equality)   ax-12 1868
      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 2077
            1.6.2  Rederive new axioms from old: theorems ax5 , ax6 , ax9from9o , ax11 , ax12   ax5 2087
            1.6.3  Legacy theorems using obsolete axioms   ax17o 2098
      1.7  Existential uniqueness
      1.8  Other axiomatizations related to classical predicate calculus
            1.8.1  Predicate calculus with all distinct variables   ax-7d 2235
            1.8.2  Aristotelian logic: Assertic syllogisms   barbara 2241
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 2265
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2270
            2.1.3  Class form not-free predicate   wnfc 2407
            2.1.4  Negated equality and membership   wne 2447
            2.1.5  Restricted quantification   wral 2544
            2.1.6  The universal class   cvv 2789
            2.1.7  Conditional equality (experimental)   wcdeq 2975
            2.1.8  Russell's Paradox   ru 2991
            2.1.9  Proper substitution of classes for sets   wsbc 2992
            2.1.10  Proper substitution of classes for sets into classes   csb 3082
            2.1.11  Define basic set operations and relations   cdif 3150
            2.1.12  Subclasses and subsets   df-ss 3167
            2.1.13  The difference, union, and intersection of two classes   difeq1 3288
            2.1.14  The empty set   c0 3456
            2.1.15  "Weak deduction theorem" for set theory   cif 3566
            2.1.16  Power classes   cpw 3626
            2.1.17  Unordered and ordered pairs   csn 3641
            2.1.18  The union of a class   cuni 3828
            2.1.19  The intersection of a class   cint 3863
            2.1.20  Indexed union and intersection   ciun 3906
            2.1.21  Disjointness   wdisj 3994
            2.1.22  Binary relations   wbr 4024
            2.1.23  Ordered-pair class abstractions (class builders)   copab 4077
            2.1.24  Transitive classes   wtr 4114
      2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 4132
            2.2.2  Derive the Axiom of Separation   axsep 4141
            2.2.3  Derive the Null Set Axiom   zfnuleu 4147
            2.2.4  Theorems requiring subset and intersection existence   nalset 4152
            2.2.5  Theorems requiring empty set existence   class2set 4177
      2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4187
            2.3.2  Derive the Axiom of Pairing   zfpair 4211
            2.3.3  Ordered pair theorem   opnz 4241
            2.3.4  Ordered-pair class abstractions (cont.)   opabid 4270
            2.3.5  Power class of union and intersection   pwin 4296
            2.3.6  Epsilon and identity relations   cep 4302
            2.3.7  Partial and complete ordering   wpo 4311
            2.3.8  Founded and well-ordering relations   wfr 4348
            2.3.9  Ordinals   word 4390
      2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4511
            2.4.2  Ordinals (continued)   ordon 4573
            2.4.3  Transfinite induction   tfi 4643
            2.4.4  The natural numbers (i.e. finite ordinals)   com 4655
            2.4.5  Peano's postulates   peano1 4674
            2.4.6  Finite induction (for finite ordinals)   find 4680
            2.4.7  Relations   cxp 4686
            2.4.8  Functions   wfun 5215
            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 10619
            5.5.4  Real number intervals   cioo 10652
            5.5.5  Finite intervals of integers   cfz 10778
            5.5.6  Half-open integer ranges   cfzo 10866
      5.6  Elementary integer functions
            5.6.1  The floor (greatest integer) function   cfl 10920
            5.6.2  The modulo (remainder) operation   cmo 10969
            5.6.3  The infinite sequence builder "seq"   om2uz0i 11006
            5.6.4  Integer powers   cexp 11100
            5.6.5  Ordered pair theorem for nonnegative integers   nn0le2msqi 11278
            5.6.6  Factorial function   cfa 11284
            5.6.7  The binomial coefficient operation   cbc 11311
            5.6.8  The ` # ` (finite set size) function   chash 11333
            5.6.9  Words over a set   cword 11399
            5.6.10  Longer string literals   cs2 11487
      5.7  Elementary real and complex functions
            5.7.1  The "shift" operation   cshi 11557
            5.7.2  Real and imaginary parts; conjugate   ccj 11577
            5.7.3  Square root; absolute value   csqr 11714
      5.8  Elementary limits and convergence
            5.8.1  Superior limit (lim sup)   clsp 11940
            5.8.2  Limits   cli 11954
            5.8.3  Finite and infinite sums   csu 12154
            5.8.4  The binomial theorem   binomlem 12283
            5.8.5  The inclusion/exclusion principle   incexclem 12291
            5.8.6  Infinite sums (cont.)   isumshft 12294
            5.8.7  Miscellaneous converging and diverging sequences   divrcnv 12307
            5.8.8  Arithmetic series   arisum 12314
            5.8.9  Geometric series   expcnv 12318
            5.8.10  Ratio test for infinite series convergence   cvgrat 12335
            5.8.11  Mertens' theorem   mertenslem1 12336
      5.9  Elementary trigonometry
            5.9.1  The exponential, sine, and cosine functions   ce 12339
            5.9.2  _e is irrational   eirrlem 12478
      5.10  Cardinality of real and complex number subsets
            5.10.1  Countability of integers and rationals   xpnnen 12483
            5.10.2  The reals are uncountable   rpnnen2lem1 12489
PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqr2irrlem 12522
