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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 mostly without distinct variables
      1.6  Predicate calculus with distinct variables
      1.7  Other axiomatizations related to classical predicate calculus
PART 2  ZF (ZERMELO-FRAENKEL) SET THEORY
      2.1  ZF Set Theory - start with the Axiom of Extensionality
      2.2  ZF Set Theory - add the Axiom of Replacement
      2.3  ZF Set Theory - add the Axiom of Power Sets
      2.4  ZF Set Theory - add the Axiom of Union
      2.5  ZF Set Theory - add the Axiom of Regularity
      2.6  ZF Set Theory - add the Axiom of Infinity
PART 3  ZFC (ZERMELO-FRAENKEL WITH CHOICE) SET THEORY
      3.1  ZFC Set Theory - add Countable Choice and Dependent Choice
      3.2  ZFC Set Theory - add the Axiom of Choice
      3.3  ZFC Axioms with no distinct variable requirements
      3.4  The Generalized Continuum Hypothesis
PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      4.1  Inaccessibles
      4.2  ZFC Set Theory plus the Tarksi-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
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 873
            1.3.8  Abbreviated conjunction and disjunction of three wff's   w3o 938
            1.3.9  Logical 'nand' (Sheffer stroke)   wnan 1292
            1.3.10  Logical 'xor'   wxo 1300
            1.3.11  True and false constants   wtru 1312
            1.3.12  Truth tables   truantru 1332
            1.3.13  Auxiliary theorems for Alan Sare's virtual deduction tool, part 1   ee22 1358
            1.3.14  Half-adders and full adders in propositional calculus   whad 1374
      1.4  Other axiomatizations of classical propositional calculus
            1.4.1  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1400
            1.4.2  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1418
            1.4.3  Derive Nicod's axiom from the standard axioms   nic-dfim 1429
            1.4.4  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1435
            1.4.5  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1454
            1.4.6  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1458
            1.4.7  Deriving the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1473
            1.4.8  Deriving the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1496
            1.4.9  Derive the Lukasiewicz axioms from the The Russell-Bernays Axioms   rb-bijust 1509
            1.4.10  Stoic logic indemonstrables (Chrysippus of Soli)   mpto1 1528
      1.5  Predicate calculus mostly without distinct variables
            1.5.1  "Pure" (equality-free) predicate calculus axioms ax-5, ax-6, ax-7, ax-gen   wal 1532
            1.5.2  Introduce equality axioms ax-8, ax-11, ax-13, and ax-14   cv 1618
            1.5.3  Axiom ax-17 - first use of the $d distinct variable statement   ax-17 1628
            1.5.4  Introduce equality axioms ax-9v and ax-12   ax-9v 1632
            1.5.5  Derive ax-12o from ax-12   ax12o10lem1 1635
            1.5.6  Derive ax-10   ax10lem16 1665
            1.5.7  Derive ax-9 from the weaker version ax-9v   ax9 1683
            1.5.8  Introduce Axiom of Existence ax-9   ax-9 1684
            1.5.9  Derive ax-4, ax-5o, and ax-6o   ax4 1691
            1.5.10  "Pure" predicate calculus including ax-4, without distinct variables   a4i 1699
            1.5.11  Equality theorems without distinct variables   ax9o 1814
            1.5.12  Axioms ax-10 and ax-11   ax10o 1835
            1.5.13  Substitution (without distinct variables)   wsb 1883
            1.5.14  Theorems using axiom ax-11   equs5a 1912
      1.6  Predicate calculus with distinct variables
            1.6.1  Derive the axiom of distinct variables ax-16   a4imv 1923
            1.6.2  Derive the obsolete axiom of variable substitution ax-11o   ax11o 1940
            1.6.3  Theorems without distinct variables that use axiom ax-11o   ax11b 1943
            1.6.4  Predicate calculus with distinct variables (cont.)   ax11v 1991
            1.6.5  More substitution theorems   equsb3lem 2064
            1.6.6  Existential uniqueness   weu 2118
      1.7  Other axiomatizations related to classical predicate calculus
            1.7.1  Predicate calculus with all distinct variables   ax-7d 2209
            1.7.2  Aristotelian logic: Assertic syllogisms   barbara 2215
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 2239
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2244
            2.1.3  Class form not-free predicate   wnfc 2381
            2.1.4  Negated equality and membership   wne 2421
            2.1.5  Restricted quantification   wral 2518
            2.1.6  The universal class   cvv 2763
            2.1.7  Conditional equality (experimental)   wcdeq 2949
            2.1.8  Russell's Paradox   ru 2965
            2.1.9  Proper substitution of classes for sets   wsbc 2966
            2.1.10  Proper substitution of classes for sets into classes   csb 3056
            2.1.11  Define basic set operations and relations   cdif 3124
            2.1.12  Subclasses and subsets   df-ss 3141
            2.1.13  The difference, union, and intersection of two classes   difeq1 3262
            2.1.14  The empty set   c0 3430
            2.1.15  "Weak deduction theorem" for set theory   cif 3539
            2.1.16  Power classes   cpw 3599
            2.1.17  Unordered and ordered pairs   csn 3614
            2.1.18  The union of a class   cuni 3801
            2.1.19  The intersection of a class   cint 3836
            2.1.20  Indexed union and intersection   ciun 3879
            2.1.21  Disjointness   wdisj 3967
            2.1.22  Binary relations   wbr 3997
            2.1.23  Ordered-pair class abstractions (class builders)   copab 4050
            2.1.24  Transitive classes   wtr 4087
      2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 4105
            2.2.2  Derive the Axiom of Separation   axsep 4114
            2.2.3  Derive the Null Set Axiom   zfnuleu 4120
            2.2.4  Theorems requiring subset and intersection existence   nalset 4125
            2.2.5  Theorems requiring empty set existence   class2set 4150
      2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4160
            2.3.2  Derive the Axiom of Pairing   zfpair 4184
            2.3.3  Ordered pair theorem   opnz 4214
            2.3.4  Ordered-pair class abstractions (cont.)   opabid 4243
            2.3.5  Power class of union and intersection   pwin 4269
            2.3.6  Epsilon and identity relations   cep 4275
            2.3.7  Partial and complete ordering   wpo 4284
            2.3.8  Founded and well-ordering relations   wfr 4321
            2.3.9  Ordinals   word 4363
