Copyright (c) 1995 Association of Mizar Users
environ vocabulary MATRIX_1, PBOOLE, ZF_REFLE, FUNCT_1, CAT_5, CAT_1, MCART_1, COMMACAT, GRCAT_1, RELAT_1, BOOLE, FUNCOP_1, PRALG_1, FRAENKEL, PARTFUN1, OPPCAT_1, GROUP_6, CAT_2, FUNCT_3, INDEX_1; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, MCART_1, RELAT_1, FUNCT_1, FUNCT_2, PARTFUN1, PBOOLE, FRAENKEL, MSUALG_1, PRALG_1, DTCONSTR, CAT_1, CAT_2, OPPCAT_1, COMMACAT, PRE_CIRC, CAT_5; constructors MSUALG_3, PRE_CIRC, OPPCAT_1, CAT_5, ISOCAT_1, DOMAIN_1; clusters FUNCT_1, RELSET_1, PRVECT_1, CAT_2, PBOOLE, PRALG_1, CAT_5, FUNCOP_1, CIRCCOMB, RELAT_1; requirements SUBSET, BOOLE; definitions TARSKI, FRAENKEL, UNIALG_1, PBOOLE, PRALG_1, CAT_5; theorems MCART_1, RELAT_1, FUNCT_1, FUNCT_2, FUNCOP_1, PARTFUN1, FUNCT_4, DTCONSTR, CAT_1, CAT_2, OPPCAT_1, ISOCAT_1, COMMACAT, CAT_5, PBOOLE, RELSET_1; schemes ZFREFLE1, MSUALG_1, CAT_5; begin :: Category Yielding Functions definition let A be non empty set; cluster non empty-yielding ManySortedSet of A; existence proof consider f being non-empty ManySortedSet of A; take f; consider a being Element of A; take a; thus a in A & f.a is not empty; end; end; definition let A be non empty set; cluster non-empty -> non empty-yielding ManySortedSet of A; coherence proof let f be ManySortedSet of A; assume A1: f is non-empty; consider a being Element of A; take a; thus a in A & f.a is not empty by A1,PBOOLE:def 16; end; end; definition let C be Categorial Category; let f be Morphism of C; redefine func f`2 -> Functor of f`11, f`12; coherence proof f`11 = cat f`11 & f`12 = cat f`12 & dom f = f`11 & cod f = f`12 by CAT_5:2,def 7; hence f`2 is Functor of f`11, f`12; end; end; theorem for C being Categorial Category, f,g being Morphism of C st dom g = cod f holds g*f = [[dom f, cod g], g`2*f`2] proof let C be Categorial Category; let f,g be Morphism of C; assume A1: dom g = cod f; A2: g`11 = dom g & g`12 = cod g & f`11 = dom f & f`12 = cod f by CAT_5:13; then consider gg being Functor of g`11, g`12 such that A3: g = [[dom g, cod g], gg] by CAT_5:def 6; consider ff being Functor of f`11, g`11 such that A4: f = [[dom f, cod f], ff] by A1,A2,CAT_5:def 6; thus g*f = [[dom f, cod g], gg*ff] by A1,A2,A3,A4,CAT_5:def 6 .= [[dom f, cod g], gg*f`2] by A4,MCART_1:7 .= [[dom f, cod g], g`2*f`2] by A3,MCART_1:7; end; theorem Th2: for C being Category, D,E being Categorial Category for F being Functor of C,D for G being Functor of C,E st F = G holds Obj F = Obj G proof let C be Category, D,E be Categorial Category; let F be Functor of C,D; let G be Functor of C,E such that A1: F = G; A2: dom Obj F = the Objects of C & dom Obj G = the Objects of C by FUNCT_2:def 1; now let x be set; assume x in the Objects of C; then reconsider a = x as Object of C; A3: a = dom id a by CAT_1:44; hence (Obj F).x = dom (F.(id a qua Morphism of C)) by CAT_1:105 .= (F.(id a qua Morphism of C))`11 by CAT_5:13 .= dom (G.(id a qua Morphism of C)) by A1,CAT_5:13 .= (Obj G).x by A3,CAT_1:105; end; hence Obj F = Obj G by A2,FUNCT_1:9; end; definition let IT be Function; attr IT is Category-yielding means: Def1: for x being set st x in dom IT holds IT.x is Category; end; definition cluster Category-yielding Function; existence proof take f = {} --> 1Cat(0,1); let x be set; assume x in dom f; hence thesis by FUNCOP_1:19; end; end; definition let X be set; cluster Category-yielding ManySortedSet of X; existence proof take f = X --> 1Cat(0,1); let x be set; assume x in dom f; then x in X by FUNCOP_1:19; hence thesis by FUNCOP_1:13; end; end; definition let A be set; mode ManySortedCategory of A is Category-yielding ManySortedSet of A; end; definition let C be Category; mode ManySortedSet of C is ManySortedSet of the Objects of C; mode ManySortedCategory of C is ManySortedCategory of the Objects