            6.1.2  Some Number sets are chains of proper subsets   nthruc 12525
            6.1.3  The divides relation   cdivides 12527
            6.1.4  The division algorithm   divalglem0 12588
            6.1.5  Bit sequences   cbits 12606
            6.1.6  The greatest common divisor operator   cgcd 12681
            6.1.7  Bézout's identity   bezoutlem1 12713
            6.1.8  Algorithms   nn0seqcvgd 12736
            6.1.9  Euclid's Algorithm   eucalgval2 12747
      6.2  Elementary prime number theory
            6.2.1  Elementary properties   cprime 12754
            6.2.2  Properties of the canonical representation of a rational   cnumer 12800
            6.2.3  Euler's theorem   codz 12827
            6.2.4  Pythagorean Triples   coprimeprodsq 12858
            6.2.5  The prime count function   cpc 12885
            6.2.6  Pocklington's theorem   prmpwdvds 12947
            6.2.7  Infinite primes theorem   unbenlem 12951
            6.2.8  Sum of prime reciprocals   prmreclem1 12959
            6.2.9  Fundamental theorem of arithmetic   1arithlem1 12966
            6.2.10  Lagrange's four-square theorem   cgz 12972
            6.2.11  Van der Waerden's theorem   cvdwa 13008
            6.2.12  Ramsey's theorem   cram 13042
            6.2.13  Decimal arithmetic (cont.)   dec2dvds 13074
            6.2.14  Specific prime numbers   4nprm 13102
            6.2.15  Very large primes   1259lem1 13125
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
            7.1.1  Basic definitions   cstr 13140
            7.1.2  Slot definitions   cplusg 13204
            7.1.3  Definition of the structure product   crest 13321
            7.1.4  Definition of the structure quotient   cordt 13394
      7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 13504
            7.2.2  Independent sets in a Moore system   mrisval 13528
            7.2.3  Algebraic closure systems   isacs 13549
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
            8.1.1  Categories   ccat 13562
            8.1.2  Opposite category   coppc 13610
            8.1.3  Monomorphisms and epimorphisms   cmon 13627
            8.1.4  Sections, inverses, isomorphisms   csect 13643
            8.1.5  Subcategories   cssc 13680
            8.1.6  Functors   cfunc 13724
            8.1.7  Full & faithful functors   cful 13772
            8.1.8  Natural transformations and the functor category   cnat 13811
      8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 13881
      8.3  Examples of categories
            8.3.1  The category of sets   csetc 13903
            8.3.2  The category of categories   ccatc 13922
      8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 13938
            8.4.2  Functor evaluation   cevlf 13979
            8.4.3  Hom functor   chof 14018
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 14070
            9.2.2  Lattices   clat 14147
            9.2.3  The dual of an ordered set   codu 14228
            9.2.4  Subset order structures   cipo 14250
            9.2.5  Distributive lattices   latmass 14287
            9.2.6  Posets and lattices as relations   cps 14297
            9.2.7  Directed sets, nets   cdir 14346
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
            10.1.1  Definition and basic properties   cmnd 14357
            10.1.2  Monoid homomorphisms and submonoids   cmhm 14409
            10.1.3  Ordered group sum operation   gsumvallem1 14444
            10.1.4  Free monoids   cfrmd 14465
      10.2  Groups
            10.2.1  Definition and basic properties   df-grp 14485
            10.2.2  Subgroups and Quotient groups   csubg 14611
            10.2.3  Elementary theory of group homomorphisms   cghm 14676
            10.2.4  Isomorphisms of groups   cgim 14717
            10.2.5  Group actions   cga 14739
            10.2.6  Symmetry groups and Cayley's Theorem   csymg 14765
            10.2.7  Centralizers and centers   ccntz 14787
            10.2.8  The opposite group   coppg 14814
            10.2.9  p-Groups and Sylow groups; Sylow's theorems   cod 14836
            10.2.10  Direct products   clsm 14941
            10.2.11  Free groups   cefg 15011
      10.3  Abelian groups
            10.3.1  Definition and basic properties   ccmn 15085
            10.3.2  Cyclic groups   ccyg 15160
            10.3.3  Group sum operation   gsumval3a 15185
            10.3.4  Internal direct products   cdprd 15227
            10.3.5  The Fundamental Theorem of Abelian Groups   ablfacrplem 15296
      10.4  Rings
            10.4.1  Multiplicative Group   cmgp 15321
            10.4.2  Definition and basic properties   crg 15333
            10.4.3  Opposite ring   coppr 15400
            10.4.4  Divisibility   cdsr 15416
            10.4.5  Ring homomorphisms   crh 15490
      10.5  Division rings and Fields
            10.5.1  Definition and basic properties   cdr 15508
            10.5.2  Subrings of a ring   csubrg 15537
            10.5.3  Absolute value (abstract algebra)   cabv 15577
            10.5.4  Star rings   cstf 15604
      10.6  Left Modules
            10.6.1  Definition and basic properties   clmod 15623
            10.6.2  Subspaces and spans in a left module   clss 15685
            10.6.3  Homomorphisms and isomorphisms of left modules   clmhm 15772
            10.6.4  Subspace sum; bases for a left module   clbs 15823
      10.7  Vector Spaces
            10.7.1  Definition and basic properties   clvec 15851
      10.8  Ideals
            10.8.1  The subring algebra; ideals   csra 15917
            10.8.2  Two-sided ideals and quotient rings   c2idl 15979