      2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4484
            2.4.2  Ordinals (continued)   ordon 4546
            2.4.3  Transfinite induction   tfi 4616
            2.4.4  The natural numbers (i.e. finite ordinals)   com 4628
            2.4.5  Peano's postulates   peano1 4647
            2.4.6  Finite induction (for finite ordinals)   find 4653
            2.4.7  Functions and relations   cxp 4659
            2.4.8  Operations   co 5792
            2.4.9  "Maps to" notation   elmpt2cl 5995
            2.4.10  Function operation   cof 6010
            2.4.11  First and second members of an ordered pair   c1st 6054
            2.4.12  Function transposition   ctpos 6167
            2.4.13  Curry and uncurry   ccur 6206
            2.4.14  Proper subset relation   crpss 6210
            2.4.15  Definite description binder (inverted iota)   cio 6223
            2.4.16  Cantor's Theorem   canth 6260
            2.4.17  Undefined values and restricted iota (description binder)   cund 6262
            2.4.18  Functions on ordinals; strictly monotone ordinal functions   iunon 6323
            2.4.19  "Strong" transfinite recursion   crecs 6355
            2.4.20  Recursive definition generator   crdg 6390
            2.4.21  Finite recursion   frfnom 6415
            2.4.22  Abian's "most fundamental" fixed point theorem   abianfplem 6438
            2.4.23  Ordinal arithmetic   c1o 6440
            2.4.24  Natural number arithmetic   nna0 6570
            2.4.25  Equivalence relations and classes   wer 6625
            2.4.26  The mapping operation   cmap 6740
            2.4.27  Infinite Cartesian products   cixp 6785
            2.4.28  Equinumerosity   cen 6828
            2.4.29  Schroeder-Bernstein Theorem   sbthlem1 6939
            2.4.30  Equinumerosity (cont.)   xpf1o 6991
            2.4.31  Pigeonhole Principle   phplem1 7008
            2.4.32  Finite sets   onomeneq 7018
            2.4.33  Finite intersections   cfi 7132
            2.4.34  Hall's marriage theorem   marypha1lem 7154
            2.4.35  Supremum   csup 7161
            2.4.36  Ordinal isomorphism, Hartog's theorem   coi 7192
            2.4.37  Hartogs function, order types, weak dominance   char 7238
      2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 7274
            2.5.2  Axiom of Infinity equivalents   inf0 7290
      2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 7307
            2.6.2  Existence of omega (the set of natural numbers)   omex 7312
            2.6.3  Cantor normal form   ccnf 7330
            2.6.4  Transitive closure   trcl 7378
            2.6.5  Rank   cr1 7402
            2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 7523
            2.6.7  Cardinal numbers   ccrd 7536
            2.6.8  Axiom of Choice equivalents   wac 7710
            2.6.9  Cardinal number arithmetic   ccda 7761
            2.6.10  The Ackermann bijection   ackbij2lem1 7813
            2.6.11  Cofinality (without Axiom of Choice)   cflem 7840
            2.6.12  Eight inequivalent definitions of finite set   sornom 7871
            2.6.13  Hereditarily size-limited sets without Choice   itunifval 8010
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 8053
            3.2.2  AC equivalents: well ordering, Zorn's lemma   numthcor 8089
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 8136
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 8164
            3.2.5  Cofinality using Axiom of Choice   alephreg 8172
      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 8272
            4.1.2  Weak universes   cwun 8290
            4.1.3  Tarski's classes   ctsk 8338
            4.1.4  Grothendieck's universes   cgru 8380
      4.2  ZFC Set Theory plus the Tarksi-Grothendieck Axiom
            4.2.1  Introduce the Tarksi-Grothendieck Axiom   ax-groth 8413
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 8416
            4.2.3  Tarski map function   ctskm 8427
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 8434
            5.1.2  Final derivation of real and complex number postulates   axaddf 8735
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 8761
      5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 8786
            5.2.2  Infinity and the extended real number system   cpnf 8832
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 8862
            5.2.4  Ordering on reals   lttr 8867
            5.2.5  Initial properties of the complex numbers   mul12 8946
      5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 8993
            5.3.2  Subtraction   cmin 9005
            5.3.3  Multiplication   muladd 9180
            5.3.4  Ordering on reals (cont.)   gt0ne0 9207
            5.3.5  Reciprocals   ixi 9365
            5.3.6  Division   cdiv 9391
            5.3.7  Ordering on reals (cont.)   elimgt0 9560
            5.3.8  Completeness Axiom and Suprema   fimaxre 9669
            5.3.9  Imaginary and complex number properties   inelr 9704
            5.3.10  Function operation analogue theorems   ofsubeq0 9711
      5.4  Integer sets
            5.4.1  Natural numbers (as a subset of complex numbers)   cn 9714
            5.4.2  Principle of mathematical induction   nnind 9732
            5.4.3  Decimal representation of numbers   c2 9763
            5.4.4  Some properties of specific numbers   0p1e1 9807
            5.4.5  The Archimedean property   nnunb 9929
            5.4.6  Nonnegative integers (as a subset of complex numbers)   cn0 9933
            5.4.7  Integers (as a subset of complex numbers)   cz 9992
            5.4.8  Decimal arithmetic   cdc 10092
            5.4.9  Upper partititions of integers   cuz 10198
            5.4.10  Well-ordering principle for bounded-below sets of integers   uzwo3 10279
            5.4.11  Rational numbers (as a subset of complex numbers)   cq 10284
            5.4.12  Existence of the set of complex numbers   rpnnen1lem1 10310
      5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 10322
            5.5.2  Infinity and the extended real number system (cont.)   cxne 10417
            5.5.3  Supremum on the extended reals   xrsupexmnf 10590
            5.5.4  Real number intervals   cioo 10623
            5.5.5  Finite intervals of integers   cfz 10749
            5.5.6  Half-open integer ranges   cfzo 10837
      5.6  Elementary integer functions
            5.6.1  The floor (greatest integer) function   cfl 10891
            5.6.2  The modulo (remainder) operation   cmo 10940
            5.6.3  The infinite sequence builder "seq"   om2uz0i 10977
            5.6.4  Integer powers   cexp 11071
            5.6.5  Ordered pair theorem for nonnegative integers   nn0le2msqi 11249