of C; end; definition let X be set, x be Category; cluster X --> x -> Category-yielding; coherence proof let a be set; assume a in dom (X --> x); then a in X by FUNCOP_1:19; hence thesis by FUNCOP_1:13; end; end; definition let X be non empty set; cluster -> non empty ManySortedSet of X; coherence proof let f be ManySortedSet of X; consider x be Element of X; dom f = X by PBOOLE:def 3; then [x,f.x] in f by FUNCT_1:def 4; hence f is non empty; end; end; definition let f be Category-yielding Function; cluster rng f -> categorial; coherence proof let x be set; assume x in rng f; then ex y being set st y in dom f & x = f.y by FUNCT_1:def 5; hence x is Category by Def1; end; end; definition let X be non empty set; let f be ManySortedCategory of X; let x be Element of X; redefine func f.x -> Category; coherence proof dom f = X by PBOOLE:def 3; hence thesis by Def1; end; end; definition let f be Function; let g be Category-yielding Function; cluster g*f -> Category-yielding; coherence proof let x be set; assume x in dom (g*f); then (g*f).x = g.(f.x) & f.x in dom g by FUNCT_1:21,22; hence (g*f).x is Category by Def1; end; end; :: Objects and Morphisms definition let F be Category-yielding Function; func Objs F -> non-empty Function means: Def2: dom it = dom F & for x being set st x in dom F for C being Category st C = F.x holds it.x = the Objects of C; existence proof defpred P[set,set] means for C being Category st C = F.$1 holds $2 = the Objects of C; A1: now let x be set; assume x in dom F; then reconsider C = F.x as Category by Def1; reconsider y = the Objects of C as set; take y; thus P[x,y]; end; consider f being Function such that A2: dom f = dom F & for x being set st x in dom F holds P[x,f.x] from NonUniqFuncEx(A1); f is non-empty proof let x be set; assume A3: x in dom f; then reconsider C = F.x as Category by A2,Def1; f.x = the Objects of C by A2,A3; hence thesis; end; hence thesis by A2; end; uniqueness proof let f1, f2 be non-empty Function such that A4: dom f1 = dom F and A5: for x being set st x in dom F for C being Category st C = F.x holds f1.x = the Objects of C and A6: dom f2 = dom F and A7: for x being set st x in dom F for C being Category st C = F.x holds f2.x = the Objects of C; now let x be set; assume A8: x in dom F; then reconsider C = F.x as Category by Def1; thus f1.x = the Objects of C by A5,A8 .= f2.x by A7,A8; end; hence thesis by A4,A6,FUNCT_1:9; end; func Mphs F -> non-empty Function means: Def3: dom it = dom F & for x being set st x in dom F for C being Category st C = F.x holds it.x = the Morphisms of C; existence proof defpred P[set,set] means for C being Category st C = F.$1 holds $2 = the Morphisms of C; A9: now let x be set; assume x in dom F; then reconsider C = F.x as Category by Def1; reconsider y = the Morphisms of C as set; take y; thus P[x,y]; end; consider f being Function such that A10: dom f = dom F & for x being set st x in dom F holds P[x,f.x] from NonUniqFuncEx(A9); f is non-empty proof let x be set; assume A11: x in dom f; then reconsider C = F.x as Category by A10,Def1; f.x = the Morphisms of C by A10,A11; hence thesis; end; hence thesis by A10; end; uniqueness proof let f1, f2 be non-empty Function such that A12: dom f1 = dom F and A13: for x being set st x in dom F for C being Category st C = F.x holds f1.x = the Morphisms of C and A14: dom f2 = dom F and A15: for x being set st x in dom F for C being Category st C = F.x holds f2.x = the Morphisms of C; now let x be set; assume A16: x in dom F; then reconsider C = F.x as Category by Def1; thus f1.x = the Morphisms of C by A13,A16 .= f2.x by A15,A16; end; hence thesis by A12,A14,FUNCT_1:9; end; end; definition let A be non empty set; let F be ManySortedCategory of A; redefine func Objs F -> non-empty ManySortedSet of A; coherence proof dom Objs F = dom F by Def2 .