            10.8.3  Principal ideal rings. Divisibility in the integers   clpidl 15989
            10.8.4  Nonzero rings   cnzr 16005
            10.8.5  Left regular elements. More kinds of ring   crlreg 16016
      10.9  Associative algebras
            10.9.1  Definition and basic properties   casa 16046
      10.10  Abstract Multivariate Polynomials
            10.10.1  Definition and basic properties   cmps 16083
            10.10.2  Polynomial evaluation   evlslem4 16241
            10.10.3  Univariate Polynomials   cps1 16246
      10.11  The complex numbers as an extensible structure
            10.11.1  Definition and basic properties   cxmt 16365
            10.11.2  Algebraic constructions based on the complexes   czrh 16447
      10.12  Hilbert spaces
            10.12.1  Definition and basic properties   cphl 16524
            10.12.2  Orthocomplements and closed subspaces   cocv 16556
            10.12.3  Orthogonal projection and orthonormal bases   cpj 16596
PART 11  BASIC TOPOLOGY
      11.1  Topology
            11.1.1  Topological spaces   ctop 16627
            11.1.2  TopBases for topologies   isbasisg 16681
            11.1.3  Examples of topologies   distop 16729
            11.1.4  Closure and interior   ccld 16749
            11.1.5  Neighborhoods   cnei 16830
            11.1.6  Limit points and perfect sets   clp 16862
            11.1.7  Subspace topologies   restrcl 16884
            11.1.8  Order topology   ordtbaslem 16914
            11.1.9  Limits and Continuity in topological spaces   ccn 16950
            11.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 17030
            11.1.11  Compactness   ccmp 17109
            11.1.12  Connectedness   ccon 17133
            11.1.13  First- and second-countability   c1stc 17159
            11.1.14  Local topological properties   clly 17186
            11.1.15  Compactly generated spaces   ckgen 17224
            11.1.16  Product topologies   ctx 17251
            11.1.17  Continuous function-builders   cnmptid 17351
            11.1.18  Quotient maps and quotient topology   ckq 17380
            11.1.19  Homeomorphisms   chmeo 17440
      11.2  Filters and filter bases
            11.2.1  Filter Bases   cfbas 17514
            11.2.2  Filters   cfil 17536
            11.2.3  Ultrafilters   cufil 17590
            11.2.4  Filter limits   cfm 17624
            11.2.5  Topological groups   ctmd 17749
            11.2.6  Infinite group sum on topological groups   ctsu 17804
            11.2.7  Topological rings, fields, vector spaces   ctrg 17834
      11.3  Metric spaces
            11.3.1  Basic metric space properties   cxme 17878
            11.3.2  Metric space balls   blfval 17943
            11.3.3  Open sets of a metric space   mopnval 17980
            11.3.4  Continuity in metric spaces   metcnp3 18082
            11.3.5  Examples of metric spaces   dscmet 18091
            11.3.6  Normed algebraic structures   cnm 18095
            11.3.7  Normed space homomorphisms (bounded linear operators)   cnmo 18210
            11.3.8  Topology on the Reals   qtopbaslem 18263
            11.3.9  Topological definitions using the reals   cii 18375
            11.3.10  Path homotopy   chtpy 18461
            11.3.11  The fundamental group   cpco 18494
      11.4  Complex metric vector spaces
            11.4.1  Complex left modules   cclm 18556
            11.4.2  Complex pre-Hilbert space   ccph 18598
            11.4.3  Convergence and completeness   ccfil 18674
            11.4.4  Baire's Category Theorem   bcthlem1 18742
            11.4.5  Banach spaces and complex Hilbert spaces   ccms 18750
            11.4.6  Minimizing Vector Theorem   minveclem1 18784
            11.4.7  Projection theorem   pjthlem1 18797
PART 12  BASIC REAL AND COMPLEX ANALYSIS
      12.1  Continuity
            12.1.1  Intermediate value theorem   pmltpclem1 18804
      12.2  Integrals
            12.2.1  Lebesgue measure   covol 18818
            12.2.2  Lebesgue integration   cmbf 18965
      12.3  Derivatives
            12.3.1  Real and Complex Differentiation   climc 19208
PART 13  BASIC REAL AND COMPLEX FUNCTIONS
      13.1  Polynomials
            13.1.1  Abstract polynomials, continued   evlslem6 19393
            13.1.2  Polynomial degrees   cmdg 19435
            13.1.3  The division algorithm for univariate polynomials   cmn1 19507
            13.1.4  Elementary properties of complex polynomials   cply 19562
            13.1.5  The Division algorithm for polynomials   cquot 19666
            13.1.6  Algebraic numbers   caa 19690
            13.1.7  Liouville's approximation theorem   aalioulem1 19708
      13.2  Sequences and series
            13.2.1  Taylor polynomials and Taylor's theorem   ctayl 19728
            13.2.2  Uniform convergence   culm 19751
            13.2.3  Power series   pserval 19782
      13.3  Basic trigonometry
            13.3.1  The exponential, sine, and cosine functions (cont.)   efcn 19815
            13.3.2  Properties of pi = 3.14159...   pilem1 19823
            13.3.3  Mapping of the exponential function   efgh 19899
            13.3.4  The natural logarithm on complex numbers   clog 19908
            13.3.5  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 20095
            13.3.6  Solutions of quadratic, cubic, and quartic equations   quad2 20131
            13.3.7  Inverse trigonometric functions   casin 20154
            13.3.8  The Birthday Problem   log2ublem1 20238
            13.3.9  Areas in R^2   carea 20246
            13.3.10  More miscellaneous converging sequences   rlimcnp 20256