            5.6.6  Factorial function   cfa 11255
            5.6.7  The binomial coefficient operation   cbc 11282
            5.6.8  The ` # ` (finite set size) function   chash 11304
            5.6.9  Words over a set   cword 11369
            5.6.10  Longer string literals   cs2 11457
      5.7  Elementary real and complex functions
            5.7.1  The "shift" operation   cshi 11527
            5.7.2  Real and imaginary parts; conjugate   ccj 11547
            5.7.3  Square root; absolute value   csqr 11684
      5.8  Elementary limits and convergence
            5.8.1  Superior limit (lim sup)   clsp 11910
            5.8.2  Limits   cli 11924
            5.8.3  Finite and infinite sums   csu 12124
            5.8.4  The binomial theorem   binomlem 12253
            5.8.5  Infinite sums (cont.)   isumshft 12261
            5.8.6  Miscellaneous converging and diverging sequences   divrcnv 12274
            5.8.7  Arithmetic series   arisum 12281
            5.8.8  Geometric series   expcnv 12285
            5.8.9  Ratio test for infinite series convergence   cvgrat 12302
            5.8.10  Mertens' theorem   mertenslem1 12303
      5.9  Elementary trigonometry
            5.9.1  The exponential, sine, and cosine functions   ce 12306
            5.9.2  _e is irrational   eirrlem 12445
      5.10  Cardinality of real and complex number subsets
            5.10.1  Countability of integers and rationals   xpnnen 12450
            5.10.2  The reals are uncountable   rpnnen2lem1 12456
PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqr2irrlem 12489
            6.1.2  Some Number sets are chains of proper subsets   nthruc 12492
            6.1.3  The divides relation   cdivides 12494
            6.1.4  The division algorithm   divalglem0 12555
            6.1.5  Bit sequences   cbits 12573
            6.1.6  The greatest common divisor operator   cgcd 12648
            6.1.7  Bézout's identity   bezoutlem1 12680
            6.1.8  Algorithms   nn0seqcvgd 12703
            6.1.9  Euclid's Algorithm   eucalgval2 12714
      6.2  Elementary prime number theory
            6.2.1  Elementary properties   cprime 12721
            6.2.2  Properties of the canonical representation of a rational   cnumer 12767
            6.2.3  Euler's theorem   codz 12794
            6.2.4  Pythagorean Triples   coprimeprodsq 12825
            6.2.5  The prime count function   cpc 12852
            6.2.6  Pocklington's theorem   prmpwdvds 12914
            6.2.7  Infinite primes theorem   unbenlem 12918
            6.2.8  Sum of prime reciprocals   prmreclem1 12926
            6.2.9  Fundamental theorem of arithmetic   1arithlem1 12933
            6.2.10  Lagrange's four-square theorem   cgz 12939
            6.2.11  Van der Waerden's theorem   cvdwa 12975
            6.2.12  Ramsey's theorem   cram 13009
            6.2.13  Decimal arithmetic (cont.)   dec2dvds 13041
            6.2.14  Specific prime numbers   4nprm 13069
            6.2.15  Very large primes   1259lem1 13092
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
            7.1.1  Basic definitions   cstr 13107
            7.1.2  Slot definitions   cplusg 13171
            7.1.3  Definition of the structure product   crest 13288
            7.1.4  Definition of the structure quotient   cordt 13361
      7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 13471
            7.2.2  Independent sets in a Moore system   mrisval 13495
            7.2.3  Algebraic closure systems   isacs 13516
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
            8.1.1  Categories   ccat 13529
            8.1.2  Opposite category   coppc 13577
            8.1.3  Monomorphisms and epimorphisms   cmon 13594
            8.1.4  Sections, inverses, isomorphisms   csect 13610
            8.1.5  Subcategories   cssc 13647
            8.1.6  Functors   cfunc 13691
            8.1.7  Full & faithful functors   cful 13739
            8.1.8  Natural transformations and the functor category   cnat 13778
      8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 13848
      8.3  Examples of categories
            8.3.1  The category of sets   csetc 13870
            8.3.2  The category of categories   ccatc 13889
      8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 13905
            8.4.2  Functor evaluation   cevlf 13946
            8.4.3  Hom functor   chof 13985
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 14037
            9.2.2  Lattices   clat 14114
            9.2.3  The dual of an ordered set   codu 14195
            9.2.4  Subset order structures   cipo 14217
            9.2.5  Distributive lattices   latmass 14254
            9.2.6  Posets and lattices as relations   cps 14264
            9.2.7  Directed sets, nets   cdir 14313
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
            10.1.1  Definition and basic properties   cmnd 14324
            10.1.2  Monoid homomorphisms and submonoids   cmhm 14376
            10.1.3  Ordered group sum operation   gsumvallem1 14411
            10.1.4  Free monoids   cfrmd 14432
      10.2  Groups
            10.2.1  Definition and basic properties   df-grp 14452
            10.2.2  Subgroups and Quotient groups   csubg 14578
            10.2.3  Elementary theory of group homomorphisms   cghm 14643
            10.2.4  Isomorphisms of groups   cgim 14684
            10.2.5  Group actions   cga 14706
            10.2.6  Symmetry groups and Cayley's Theorem   csymg 14732
            10.2.7  Centralizers and centers   ccntz 14754
            10.2.8  The opposite group   coppg 14781
            10.2.9  p-Groups and Sylow groups; Sylow's theorems   cod 14803
            10.2.10  Direct products   clsm 14908
            10.2.11  Free groups   cefg 14978
      10.3  Abelian groups
            10.3.1  Definition and basic properties   ccmn 15052
            10.3.2  Cyclic groups   ccyg 15127
            10.3.3  Group sum operation   gsumval3a 15152
            10.3.4  Internal direct products   cdprd 15194
            10.3.5  The Fundamental Theorem of Abelian Groups   ablfacrplem 15263
      10.4  Rings
            10.4.1  Multiplicative Group   cmgp 15288
            10.4.2  Definition and basic properties   crg 15300
            10.4.3  Opposite ring   coppr 15367
            10.4.4  Divisibility   cdsr 15383
            10.4.5  Ring homomorphisms   crh 15457
      10.5  Division rings and Fields
            10.5.1  Definition and basic properties   cdr 15475
            10.5.2  Subrings of a ring   csubrg 15504
            10.5.3  Absolute value (abstract algebra)   cabv 15544
            10.5.4  Star rings   cstf 15571
      10.6  Left Modules
            10.6.1  Definition and basic properties   clmod 15590