= A by PBOOLE:def 3; hence thesis by PBOOLE :def 3; end; func Mphs F -> non-empty ManySortedSet of A; coherence proof dom Mphs F = dom F by Def3 .= A by PBOOLE:def 3; hence thesis by PBOOLE :def 3; end; end; theorem for X being set, C being Category holds Objs (X --> C) = X --> the Objects of C & Mphs (X --> C) = X --> the Morphisms of C proof let X be set, C be Category; A1: dom Objs (X --> C) = dom (X --> C) & dom Mphs (X --> C) = dom (X --> C) & dom (X --> C) = X by Def2,Def3,FUNCOP_1:19; now let a be set; assume A2: a in dom Objs (X --> C); then (X --> C).a = C by A1,FUNCOP_1:13; hence (Objs (X --> C)).a = the Objects of C by A1,A2,Def2; end; hence Objs (X --> C) = X --> the Objects of C by A1,FUNCOP_1:17; now let a be set; assume A3: a in dom Mphs (X --> C); then (X --> C).a = C by A1,FUNCOP_1:13; hence (Mphs (X --> C)).a = the Morphisms of C by A1,A3,Def3; end; hence thesis by A1,FUNCOP_1:17; end; begin :: Pairs of Many Sorted Sets definition let A,B be set; mode ManySortedSet of A,B means: Def4: ex f being (ManySortedSet of A), g being ManySortedSet of B st it = [f,g]; existence proof consider f being (ManySortedSet of A), g being ManySortedSet of B; take [f,g], f, g; thus thesis; end; end; definition let A,B be set; let f be ManySortedSet of A; let g be ManySortedSet of B; redefine func [f,g] -> ManySortedSet of A,B; coherence proof take f,g; thus thesis; end; end; definition let A, B be set; let X be ManySortedSet of A,B; redefine func X`1 -> ManySortedSet of A; coherence proof consider f being (ManySortedSet of A), g being ManySortedSet of B such that A1: X = [f,g] by Def4; thus thesis by A1,MCART_1:7; end; func X`2 -> ManySortedSet of B; coherence proof consider f being (ManySortedSet of A), g being ManySortedSet of B such that A2: X = [f,g] by Def4; thus thesis by A2,MCART_1:7; end; end; definition let A,B be set; let IT be ManySortedSet of A,B; attr IT is Category-yielding_on_first means: Def5: IT`1 is Category-yielding; attr IT is Function-yielding_on_second means: Def6: IT`2 is Function-yielding; end; definition let A,B be set; cluster Category-yielding_on_first Function-yielding_on_second ManySortedSet of A,B; existence proof consider f being ManySortedCategory of A; consider g being ManySortedFunction of B; reconsider X = [f,g] as ManySortedSet of A,B; take X; thus X`1 is Category-yielding & X`2 is Function-yielding by MCART_1 :7; end; end; definition let A, B be set; let X be Category-yielding_on_first ManySortedSet of A,B; redefine func X`1 -> ManySortedCategory of A; coherence by Def5; end; definition let A, B be set; let X be Function-yielding_on_second ManySortedSet of A,B; redefine func X`2 -> ManySortedFunction of B; coherence by Def6; end; definition let f be Function-yielding Function; cluster rng f -> functional; coherence proof let x be set; assume x in rng f; then ex y being set st y in dom f & x = f.y by FUNCT_1:def 5; hence x is Function; end; end; definition let A,B be set; let f be ManySortedCategory of A; let g be ManySortedFunction of B; redefine func [f,g] -> Category-yielding_on_first Function-yielding_on_second ManySortedSet of A,B; coherence proof [f,g]`1 = f & [f,g]`2 = g by MCART_1:7; hence thesis by Def5,Def6; end; end; definition let A be non empty set; let F,G be ManySortedCategory of A; mode ManySortedFunctor of F,G -> ManySortedFunction of A means: Def7: for a being Element of A holds it.a is Functor of F.a, G.a; existence proof defpred P[set,set] means ex a' being Element of A, t being Functor of F.a', G.a' st $1 = a' & $2 = t; A1: now let a be set; assume a in A; then reconsider a' = a as Element of A; consider f being Functor of F.a', G.a'; reconsider f' = f as set; take f'; thus P[a,f']; end; consider f being Function such that A2: dom