            13.3.11  Inequality of arithmetic and geometric means   cvxcl 20275
            13.3.12  Euler-Mascheroni constant   cem 20282
      13.4  Basic number theory
            13.4.1  Wilson's theorem   wilthlem1 20302
            13.4.2  The Fundamental Theorem of Algebra   ftalem1 20306
            13.4.3  The Basel problem (ζ(2) = π2/6)   basellem1 20314
            13.4.4  Number-theoretical functions   ccht 20324
            13.4.5  Perfect Number Theorem   mersenne 20462
            13.4.6  Characters of Z/nZ   cdchr 20467
            13.4.7  Bertrand's postulate   bcctr 20510
            13.4.8  Legendre symbol   clgs 20529
            13.4.9  Quadratic Reciprocity   lgseisenlem1 20584
            13.4.10  All primes 4n+1 are the sum of two squares   2sqlem1 20598
            13.4.11  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 20614
            13.4.12  The Prime Number Theorem   mudivsum 20675
            13.4.13  Ostrowski's theorem   abvcxp 20760
PART 14  MISCELLANEA
      14.1  Definitional Examples
      14.2  Natural deduction examples
      14.3  Humor
            14.3.1  April Fool's theorem   avril1 20830
      14.4  (Future - to be reviewed and classified)
            14.4.1  Planar incidence geometry   cplig 20836
            14.4.2  Algebra preliminaries   crpm 20841
            14.4.3  Transitive closure   ctcl 20843
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 20847
            15.1.2  Definition and basic properties of Abelian groups   cablo 20942
            15.1.3  Subgroups   csubgo 20962
            15.1.4  Operation properties   cass 20973
            15.1.5  Group-like structures   cmagm 20979
            15.1.6  Examples of Abelian groups   ablosn 21008
            15.1.7  Group homomorphism and isomorphism   cghom 21018
      15.2  Additional material on Rings and Fields
            15.2.1  Definition and basic properties   crngo 21036
            15.2.2  Examples of rings   cnrngo 21064
            15.2.3  Division Rings   cdrng 21066
            15.2.4  Star Fields   csfld 21069
            15.2.5  Fields and Rings   ccm2 21071
PART 16  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      16.1  Complex vector spaces
            16.1.1  Definition and basic properties   cvc 21095
            16.1.2  Examples of complex vector spaces   cncvc 21133
      16.2  Normed complex vector spaces
            16.2.1  Definition and basic properties   cnv 21134
            16.2.2  Examples of normed complex vector spaces   cnnv 21239
            16.2.3  Induced metric of a normed complex vector space   imsval 21248
            16.2.4  Inner product   cdip 21267
            16.2.5  Subspaces   css 21291
      16.3  Operators on complex vector spaces
            16.3.1  Definitions and basic properties   clno 21312
      16.4  Inner product (pre-Hilbert) spaces
            16.4.1  Definition and basic properties   ccphlo 21384
            16.4.2  Examples of pre-Hilbert spaces   cncph 21391
            16.4.3  Properties of pre-Hilbert spaces   isph 21394
      16.5  Complex Banach spaces
            16.5.1  Definition and basic properties   ccbn 21435
            16.5.2  Examples of complex Banach spaces   cnbn 21442
            16.5.3  Uniform Boundedness Theorem   ubthlem1 21443
            16.5.4  Minimizing Vector Theorem   minvecolem1 21447
      16.6  Complex Hilbert spaces
            16.6.1  Definition and basic properties   chlo 21458
            16.6.2  Standard axioms for a complex Hilbert space   hlex 21471
            16.6.3  Examples of complex Hilbert spaces   cnchl 21489
            16.6.4  Subspaces   ssphl 21490
            16.6.5  Hellinger-Toeplitz Theorem   htthlem 21491
      16.7  Conventions
PART 17  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      17.1  Axiomatization of complex pre-Hilbert spaces
            17.1.1  Basic Hilbert space definitions   chil 21495
            17.1.2  Preliminary ZFC lemmas   df-hnorm 21544
            17.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 21557
            17.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 21575
            17.1.5  Vector operations   hvmulex 21587
            17.1.6  Inner product postulates for a Hilbert space   ax-hfi 21654
      17.2  Inner product and norms
            17.2.1  Inner product   his5 21661
            17.2.2  Norms   dfhnorm2 21697
            17.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 21735
            17.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 21754
      17.3  Cauchy sequences and completeness axiom
            17.3.1  Cauchy sequences and limits   hcau 21759
            17.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 21769
            17.3.3  Completeness postulate for a Hilbert space   ax-hcompl 21777
            17.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 21778
      17.4  Subspaces and projections
            17.4.1  Subspaces   df-sh 21782
            17.4.2  Closed subspaces   df-ch 21797
            17.4.3  Orthocomplements   df-oc 21827
            17.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 21883
            17.4.5  Projection theorem   pjhthlem1 21966
            17.4.6  Projectors   df-pjh 21970
      17.5  Properties of Hilbert subspaces
            17.5.1  Orthomodular law   omlsilem 21977
            17.5.2  Projectors (cont.)   pjhtheu2 21991
            17.5.3  Hilbert lattice operations   sh0le 22015
            17.5.4  Span (cont.) and one-dimensional subspaces   spansn0 22116
            17.5.5  Commutes relation for Hilbert lattice elements   df-cm 22158