            10.6.2  Subspaces and spans in a left module   clss 15652
            10.6.3  Homomorphisms and isomorphisms of left modules   clmhm 15739
            10.6.4  Subspace sum; bases for a left module   clbs 15790
      10.7  Vector Spaces
            10.7.1  Definition and basic properties   clvec 15818
      10.8  Ideals
            10.8.1  The subring algebra; ideals   csra 15884
            10.8.2  Two-sided ideals and quotient rings   c2idl 15946
            10.8.3  Principal ideal rings. Divisibility in the integers   clpidl 15956
            10.8.4  Nonzero rings   cnzr 15972
            10.8.5  Left regular elements. More kinds of ring   crlreg 15983
      10.9  Associative algebras
            10.9.1  Definition and basic properties   casa 16013
      10.10  Abstract Multivariate Polynomials
            10.10.1  Definition and basic properties   cmps 16050
            10.10.2  Polynomial evaluation   evlslem4 16208
            10.10.3  Univariate Polynomials   cps1 16213
      10.11  The complex numbers as an extensible structure
            10.11.1  Definition and basic properties   cxmt 16332
            10.11.2  Algebraic constructions based on the complexes   czrh 16414
      10.12  Hilbert spaces
            10.12.1  Definition and basic properties   cphl 16491
            10.12.2  Orthocomplements and closed subspaces   cocv 16523
            10.12.3  Orthogonal projection and orthonormal bases   cpj 16563
PART 11  BASIC TOPOLOGY
      11.1  Topology
            11.1.1  Topological spaces   ctop 16594
            11.1.2  TopBases for topologies   isbasisg 16648
            11.1.3  Examples of topologies   distop 16696
            11.1.4  Closure and interior   ccld 16716
            11.1.5  Neighborhoods   cnei 16797
            11.1.6  Limit points and perfect sets   clp 16829
            11.1.7  Subspace topologies   restrcl 16851
            11.1.8  Order topology   ordtbaslem 16881
            11.1.9  Limits and Continuity in topological spaces   ccn 16917
            11.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 16997
            11.1.11  Compactness   ccmp 17076
            11.1.12  Connectedness   ccon 17100
            11.1.13  First- and second-countability   c1stc 17126
            11.1.14  Local topological properties   clly 17153
            11.1.15  Compactly generated spaces   ckgen 17191
            11.1.16  Product topologies   ctx 17218
            11.1.17  Continuous function-builders   cnmptid 17318
            11.1.18  Quotient maps and quotient topology   ckq 17347
            11.1.19  Homeomorphisms   chmeo 17407
      11.2  Filters and filter bases
            11.2.1  Filter Bases   cfbas 17481
            11.2.2  Filters   cfil 17503
            11.2.3  Ultrafilters   cufil 17557
            11.2.4  Filter limits   cfm 17591
            11.2.5  Topological groups   ctmd 17716
            11.2.6  Infinite group sum on topological groups   ctsu 17771
            11.2.7  Topological rings, fields, vector spaces   ctrg 17801
      11.3  Metric spaces
            11.3.1  Basic metric space properties   cxme 17845
            11.3.2  Metric space balls   blfval 17910
            11.3.3  Open sets of a metric space   mopnval 17947
            11.3.4  Continuity in metric spaces   metcnp3 18049
            11.3.5  Examples of metric spaces   dscmet 18058
            11.3.6  Normed algebraic structures   cnm 18062
            11.3.7  Normed space homomorphisms (bounded linear operators)   cnmo 18177
            11.3.8  Topology on the Reals   qtopbaslem 18230
            11.3.9  Topological definitions using the reals   cii 18342
            11.3.10  Path homotopy   chtpy 18428
            11.3.11  The fundamental group   cpco 18461
            11.3.12  Complex left modules   cclm 18523
            11.3.13  Complex pre-Hilbert space   ccph 18565
            11.3.14  Convergence and completeness   ccfil 18641
            11.3.15  Baire's Category Theorem   bcthlem1 18709
            11.3.16  Banach spaces and complex Hilbert spaces   ccms 18717
            11.3.17  Minimizing Vector Theorem   minveclem1 18751
            11.3.18  Projection theorem   pjthlem1 18764
PART 12  BASIC REAL AND COMPLEX ANALYSIS
      12.1  Continuity
            12.1.1  Intermediate value theorem   pmltpclem1 18771
      12.2  Integrals
            12.2.1  Lebesgue measure   covol 18785
            12.2.2  Lebesgue integration   cmbf 18932
      12.3  Derivatives
            12.3.1  Real and Complex Differentiation   climc 19175
PART 13  BASIC REAL AND COMPLEX FUNCTIONS
      13.1  Polynomials
            13.1.1  Abstract polynomials, continued   evlslem6 19360
            13.1.2  Polynomial degrees   cmdg 19402
            13.1.3  The division algorithm for univariate polynomials   cmn1 19474
            13.1.4  Elementary properties of complex polynomials   cply 19529
            13.1.5  The Division algorithm for polynomials   cquot 19633
            13.1.6  Algebraic numbers   caa 19657
            13.1.7  Liouville's approximation theorem   aalioulem1 19675
      13.2  Sequences and series
            13.2.1  Taylor polynomials and Taylor's theorem   ctayl 19695
            13.2.2  Uniform convergence   culm 19718
            13.2.3  Power series   pserval 19749
      13.3  Basic trigonometry
            13.3.1  The exponential, sine, and cosine functions (cont.)   efcn 19782
            13.3.2  Properties of pi = 3.14159...   pilem1 19790
            13.3.3  Mapping of the exponential function   efgh 19866
            13.3.4  The natural logarithm on complex numbers   clog 19875
            13.3.5  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 20062
            13.3.6  Solutions of quadratic, cubic, and quartic equations   quad2 20098
            13.3.7  Inverse trigonometric functions   casin 20121
            13.3.8  The Birthday Problem   log2ublem1 20205
            13.3.9  Areas in R^2   carea 20213
            13.3.10  More miscellaneous converging sequences   rlimcnp 20223
            13.3.11  Inequality of arithmetic and geometric means   cvxcl 20242
            13.3.12  Euler-Mascheroni constant   cem 20249
      13.4  Basic number theory
            13.4.1  Wilson's theorem   wilthlem1 20269
            13.4.2  The Fundamental Theorem of Algebra   ftalem1 20273
            13.4.3  The Basel problem (ζ(2) = π2/6)   basellem1 20281
            13.4.4  Number-theoretical functions   ccht 20291
            13.4.5  Perfect Number Theorem   mersenne 20429
            13.4.6  Characters of Z/nZ   cdchr 20434
            13.4.7  Bertrand's postulate   bcctr 20477
            13.4.8  Legendre symbol   clgs 20496