f = A & for a being set st a in A holds P[a,f.a] from NonUniqFuncEx(A1); f is Function-yielding proof let a be set; assume a in dom f; then ex a' being Element of A, t being Functor of F.a', G.a' st a = a' & f.a = t by A2; hence thesis; end; then reconsider f as ManySortedFunction of A by A2,PBOOLE:def 3; take f; let a be Element of A; ex a' being Element of A, t being Functor of F.a', G.a' st a = a' & f.a = t by A2; hence thesis; end; end; scheme LambdaMSFr {A() -> non empty set, C1, C2() -> ManySortedCategory of A(), F(set) -> set}: ex F being ManySortedFunctor of C1(), C2() st for a being Element of A() holds F.a = F(a) provided A1: for a being Element of A() holds F(a) is Functor of C1().a, C2().a proof deffunc G(set)=F($1); consider f being ManySortedSet of A() such that A2: for a being set st a in A() holds f.a = G(a) from MSSLambda; f is Function-yielding proof let a be set; assume a in dom f; then reconsider a as Element of A() by PBOOLE:def 3; f.a = F(a) by A2; hence thesis by A1; end; then reconsider f as ManySortedFunction of A(); f is ManySortedFunctor of C1(), C2() proof let a be Element of A(); f.a = F(a) by A2; hence thesis by A1; end; then reconsider f as ManySortedFunctor of C1(), C2(); take f; thus thesis by A2; end; definition let A be non empty set; let F,G be ManySortedCategory of A; let f be ManySortedFunctor of F,G; let a be Element of A; redefine func f.a -> Functor of F.a, G.a; coherence by Def7; end; begin :: Indexing definition let A,B be non empty set; let F,G be Function of B,A; mode Indexing of F,G -> Category-yielding_on_first ManySortedSet of A,B means: Def8: it`2 is ManySortedFunctor of it`1*F, it`1*G; existence proof consider g being ManySortedCategory of A; consider f being ManySortedFunctor of g*F, g*G; take I = [g,f]; I`1 = g & I`2 = f by MCART_1:7; hence thesis; end; end; theorem Th4: for A,B being non empty set, F,G being Function of B,A for I being Indexing of F,G for m being Element of B holds I`2.m is Functor of I`1.(F.m), I`1.(G.m) proof let A,B be non empty set, F,G be Function of B,A; let I be Indexing of F,G; let m be Element of B; reconsider H = I`2 as ManySortedFunctor of I`1*F, I`1*G by Def8; A1: H.m is Functor of (I`1*F).m, (I`1*G).m; dom (I`1*F) = B & dom (I`1*G) = B by PBOOLE:def 3; then (I`1*F).m = I`1.(F.m) & (I`1*G).m = I`1.(G.m) by FUNCT_1:22; hence I`2.m is Functor of I`1.(F.m), I`1.(G.m) by A1; end; theorem Th5: for C being Category, I being Indexing of the Dom of C, the Cod of C for m being Morphism of C holds I`2.m is Functor of I`1.dom m, I`1.cod m proof let C be Category, I be Indexing of the Dom of C, the Cod of C; let m be Morphism of C; dom m = (the Dom of C).m & cod m = (the Cod of C).m by CAT_1:def 2,def 3; hence thesis by Th4; end; definition let A,B be non empty set; let F,G be Function of B,A; let I be Indexing of F,G; redefine func I`2 -> ManySortedFunctor of I`1*F,I`1*G; coherence by Def8; end; Lm1: now let A,B be non empty set; let F,G be Function of B,A; let I be Indexing of F,G; A1: dom I`1 = A by PBOOLE:def 3; consider C being full (strict Categorial Category) such that A2: the Objects of C = rng I`1 by CAT_5:20; take C; thus for a be Element of A holds I`1.a is Object of C by A1,A2,FUNCT_1:def 5; let b be Element of B; I`1.(F.b) is Object of C & I`1.(G.b) is Object of C & I`2.b is Functor of I`1.(F.b), I`1.(G.b) by A1,A2,Th4,FUNCT_1:def 5; hence [[I`1.(F.b),I`1.(G.b)],I`2.b] is Morphism of C by CAT_5:def 8; end; definition let A,B be non empty set; let F,G be Function of B,A; let I be Indexing of F,G; mode TargetCat of I -> Categorial Category means: Def9: (for a being Element of A holds I`1.a is Object of it) & (for b being Element of B holds [[I`1.(F.b),I`1.