            17.5.6  Foulis-Holland theorem   fh1 22193
            17.5.7  Quantum Logic Explorer axioms   qlax1i 22202
            17.5.8  Orthogonal subspaces   chscllem1 22212
            17.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 22229
            17.5.10  Projectors (cont.)   pjorthi 22244
            17.5.11  Mayet's equation E_3   mayete3i 22303
      17.6  Operators on Hilbert spaces
            17.6.1  Operator sum, difference, and scalar multiplication   df-hosum 22306
            17.6.2  Zero and identity operators   df-h0op 22324
            17.6.3  Operations on Hilbert space operators   hoaddcl 22334
            17.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 22415
            17.6.5  Linear and continuous functionals and norms   df-nmfn 22421
            17.6.6  Adjoint   df-adjh 22425
            17.6.7  Dirac bra-ket notation   df-bra 22426
            17.6.8  Positive operators   df-leop 22428
            17.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 22429
            17.6.10  Theorems about operators and functionals   nmopval 22432
            17.6.11  Riesz lemma   riesz3i 22638
            17.6.12  Adjoints (cont.)   cnlnadjlem1 22643
            17.6.13  Quantum computation error bound theorem   unierri 22680
            17.6.14  Dirac bra-ket notation (cont.)   branmfn 22681
            17.6.15  Positive operators (cont.)   leopg 22698
            17.6.16  Projectors as operators   pjhmopi 22722
      17.7  States on an Hilbert lattice and Godowski's equation
            17.7.1  States on a Hilbert lattice   df-st 22787
            17.7.2  Godowski's equation   golem1 22847
      17.8  Cover relation, atoms, exchange axiom, and modular symmetry
            17.8.1  Covers relation; modular pairs   df-cv 22855
            17.8.2  Atoms   df-at 22914
            17.8.3  Superposition principle   superpos 22930
            17.8.4  Atoms, exchange and covering properties, atomicity   chcv1 22931
            17.8.5  Irreducibility   chirredlem1 22966
            17.8.6  Atoms (cont.)   atcvat3i 22972
            17.8.7  Modular symmetry   mdsymlem1 22979
PART 18  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      18.1  Mathboxes for user contributions
            18.1.1  Mathbox guidelines   mathbox 23018
      18.2  Mathbox for Stefan Allan
      18.3  Mathbox for Thierry Arnoux
            18.3.1  Bertrand's Ballot Problem   ballotlemoex 23040
      18.4  Mathbox for Mario Carneiro
            18.4.1  Miscellaneous stuff   quartfull 23093
            18.4.2  Zeta function   czeta 23094
            18.4.3  Gamma function   clgam 23097
            18.4.4  Derangements and the Subfactorial   deranglem 23104
            18.4.5  The Erdős-Szekeres theorem   erdszelem1 23129
            18.4.6  The Kuratowski closure-complement theorem   kur14lem1 23144
            18.4.7  Retracts and sections   cretr 23155
            18.4.8  Path-connected and simply connected spaces   cpcon 23157
            18.4.9  Covering maps   ccvm 23193
            18.4.10  Undirected multigraphs   cumg 23267
            18.4.11  Normal numbers   snmlff 23319
            18.4.12  Godel-sets of formulas   cgoe 23323
            18.4.13  Models of ZF   cgze 23351
            18.4.14  Splitting fields   citr 23365
            18.4.15  p-adic number fields   czr 23381
      18.5  Mathbox for Paul Chapman
            18.5.1  Group homomorphism and isomorphism   ghomgrpilem1 23399
            18.5.2  Real and complex numbers (cont.)   climuzcnv 23411
            18.5.3  Miscellaneous theorems   elfzm12 23415
      18.6  Mathbox for Drahflow
      18.7  Mathbox for Scott Fenton
            18.7.1  ZFC Axioms in primitive form   axextprim 23454
            18.7.2  Untangled classes   untelirr 23461
            18.7.3  Extra propositional calculus theorems   3orel1 23468
            18.7.4  Misc. Useful Theorems   nepss 23479
            18.7.5  Properties of reals and complexes   sqdivzi 23485
            18.7.6  Greatest common divisor and divisibility   pdivsq 23508
            18.7.7  Properties of relationships   brtp 23512
            18.7.8  Properties of functions and mappings   funpsstri 23525
            18.7.9  Epsilon induction   setinds 23538
            18.7.10  Ordinal numbers   elpotr 23541
            18.7.11  Defined equality axioms   axextdfeq 23558
            18.7.12  Hypothesis builders   hbntg 23566
            18.7.13  The Predecessor Class   cpred 23571
            18.7.14  (Trans)finite Recursion Theorems   tfisg 23608
            18.7.15  Well-founded induction   tz6.26 23609
            18.7.16  Transitive closure under a relationship   ctrpred 23624
            18.7.17  Founded Induction   frmin 23646
            18.7.18  Ordering Ordinal Sequences   orderseqlem 23656
            18.7.19  Well-founded recursion   wfr3g 23659
            18.7.20  Transfinite recursion via Well-founded recursion   tfrALTlem 23680
            18.7.21  Founded Recursion   frr3g 23684
            18.7.22  Surreal Numbers   csur 23698
            18.7.23  Surreal Numbers: Ordering   axsltsolem1 23725
            18.7.24  Surreal Numbers: Birthday Function   axbday 23732
            18.7.25  Surreal Numbers: Density   axdenselem1 23739
            18.7.26  Surreal Numbers: Full-Eta Property   axfelem1 23750
            18.7.27  Symmetric difference   csymdif 23772
            18.7.28  Quantifier-free definitions   ctxp 23784
            18.7.29  Alternate ordered pairs   caltop 23900
            18.7.30  Tarskian geometry   cee 23926
            18.7.31  Tarski's axioms for geometry   axdimuniq 23951