            13.4.9  Quadratic Reciprocity   lgseisenlem1 20551
            13.4.10  All primes 4n+1 are the sum of two squares   2sqlem1 20565
            13.4.11  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 20581
            13.4.12  The Prime Number Theorem   mudivsum 20642
            13.4.13  Ostrowski's theorem   abvcxp 20727
PART 14  MISCELLANEA
      14.1  Definitional Examples
      14.2  Natural deduction examples
      14.3  Humor
            14.3.1  April Fool's theorem   avril1 20797
      14.4  (Future - to be reviewed and classified)
            14.4.1  Planar incidence geometry   cplig 20803
            14.4.2  Algebra preliminaries   crpm 20808
            14.4.3  Transitive closure   ctcl 20810
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 20814
            15.1.2  Definition and basic properties of Abelian groups   cablo 20909
            15.1.3  Subgroups   csubgo 20929
            15.1.4  Operation properties   cass 20940
            15.1.5  Group-like structures   cmagm 20946
            15.1.6  Examples of Abelian groups   ablosn 20975
            15.1.7  Group homomorphism and isomorphism   cghom 20985
      15.2  Additional material on Rings and Fields
            15.2.1  Definition and basic properties   crngo 21003
            15.2.2  Examples of rings   cnrngo 21031
            15.2.3  Division Rings   cdrng 21033
            15.2.4  Star Fields   csfld 21036
            15.2.5  Fields and Rings   ccm2 21038
PART 16  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      16.1  Complex vector spaces
            16.1.1  Definition and basic properties   cvc 21062
            16.1.2  Examples of complex vector spaces   cncvc 21100
      16.2  Normed complex vector spaces
            16.2.1  Definition and basic properties   cnv 21101
            16.2.2  Examples of normed complex vector spaces   cnnv 21206
            16.2.3  Induced metric of a normed complex vector space   imsval 21215
            16.2.4  Inner product   cdip 21234
            16.2.5  Subspaces   css 21258
      16.3  Operators on complex vector spaces
            16.3.1  Definitions and basic properties   clno 21279
      16.4  Inner product (pre-Hilbert) spaces
            16.4.1  Definition and basic properties   ccphlo 21351
            16.4.2  Examples of pre-Hilbert spaces   cncph 21358
            16.4.3  Properties of pre-Hilbert spaces   isph 21361
      16.5  Complex Banach spaces
            16.5.1  Definition and basic properties   ccbn 21402
            16.5.2  Examples of complex Banach spaces   cnbn 21409
            16.5.3  Uniform Boundedness Theorem   ubthlem1 21410
            16.5.4  Minimizing Vector Theorem   minvecolem1 21414
      16.6  Complex Hilbert spaces
            16.6.1  Definition and basic properties   chlo 21425
            16.6.2  Standard axioms for a complex Hilbert space   hlex 21438
            16.6.3  Examples of complex Hilbert spaces   cnchl 21456
            16.6.4  Subspaces   ssphl 21457
            16.6.5  Hellinger-Toeplitz Theorem   htthlem 21458
PART 17  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      17.1  Axiomatization of complex pre-Hilbert spaces
            17.1.1  Basic Hilbert space definitions   chil 21460
            17.1.2  Preliminary ZFC lemmas   df-hnorm 21509
            17.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 21522
            17.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 21540
            17.1.5  Vector operations   hvmulex 21552
            17.1.6  Inner product postulates for a Hilbert space   ax-hfi 21619
      17.2  Inner product and norms
            17.2.1  Inner product   his5 21626
            17.2.2  Norms   dfhnorm2 21662
            17.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 21700
            17.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 21719
      17.3  Cauchy sequences and completeness axiom
            17.3.1  Cauchy sequences and limits   hcau 21724
            17.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 21734
            17.3.3  Completeness postulate for a Hilbert space   ax-hcompl 21742
            17.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 21743
      17.4  Subspaces and projections
            17.4.1  Subspaces   df-sh 21747
            17.4.2  Closed subspaces   df-ch 21762
            17.4.3  Orthocomplements   df-oc 21792
            17.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 21848
            17.4.5  Projection theorem   pjhthlem1 21931
            17.4.6  Projectors   df-pjh 21935
      17.5  Properties of Hilbert subspaces
            17.5.1  Orthomodular law   omlsilem 21942
            17.5.2  Projectors (cont.)   pjhtheu2 21956
            17.5.3  Hilbert lattice operations   sh0le 21980
            17.5.4  Span (cont.) and one-dimensional subspaces   spansn0 22081
            17.5.5  Commutes relation for Hilbert lattice elements   df-cm 22123
            17.5.6  Foulis-Holland theorem   fh1 22158
            17.5.7  Quantum Logic Explorer axioms   qlax1i 22167
            17.5.8  Orthogonal subspaces   chscllem1 22177
            17.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 22194
            17.5.10  Projectors (cont.)   pjorthi 22209
            17.5.11  Mayet's equation E_3   mayete3i 22268
      17.6  Operators on Hilbert spaces
            17.6.1  Operator sum, difference, and scalar multiplication   df-hosum 22271
            17.6.2  Zero and identity operators   df-h0op 22289
            17.6.3  Operations on Hilbert space operators   hoaddcl 22299
            17.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 22380
            17.6.5  Linear and continuous functionals and norms   df-nmfn 22386
            17.6.6  Adjoint   df-adjh 22390
            17.6.7  Dirac bra-ket notation   df-bra 22391
            17.6.8  Positive operators   df-leop 22393
            17.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 22394
            17.6.10  Theorems about operators and functionals   nmopval 22397
            17.6.11  Riesz lemma   riesz3i 22603
            17.6.12  Adjoints (cont.)   cnlnadjlem1 22608
            17.6.13  Quantum computation error bound theorem   unierri 22645
            17.6.14  Dirac bra-ket notation (cont.)   branmfn 22646
            17.6.15  Positive operators (cont.)   leopg 22663
            17.6.16  Projectors as operators   pjhmopi 22687
      17.7  States on an Hilbert lattice and Godowski's equation
            17.7.1  States on a Hilbert lattice   df-st 22752
            17.7.2  Godowski's equation   golem1 22812
      17.8  Cover relation, atoms, exchange axiom, and modular symmetry
            17.8.1  Covers relation; modular pairs   df-cv 22820