            18.7.32  Congruence properties   cofs 24015
            18.7.33  Betweenness properties   btwntriv2 24045
            18.7.34  Segment Transportation   ctransport 24062
            18.7.35  Properties relating betweenness and congruence   cifs 24068
            18.7.36  Connectivity of betweenness   btwnconn1lem1 24120
            18.7.37  Segment less than or equal to   csegle 24139
            18.7.38  Outside of relationship   coutsideof 24152
            18.7.39  Lines and Rays   cline2 24167
            18.7.40  Bernoulli polynomials and sums of k-th powers   cbp 24191
            18.7.41  Rank theorems   rankung 24206
            18.7.42  Hereditarily Finite Sets   chf 24212
      18.8  Mathbox for Anthony Hart
            18.8.1  Propositional Calculus   tb-ax1 24227
            18.8.2  Predicate Calculus   quantriv 24249
            18.8.3  Misc. Single Axiom Systems   meran1 24260
            18.8.4  Connective Symmetry   negsym1 24266
      18.9  Mathbox for Chen-Pang He
            18.9.1  Ordinal topology   ontopbas 24277
      18.10  Mathbox for Jeff Hoffman
            18.10.1  Inferences for finite induction on generic function values   fveleq 24300
            18.10.2  gdc.mm   nnssi2 24304
      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 24342
            18.13.2  Propositional and predicate calculus   neleq12d 24343
            18.13.3  Linear temporal logic   wbox 24380
            18.13.4  Operations   ssoprab2g 24442
            18.13.5  General Set Theory   uninqs 24449
            18.13.6  The "maps to" notation   cmpfunOLD 24553
            18.13.7  Cartesian Products   cpro 24561
            18.13.8  Operations on subsets and functions   ccst 24583
            18.13.9  Arithmetic   3timesi 24589
            18.13.10  Lattice (algebraic definition)   clatalg 24592
            18.13.11  Currying and Partial Mappings   ccur1 24605
            18.13.12  Order theory (Extensible Structure Builder)   corhom 24618
            18.13.13  Order theory   cpresetrel 24626
            18.13.14  Finite composites ( i. e. finite sums, products ... )   cprd 24709
            18.13.15  Operation properties   ccm1 24742
            18.13.16  Groups and related structures   ridlideq 24746
            18.13.17  Free structures   csubsmg 24794
            18.13.18  Translations   trdom2 24802
            18.13.19  Fields and Rings   com2i 24827
            18.13.20  Ideals   cidln 24854
            18.13.21  Generic modules and vector spaces (New Structure builder)   cact 24858
            18.13.22  Generic modules and vector spaces   cvec 24860
            18.13.23  Real vector spaces   cvr 24900
            18.13.24  Matrices   cmmat 24904
            18.13.25  Affine spaces   craffsp 24910
            18.13.26  Intervals of reals and extended reals   bsi 24912
            18.13.27  Topology   topnem 24923
            18.13.28  Continuous functions   cnrsfin 24936
            18.13.29  Homeomorphisms   dmhmph 24944
            18.13.30  Initial and final topologies   intopcoaconlem3b 24949
            18.13.31  Filters   efilcp 24963
            18.13.32  Limits   plimfil 24969
            18.13.33  Uniform spaces   cunifsp 24996
            18.13.34  Separated spaces: T0, T1, T2 (Hausdorff) ...   hst1 24998
            18.13.35  Compactness   indcomp 25000
            18.13.36  Connectedness   singempcon 25004
            18.13.37  Topological fields   ctopfld 25008
            18.13.38  Standard topology on RR   intrn 25010
            18.13.39  Standard topology of intervals of RR   stoi 25012
            18.13.40  Cantor's set   cntrset 25013
            18.13.41  Pre-calculus and Cartesian geometry   dmse1 25014
            18.13.42  Extended Real numbers   nolimf 25030
            18.13.43  ( RR ^ N ) and ( CC ^ N )   cplcv 25055
            18.13.44  Calculus   cintvl 25107
            18.13.45  Directed multi graphs   cmgra 25119
            18.13.46  Category and deductive system underlying "structure"   calg 25122
            18.13.47  Deductive systems   cded 25145
            18.13.48  Categories   ccatOLD 25163
            18.13.49  Homsets   chomOLD 25196
            18.13.50  Monomorphisms, Epimorphisms, Isomorphisms   cepiOLD 25214
            18.13.51  Functors   cfuncOLD 25242
            18.13.52  Subcategories   csubcat 25254
            18.13.53  Terminal and initial objects   ciobj 25271
            18.13.54  Sources and sinks   csrce 25276
            18.13.55  Limits and co-limits   clmct 25285
            18.13.56  Product and sum of two objects   cprodo 25288
            18.13.57  Tarski's classes   ctar 25292
            18.13.58  Category Set   ccmrcase 25321
            18.13.59  Grammars, Logics, Machines and Automata   ckln 25391
            18.13.60  Words   cwrd 25392
            18.13.61  Planar geometry   cpoints 25467
      18.14  Mathbox for Jeff Hankins
            18.14.1  Miscellany   a1i13 25611
            18.14.2  Basic topological facts   topbnd 25653
            18.14.3  Topology of the real numbers   reconnOLD 25666
            18.14.4  Refinements   cfne 25670
            18.14.5  Neighborhood bases determine topologies   neibastop1 25719
            18.14.6  Lattice structure of topologies   topmtcl 25723
            18.14.7  Filter bases   fgmin 25730
            18.14.8  Directed sets, nets   tailfval 25732
      18.15  Mathbox for Jeff Madsen
            18.15.1  Logic and set theory   anim12da 25743
            18.15.2  Real and complex numbers; integers   fimaxreOLD 25841
            18.15.3  Sequences and sums   sdclem2 25863
            18.15.4  Topology   unopnOLD 25875
            18.15.5  Metric spaces   metf1o 25880
            18.15.6  Continuous maps and homeomorphisms   constcncf 25889
            18.15.7  Product topologies   txtopiOLD 25897
            18.15.8  Boundedness   ctotbnd 25901
            18.15.9  Isometries   cismty 25933
            18.15.10  Heine-Borel Theorem   heibor1lem 25944
            18.15.11  Banach Fixed Point Theorem   bfplem1 25957
            18.15.12  Euclidean space   crrn 25960
            18.15.13  Intervals (continued)   ismrer1 25973
            18.15.14  Groups and related structures   exidcl 25977
            18.15.15  Rings   rngonegcl 25987
            18.15.16  Ring homomorphisms   crnghom 26002
            18.15.17  Commutative rings   ccring 26031
            18.15.18  Ideals   cidl 26043
            18.15.19  Prime rings and integral domains   cprrng 26082
            18.15.20  Ideal generators   cigen 26095
      18.16  Mathbox for Rodolfo Medina
            18.16.1  Partitions   prtlem60 26114
      18.17  Mathbox for Stefan O'Rear
            18.17.1  Additional elementary logic and set theory   nelss 26162
            18.17.2  Additional theory of functions   fninfp 26165
            18.17.3  Extensions beyond function theory   gsumvsmul 26175
            18.17.4  Additional topology   elrfi 26180
            18.17.5  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 26184
            18.17.6  Algebraic closure systems   cnacs 26188
            18.17.7  Miscellanea 1. Map utilities   constmap 26199
            18.17.8  Miscellanea for polynomials   ofmpteq 26208
            18.17.9  Multivariate polynomials over the integers   cmzpcl 26210
            18.17.10  Miscellanea for Diophantine sets 1   coeq0 26242
            18.17.11  Diophantine sets 1: definitions   cdioph 26245
            18.17.12  Diophantine sets 2 miscellanea   ellz1 26257
            18.17.13  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 26263
            18.17.14  Diophantine sets 3: construction   diophrex 26266
            18.17.15  Diophantine sets 4 miscellanea   2sbcrex 26275
            18.17.16  Diophantine sets 4: Quantification   rexrabdioph 26286
            18.17.17  Diophantine sets 5: Arithmetic sets   rabdiophlem1 26293
            18.17.18  Diophantine sets 6 miscellanea   fz1ssnn 26303
            18.17.19  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 26305
            18.17.20  Pigeonhole Principle and cardinality helpers   fphpd 26310
            18.17.21  A non-closed set of reals is infinite   rencldnfilem 26314
            18.17.22  Miscellanea for Lagrange's theorem   icodiamlt 26316
            18.17.23  Lagrange's rational approximation theorem   irrapxlem1 26318
            18.17.24  Pell equations 1: A nontrivial solution always exists   pellexlem1 26325
            18.17.25  Pell equations 2: Algebraic number theory of the solution set   csquarenn 26332
            18.17.26  Pell equations 3: characterizing fundamental solution   infmrgelbi 26374
            18.17.27  Logarithm laws generalized to an arbitrary base   reglogcl 26386
            18.17.28  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 26394
            18.17.29  X and Y sequences 1: Definition and recurrence laws   crmx 26396
            18.17.30  Ordering and induction lemmas for the integers   monotuz 26437
            18.17.31  X and Y sequences 2: Order properties   rmxypos 26445
            18.17.32  Congruential equations   congtr 26463
            18.17.33  Alternating congruential equations   acongid 26473
            18.17.34  Additional theorems on integer divisibility   bezoutr 26483
            18.17.35  X and Y sequences 3: Divisibility properties   jm2.18 26492
            18.17.36  X and Y sequences 4: Diophantine representability of Y   jm2.27a 26509
            18.17.37  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 26519
            18.17.38  Uncategorized stuff not associated with a major project   setindtr 26528
            18.17.39  More equivalents of the Axiom of Choice   axac10 26537
            18.17.40  Finitely generated left modules   clfig 26576
            18.17.41  Noetherian left modules I   clnm 26584
            18.17.42  Addenda for structure powers   pwssplit0 26598
            18.17.43  Direct sum of left modules   cdsmm 26608
            18.17.44  Free modules   cfrlm 26623
            18.17.45  Every set admits a group structure iff choice   unxpwdom3 26667
            18.17.46  Independent sets and families   clindf 26685
            18.17.47  Characterization of free modules   lmimlbs 26717
            18.17.48  Noetherian rings and left modules II   clnr 26724
            18.17.49  Hilbert's Basis Theorem   cldgis 26736
            18.17.50  Additional material on polynomials [DEPRECATED]   cmnc 26746
            18.17.51  Degree and minimal polynomial of algebraic numbers   cdgraa 26756
            18.17.52  Algebraic integers I   citgo 26773
            18.17.53  Finite cardinality [SO]   en1uniel 26791
            18.17.54  Words in monoids and ordered group sum   issubmd 26794
            18.17.55  Transpositions in the symmetric group   cpmtr 26795
            18.17.56  The sign of a permutation   cpsgn 26825
            18.17.57  The matrix algebra   cmmul 26850
            18.17.58  The determinant   cmdat 26894
            18.17.59  Endomorphism algebra   cmend 26900
            18.17.60  Subfields   csdrg 26914
            18.17.61  Cyclic groups and order   idomrootle 26922
            18.17.62  Cyclotomic polynomials   ccytp 26932