            17.8.2  Atoms   df-at 22879
            17.8.3  Superposition principle   superpos 22895
            17.8.4  Atoms, exchange and covering properties, atomicity   chcv1 22896
            17.8.5  Irreducibility   chirredlem1 22931
            17.8.6  Atoms (cont.)   atcvat3i 22937
            17.8.7  Modular symmetry   mdsymlem1 22944
PART 18  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      18.1  Mathboxes for user contributions
            18.1.1  Mathbox guidelines   mathbox 22983
      18.2  Mathbox for Stefan Allan
      18.3  Mathbox for Thierry Arnoux
            18.3.1  Bertrand's Ballot Problem   ballotlemoex 23006
      18.4  Mathbox for Mario Carneiro
            18.4.1  Miscellaneous stuff   quartfull 23059
            18.4.2  Zeta function   czeta 23060
            18.4.3  Gamma function   clgam 23063
            18.4.4  Derangements and the Subfactorial   deranglem 23070
            18.4.5  The Erdős-Szekeres theorem   erdszelem1 23095
            18.4.6  The Kuratowski closure-complement theorem   kur14lem1 23110
            18.4.7  Retracts and sections   cretr 23121
            18.4.8  Path-connected and simply connected spaces   cpcon 23123
            18.4.9  Covering maps   ccvm 23159
            18.4.10  Undirected multigraphs   cumg 23233
            18.4.11  Normal numbers   snmlff 23285
            18.4.12  Godel-sets of formulas   cgoe 23289
            18.4.13  Models of ZF   cgze 23317
            18.4.14  Splitting fields   citr 23331
            18.4.15  p-adic number fields   czr 23347
      18.5  Mathbox for Paul Chapman
            18.5.1  Group homomorphism and isomorphism   ghomgrpilem1 23365
            18.5.2  Real and complex numbers (cont.)   climuzcnv 23377
            18.5.3  Miscellaneous theorems   elfzm12 23381
      18.6  Mathbox for Drahflow
      18.7  Mathbox for Scott Fenton
            18.7.1  ZFC Axioms in primitive form   axextprim 23420
            18.7.2  Untangled classes   untelirr 23427
            18.7.3  Extra propositional calculus theorems   3orel1 23434
            18.7.4  Misc. Useful Theorems   nepss 23445
            18.7.5  Properties of reals and complexes   sqdivzi 23451
            18.7.6  Greatest common divisor and divisibility   pdivsq 23474
            18.7.7  Properties of relationships   brtp 23478
            18.7.8  Properties of functions and mappings   funpsstri 23491
            18.7.9  Epsilon induction   setinds 23504
            18.7.10  Ordinal numbers   elpotr 23507
            18.7.11  Defined equality axioms   axextdfeq 23524
            18.7.12  Hypothesis builders   hbntg 23532
            18.7.13  The Predecessor Class   cpred 23537
            18.7.14  (Trans)finite Recursion Theorems   tfisg 23574
            18.7.15  Well-founded induction   tz6.26 23575
            18.7.16  Transitive closure under a relationship   ctrpred 23590
            18.7.17  Founded Induction   frmin 23612
            18.7.18  Ordering Ordinal Sequences   orderseqlem 23622
            18.7.19  Well-founded recursion   wfr3g 23625
            18.7.20  Transfinite recursion via Well-founded recursion   tfrALTlem 23646
            18.7.21  Founded Recursion   frr3g 23650
            18.7.22  Surreal Numbers   csur 23664
            18.7.23  Surreal Numbers: Ordering   axsltsolem1 23691
            18.7.24  Surreal Numbers: Birthday Function   axbday 23698
            18.7.25  Surreal Numbers: Density   axdenselem1 23705
            18.7.26  Surreal Numbers: Full-Eta Property   axfelem1 23716
            18.7.27  Symmetric difference   csymdif 23738
            18.7.28  Quantifier-free definitions   ctxp 23750
            18.7.29  Alternate ordered pairs   caltop 23866
            18.7.30  Tarskian geometry   cee 23892
            18.7.31  Tarski's axioms for geometry   axdimuniq 23917
            18.7.32  Congruence properties   cofs 23981
            18.7.33  Betweenness properties   btwntriv2 24011
            18.7.34  Segment Transportation   ctransport 24028
            18.7.35  Properties relating betweenness and congruence   cifs 24034
            18.7.36  Connectivity of betweenness   btwnconn1lem1 24086
            18.7.37  Segment less than or equal to   csegle 24105
            18.7.38  Outside of relationship   coutsideof 24118
            18.7.39  Lines and Rays   cline2 24133
            18.7.40  Bernoulli polynomials and sums of k-th powers   cbp 24157
            18.7.41  Rank theorems   rankung 24172
            18.7.42  Hereditarily Finite Sets   chf 24178
      18.8  Mathbox for Anthony Hart
            18.8.1  Propositional Calculus   tb-ax1 24193
            18.8.2  Predicate Calculus   quantriv 24215
            18.8.3  Misc. Single Axiom Systems   meran1 24226
            18.8.4  Connective Symmetry   negsym1 24232
      18.9  Mathbox for Chen-Pang He
            18.9.1  Ordinal topology   ontopbas 24243
      18.10  Mathbox for Jeff Hoffman
            18.10.1  Inferences for finite induction on generic function values   fveleq 24266
            18.10.2  gdc.mm   nnssi2 24270
      18.11  Mathbox for Wolf Lammen
      18.12  Mathbox for Frédéric Liné
            18.12.1  Theorems from other workspaces   tpssg 24299
            18.12.2  Propositional and predicate calculus   neleq12d 24300
            18.12.3  Linear temporal logic   wbox 24337
            18.12.4  Operations   ssoprab2g 24399
            18.12.5  General Set Theory   uninqs 24406
            18.12.6  The "maps to" notation   cmpfun 24510
            18.12.7  Cartesian Products   cpro 24518
            18.12.8  Operations on subsets and functions   ccst 24540
            18.12.9  Arithmetic   3timesi 24546
            18.12.10  Lattice (algebraic definition)   clatalg 24549
            18.12.11  Currying and Partial Mappings   ccur1 24562
            18.12.12  Order theory (Extensible Structure Builder)   corhom 24575
            18.12.13  Order theory   cpresetrel 24583
            18.12.14  Finite composites ( i. e. finite sums, products ... )   cprd 24666
            18.12.15  Operation properties   ccm1 24699
            18.12.16  Groups and related structures   ridlideq 24703
            18.12.17  Free structures   csubsmg 24751
            18.12.18  Translations   trdom2 24759
            18.12.19  Fields and Rings   com2i 24784
            18.12.20  Ideals   cidln 24811
            18.12.21  Generic modules and vector spaces (New Structure builder)   cact 24815
            18.12.22  Generic modules and vector spaces   cvec 24817
            18.12.23  Real vector spaces   cvr 24857
            18.12.24  Matrices   cmmat 24861
            18.12.25  Affine spaces   craffsp 24867