            18.17.63  Miscellaneous topology   fgraphopab 26940
      18.18  Mathbox for Steve Rodriguez
            18.18.1  Miscellanea   iso0 26947
            18.18.2  Function operations   caofcan 26951
            18.18.3  Calculus   lhe4.4ex1a 26957
      18.19  Mathbox for Andrew Salmon
            18.19.1  Principia Mathematica * 10   pm10.12 26964
            18.19.2  Principia Mathematica * 11   2alanimi 26978
            18.19.3  Predicate Calculus   sbeqal1 27008
            18.19.4  Principia Mathematica * 13 and * 14   pm13.13a 27018
            18.19.5  Set Theory   elnev 27049
            18.19.6  Arithmetic   addcomgi 27072
            18.19.7  Geometry   cplusr 27073
      18.20  Mathbox for Glauco Siliprandi
            18.20.1  Miscellanea   ssrexf 27095
            18.20.2  Finite multiplication of numbers and finite multiplication of functions   fmul01 27121
            18.20.3  Limits   clim1fr1 27138
            18.20.4  Derivatives   dvsinexp 27151
            18.20.5  Integrals   ioovolcl 27153
            18.20.6  Stone Weierstrass theorem - real version   stoweidlem1 27161
            18.20.7  Wallis' product for π   wallispilem1 27225
            18.20.8  Stirling's approximation formula for ` n ` factorial   stirlinglem1 27234
      18.21  Mathbox for Jarvin Udandy
      18.22  Mathbox for Alexander van der Vekens
            18.22.1  Double restricted existential uniqueness   r19.32 27336
                  18.22.1.1  Restricted quantification (extension)   r19.32 27336
                  18.22.1.2  The empty set (extension)   raaan2 27344
                  18.22.1.3  Restricted uniqueness and "at most one" quantification   rmoimi 27345
                  18.22.1.4  Analogs to Existential uniqueness (double quantification)   2reurex 27350
            18.22.2  Alternative definitions of function's and operation's values   wdfat 27362
                  18.22.2.1  Restricted quantification (extension)   ralbinrald 27368
                  18.22.2.2  The universal class (extension)   nvelim 27369
                  18.22.2.3  Relations (extension)   sbcrel 27370
                  18.22.2.4  Functions (extension)   sbcfun 27376
                  18.22.2.5  Predicate "defined at"   dfateq12d 27383
                  18.22.2.6  Alternative definition of the value of a function   dfafv2 27386
                  18.22.2.7  Alternative definition of the value of an operation   aoveq123d 27429
      18.23  Mathbox for David A. Wheeler
            18.23.1  Natural deduction   19.8ad 27459
            18.23.2  Greater than, greater than or equal to.   cge-real 27462
            18.23.3  Hyperbolic trig functions   csinh 27472
            18.23.4  Reciprocal trig functions (sec, csc, cot)   csec 27483
            18.23.5  Identities for "if"   ifnmfalse 27505
            18.23.6  Not-member-of   AnelBC 27506
            18.23.7  Decimal point   cdp2 27507
            18.23.8  Signum (sgn or sign) function   csgn 27515
            18.23.9  Ceiling function   ccei 27525
            18.23.10  Logarithm laws generalized to an arbitrary base - logb   clogb 27529
            18.23.11  Logarithm laws generalized to an arbitrary base - log_   clog_ 27540
            18.23.12  Miscellaneous   5m4e1 27542
      18.24  Mathbox for Alan Sare
            18.24.1  Conventional Metamath proofs, some derived from VD proofs   iidn3 27545
            18.24.2  What is Virtual Deduction?   wvd1 27620
            18.24.3  Virtual Deduction Theorems   df-vd1 27621
            18.24.4  Theorems proved using virtual deduction   trsspwALT 27872
            18.24.5  Theorems proved using virtual deduction with mmj2 assistance   simplbi2VD 27902
            18.24.6  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 27969
            18.24.7  Theorems proved using conjunction-form virtual deduction   elpwgdedVD 27973
            18.24.8  Theorems with VD proofs in conventional notation derived from VD proofs   suctrALT3 27980
            18.24.9  Theorems with a proof in conventional notation automatically derived   notnot2ALT2 27983
      18.25  Mathbox for Jonathan Ben-Naim
            18.25.1  First order logic and set theory   bnj170 28002
            18.25.2  Well founded induction and recursion   bnj110 28169
            18.25.3  The existence of a minimal element in certain classes   bnj69 28319
            18.25.4  Well-founded induction   bnj1204 28321
            18.25.5  Well-founded recursion, part 1 of 3   bnj60 28371
            18.25.6  Well-founded recursion, part 2 of 3   bnj1500 28377
            18.25.7  Well-founded recursion, part 3 of 3   bnj1522 28381
      18.26  Mathbox for Norm Megill
            18.26.1  Obsolete experiments to study ax-12o   ax12-2 28382
            18.26.2  Miscellanea   cnaddcom 28440
            18.26.3  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 28443
            18.26.4  Functionals and kernels of a left vector space (or module)   clfn 28526
            18.26.5  Opposite rings and dual vector spaces   cld 28592
            18.26.6  Ortholattices and orthomodular lattices   cops 28641
            18.26.7  Atomic lattices with covering property   ccvr 28731
            18.26.8  Hilbert lattices   chlt 28819
            18.26.9  Projective geometries based on Hilbert lattices   clln 28959
            18.26.10  Construction of a vector space from a Hilbert lattice   cdlema1N 29259
            18.26.11  Construction of involution and inner product from a Hilbert lattice   clpoN 30949

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