            18.12.26  Intervals of reals and extended reals   bsi 24869
            18.12.27  Topology   topnem 24880
            18.12.28  Continuous functions   cnrsfin 24893
            18.12.29  Homeomorphisms   dmhmph 24901
            18.12.30  Initial and final topologies   intopcoaconlem3b 24906
            18.12.31  Filters   efilcp 24920
            18.12.32  Limits   plimfil 24926
            18.12.33  Uniform spaces   cunifsp 24953
            18.12.34  Separated spaces: T0, T1, T2 (Hausdorff) ...   hst1 24955
            18.12.35  Compactness   indcomp 24957
            18.12.36  Connectedness   singempcon 24961
            18.12.37  Topological fields   ctopfld 24965
            18.12.38  Standard topology on RR   intrn 24967
            18.12.39  Standard topology of intervals of RR   stoi 24969
            18.12.40  Cantor's set   cntrset 24970
            18.12.41  Pre-calculus and Cartesian geometry   dmse1 24971
            18.12.42  Extended Real numbers   nolimf 24987
            18.12.43  ( RR ^ N ) and ( CC ^ N )   cplcv 25012
            18.12.44  Calculus   cintvl 25064
            18.12.45  Directed multi graphs   cmgra 25076
            18.12.46  Category and deductive system underlying "structure"   calg 25079
            18.12.47  Deductive systems   cded 25102
            18.12.48  Categories   ccatOLD 25120
            18.12.49  Homsets   chomOLD 25153
            18.12.50  Monomorphisms, Epimorphisms, Isomorphisms   cepiOLD 25171
            18.12.51  Functors   cfuncOLD 25199
            18.12.52  Subcategories   csubcat 25211
            18.12.53  Terminal and initial objects   ciobj 25228
            18.12.54  Sources and sinks   csrce 25233
            18.12.55  Limits and co-limits   clmct 25242
            18.12.56  Product and sum of two objects   cprodo 25245
            18.12.57  Tarski's classes   ctar 25249
            18.12.58  Category Set   ccmrcase 25278
            18.12.59  Grammars, Logics, Machines and Automata   ckln 25348
            18.12.60  Words   cwrd 25349
            18.12.61  Planar geometry   cpoints 25424
      18.13  Mathbox for Jeff Hankins
            18.13.1  Miscellany   a1i13 25568
            18.13.2  Basic topological facts   topbnd 25610
            18.13.3  Topology of the real numbers   reconnOLD 25623
            18.13.4  Refinements   cfne 25627
            18.13.5  Neighborhood bases determine topologies   neibastop1 25676
            18.13.6  Lattice structure of topologies   topmtcl 25680
            18.13.7  Filter bases   fgmin 25687
            18.13.8  Directed sets, nets   tailfval 25689
      18.14  Mathbox for Jeff Madsen
            18.14.1  Logic and set theory   anim12da 25700
            18.14.2  Real and complex numbers; integers   fimaxreOLD 25798
            18.14.3  Sequences and sums   sdclem2 25820
            18.14.4  Topology   unopnOLD 25832
            18.14.5  Metric spaces   metf1o 25837
            18.14.6  Continuous maps and homeomorphisms   constcncf 25846
            18.14.7  Product topologies   txtopiOLD 25854
            18.14.8  Boundedness   ctotbnd 25858
            18.14.9  Isometries   cismty 25890
            18.14.10  Heine-Borel Theorem   heibor1lem 25901
            18.14.11  Banach Fixed Point Theorem   bfplem1 25914
            18.14.12  Euclidean space   crrn 25917
            18.14.13  Intervals (continued)   ismrer1 25930
            18.14.14  Groups and related structures   exidcl 25934
            18.14.15  Rings   rngonegcl 25944
            18.14.16  Ring homomorphisms   crnghom 25959
            18.14.17  Commutative rings   ccring 25988
            18.14.18  Ideals   cidl 26000
            18.14.19  Prime rings and integral domains   cprrng 26039
            18.14.20  Ideal generators   cigen 26052
      18.15  Mathbox for Rodolfo Medina
            18.15.1  Partitions   prtlem60 26071
      18.16  Mathbox for Stefan O'Rear
            18.16.1  Additional elementary logic and set theory   nelss 26119
            18.16.2  Additional theory of functions   fninfp 26122
            18.16.3  Extensions beyond function theory   gsumvsmul 26132
            18.16.4  Additional topology   elrfi 26137
            18.16.5  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 26141
            18.16.6  Algebraic closure systems   cnacs 26145
            18.16.7  Miscellanea 1. Map utilities   constmap 26156
            18.16.8  Miscellanea for polynomials   ofmpteq 26165
            18.16.9  Multivariate polynomials over the integers   cmzpcl 26167
            18.16.10  Miscellanea for Diophantine sets 1   coeq0 26199
            18.16.11  Diophantine sets 1: definitions   cdioph 26202
            18.16.12  Diophantine sets 2 miscellanea   ellz1 26214
            18.16.13  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 26220
            18.16.14  Diophantine sets 3: construction   diophrex 26223
            18.16.15  Diophantine sets 4 miscellanea   2sbcrex 26232
            18.16.16  Diophantine sets 4: Quantification   rexrabdioph 26243
            18.16.17  Diophantine sets 5: Arithmetic sets   rabdiophlem1 26250
            18.16.18  Diophantine sets 6 miscellanea   fz1ssnn 26260
            18.16.19  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 26262
            18.16.20  Pigeonhole Principle and cardinality helpers   fphpd 26267
            18.16.21  A non-closed set of reals is infinite   rencldnfilem 26271
            18.16.22  Miscellanea for Lagrange's theorem   icodiamlt 26273
            18.16.23  Lagrange's rational approximation theorem   irrapxlem1 26275
            18.16.24  Pell equations 1: A nontrivial solution always exists   pellexlem1 26282
            18.16.25  Pell equations 2: Algebraic number theory of the solution set   csquarenn 26289
            18.16.26  Pell equations 3: characterizing fundamental solution   infmrgelbi 26331
            18.16.27  Logarithm laws generalized to an arbitrary base   reglogcl 26343
            18.16.28  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 26351
            18.16.29  X and Y sequences 1: Definition and recurrence laws   crmx 26353
            18.16.30  Ordering and induction lemmas for the integers   monotuz 26394
            18.16.31  X and Y sequences 2: Order properties   rmxypos 26402
            18.16.32  Congruential equations   congtr 26420
            18.16.33  Alternating congruential equations   acongid 26430
            18.16.34  Additional theorems on integer divisibility   bezoutr 26440
            18.16.35  X and Y sequences 3: Divisibility properties   jm2.18 26449
            18.16.36  X and Y sequences 4: Diophantine representability of Y   jm2.27a 26466
            18.16.37  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 26476
            18.16.38  Uncategorized stuff not associated with a major project   setindtr 26485
            18.16.39  More equivalents of the Axiom of Choice   axac10 26494
            18.16.40  Finitely generated left modules   clfig 26533
            18.16.41  Noetherian left modules I   clnm 26541
            18.16.42  Addenda for structure powers   pwssplit0 26555
            18.16.43  Direct sum of left modules   cdsmm 26565
            18.16.44  Free modules   cfrlm 26580
            18.16.45  Every set admits a group structure iff choice   unxpwdom3 26624
            18.16.46  Independent sets and families   clindf 26642
            18.16.47  Characterization of free modules   lmimlbs 26674
            18.16.48  Noetherian rings and left modules II   clnr 26681
            18.16.49  Hilbert's Basis Theorem   cldgis 26693
            18.16.50  Additional material on polynomials [DEPRECATED]   cmnc 26703
            18.16.51  Degree and minimal polynomial of algebraic numbers   cdgraa 26713
            18.16.52  Algebraic integers I   citgo 26730
            18.16.53  Finite cardinality [SO]   en1uniel 26748
            18.16.54  Words in monoids and ordered group sum   issubmd 26751
            18.16.55  Transpositions in the symmetric group   cpmtr 26752
            18.16.56  The sign of a permutation   cpsgn 26782
            18.16.57  The matrix algebra   cmmul 26807
            18.16.58  The determinant   cmdat 26851
            18.16.59  Endomorphism algebra   cmend 26857
            18.16.60  Subfields   csdrg 26871
            18.16.61  Cyclic groups and order   idomrootle 26879
            18.16.62  Cyclotomic polynomials   ccytp 26889
            18.16.63  Miscellaneous topology   fgraphopab 26897
      18.17  Mathbox for Steve Rodriguez
            18.17.1  Miscellanea   iso0 26904
            18.17.2  Function operations   caofcan 26908
            18.17.3  Calculus   lhe4.4ex1a 26914
      18.18  Mathbox for Andrew Salmon
            18.18.1  Principia Mathematica * 10   pm10.12 26921
            18.18.2  Principia Mathematica * 11   2alanimi 26935
            18.18.3  Predicate Calculus   sbeqal1 26965
            18.18.4  Principia Mathematica * 13 and * 14   pm13.13a 26976
            18.18.5  Set Theory   elnev 27007
            18.18.6  Arithmetic   addcomgi 27030
            18.18.7  Geometry   cplusr 27031
      18.19  Mathbox for Glauco Siliprandi
            18.19.1  Miscellanea   ssrexf 27053
            18.19.2  Finite multiplication of numbers and finite multiplication of functions   fmul01 27079
            18.19.3  Limits   clim1fr1 27096
            18.19.4  Derivatives   dvsinexp 27109
            18.19.5  Integrals   ioovolcl 27111
            18.19.6  Stone Weierstrass theorem - real version   stoweidlem1 27119
            18.19.7  Wallis' product for π   wallispilem1 27183
            18.19.8  Stirling's approximation formula for ` n ` factorial   stirlinglem1 27192
      18.20  Mathbox for Jarvin Udandy
      18.21  Mathbox for Alexander van der Vekens
            18.21.1  Restricted quantification (extension)   r19.32 27294
            18.21.2  The empty set (extension)   raaan2 27302
            18.21.3  Restricted uniqueness and "at most one" quantification   rmoimi 27303
            18.21.4  Analogs to 1.6.6 Existential uniqueness (double quantification)   2reurex 27308
      18.22  Mathbox for David A. Wheeler
            18.22.1  Natural deduction   19.8ad 27320
            18.22.2  Greater than, greater than or equal to.   cge-real 27323
            18.22.3  Hyperbolic trig functions   csinh 27333
            18.22.4  Reciprocal trig functions (sec, csc, cot)   csec 27344
            18.22.5  Identities for "if"   ifnmfalse 27366
            18.22.6  Not-member-of   AnelBC 27367
            18.22.7  Decimal point   cdp2 27368
            18.22.8  Signum (sgn or sign) function   csgn 27376
            18.22.9  Ceiling function   ccei 27386
            18.22.10  Logarithm laws generalized to an arbitrary base   clogb 27390
            18.22.11  Miscellaneous   5m4e1 27395
      18.23  Mathbox for Alan Sare
            18.23.1  Conventional Metamath proofs, some derived from VD proofs   iidn3 27398
            18.23.2  What is Virtual Deduction?   wvd1 27473
            18.23.3  Virtual Deduction Theorems   df-vd1 27474
            18.23.4  Theorems proved using virtual deduction   trsspwALT 27725
            18.23.5  Theorems proved using virtual deduction with mmj2 assistance   simplbi2VD 27755
            18.23.6  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 27822
            18.23.7  Theorems proved using conjunction-form virtual deduction   elpwgdedVD 27826
            18.23.8  Theorems with VD proofs in conventional notation derived from VD proofs   suctrALT3 27833
            18.23.9  Theorems with a proof in conventional notation automatically derived   notnot2ALT2 27836
      18.24  Mathbox for Jonathan Ben-Naim
            18.24.1  First order logic and set theory   bnj170 27855
            18.24.2  Well founded induction and recursion   bnj110 28022
            18.24.3  The existence of a minimal element in certain classes   bnj69 28172
            18.24.4  Well-founded induction   bnj1204 28174
            18.24.5  Well-founded recursion, part 1 of 3   bnj60 28224
            18.24.6  Well-founded recursion, part 2 of 3   bnj1500 28230
            18.24.7  Well-founded recursion, part 3 of 3   bnj1522 28234
      18.25  Mathbox for Norm Megill
            18.25.1  Study of ax-6, ax-7, ax-11, ax-12   equidK 28235
            18.25.2  Derive ax-12o from ax-12   ax12vX 28281
            18.25.3  Derive ax-10   ax10lem16X 28335
            18.25.4  Derive ax-9 from the weaker version ax-9v   ax9X 28352
            18.25.5  Obsolete experiments to study ax-12o   ax12-2 28353
            18.25.6  Miscellanea   cnaddcom 28411
            18.25.7  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 28414
            18.25.8  Functionals and kernels of a left vector space (or module)   clfn 28497
            18.25.9  Opposite rings and dual vector spaces   cld 28563
            18.25.10  Ortholattices and orthomodular lattices   cops 28612
            18.25.11  Atomic lattices with covering property   ccvr 28702
            18.25.12  Hilbert lattices   chlt 28790
            18.25.13  Projective geometries based on Hilbert lattices   clln 28930
            18.25.14  Construction of a vector space from a Hilbert lattice   cdlema1N 29230
            18.25.15  Construction of involution and inner product from a Hilbert lattice   clpoN 30920

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