(G.b)],I`2.b] is Morphism of it); existence proof ex C being full (strict Categorial Category) st (for a being Element of A holds I`1.a is Object of C) & (for b being Element of B holds [[I`1.(F.b),I`1.(G.b)],I`2.b] is Morphism of C) by Lm1; hence thesis; end; end; definition let A,B be non empty set; let F,G be Function of B,A; let I be Indexing of F,G; cluster full strict TargetCat of I; existence proof consider C being full (strict Categorial Category) such that A1: (for a being Element of A holds I`1.a is Object of C) & (for b being Element of B holds [[I`1.(F.b),I`1.(G.b)],I`2.b] is Morphism of C) by Lm1; C is TargetCat of I by A1,Def9; hence thesis; end; end; Lm2: now let C be Category; id C = id the Morphisms of C by CAT_1:def 21; hence id C = (id C)* id C by FUNCT_2:23; end; definition :: This is very technical definition made for having the indexing and :: coindexing as the same mode. let A,B be non empty set; let F,G be Function of B,A; let c be PartFunc of [:B,B:], B; let i be Function of A,B; given C being Category such that A1: C = CatStr(#A, B, F, G, c, i#); set I1 = A --> C; set I2 = B --> id C; A2: [I1,I2]`1 = I1 & [I1,I2]`2 = I2 by MCART_1:7; I2 is ManySortedFunctor of I1*F, I1*G proof let a be Element of B; I1.(F.a) = C & I1.(G.a) = C & dom (I1*F) = B & dom (I1*G) = B by FUNCOP_1:13,PBOOLE:def 3; then I2.a = id C & (I1*F).a = C & (I1*G).a = C by FUNCOP_1:13,FUNCT_1:22; hence thesis; end; then reconsider I = [I1,I2] as Indexing of F,G by A2,Def8; mode Indexing of F, G, c, i -> Indexing of F,G means: Def10: (for a being Element of A holds it`2.(i.a) = id (it`1.a)) & for m1, m2 being Element of B st F.m2 = G.m1 holds it`2.(c.[m2,m1]) = (it`2.m2)*(it`2.m1); existence proof take I; hereby let a be Element of A; thus I`2.(i.a) = id C by A2,FUNCOP_1:13 .= id (I`1.a) by A2,FUNCOP_1:13; end; let m1, m2 be Element of B; assume F.m2 = G.m1; then [m2,m1] in dom c by A1,CAT_1:def 8; then c.[m2,m1] in B by PARTFUN1:27; then I`2.(c.[m2,m1]) = id C & I`2.m1 = id C & I`2.m2 = id C by A2,FUNCOP_1 :13; hence thesis by Lm2; end; end; definition let C be Category; mode Indexing of C is Indexing of the Dom of C, the Cod of C, the Comp of C, the Id of C; mode coIndexing of C is Indexing of the Cod of C, the Dom of C, ~(the Comp of C), the Id of C; end; theorem Th6: for C being Category, I being Indexing of the Dom of C, the Cod of C holds I is Indexing of C iff (for a being Object of C holds I`2.id a = id (I`1.a)) & for m1, m2 being Morphism of C st dom m2 = cod m1 holds I`2.(m2*m1) = (I`2.m2)*(I`2.m1) proof let C be Category; reconsider D = the CatStr of C as Category by CAT_5:1; A1: D = CatStr(# the Objects of C, the Morphisms of C, the Dom of C, the Cod of C, the Comp of C, the Id of C#); let I be Indexing of the Dom of C, the Cod of C; hereby assume A2: I is Indexing of C; hereby let a be Object of C; thus I`2.id a = I`2.((the Id of C).a) by CAT_1:def 5 .= id (I`1.a) by A1,A2,Def10; end; let m1, m2 be Morphism of C; assume A3: dom m2 = cod m1; dom m2 = (the Dom of C).m2 & cod m1 = (the Cod of C).m1 by CAT_1:def 2,def 3; then I`2.((the Comp of C).[m2,m1]) = (I`2.m2)*(I`2.m1) by A1,A2,A3,Def10; hence I`2.(m2*m1) = (I`2.m2)*(I`2.m1) by A3,CAT_1:41; end; assume A4: (for a being Object of C holds I`2.id a = id (I`1.a)) & for m1, m2 being Morphism of C st dom m2 = cod m1 holds I`2.(m2*m1) = (I`2.m2)*(I`2.m1); thus ex D being Category st D = CatStr(# the Objects of C, the Morphisms of C, the Dom of C, the Cod of C, the Comp of C, the Id of C#) by A1; hereby let a be Object of C; thus I`2.((the Id of C).a) = I`2.id a by CAT_1:def 5 .= id (I`1.a) by A4; end; let m1, m2 be Morphism of C; assume (the Dom of C).m2 = (the Cod of C).m1; then (the Dom of C).m2 = cod m1 by CAT_1:def 3; then A5: dom m2 = cod m1 by CAT_1:def 2; then I`2.(m2*m1) = (I`2.m2)*(I`2.m1) by A4; hence I`2.((the Comp of C).[m2,m1]) = (I`2.m2)*(I`2.m1) by A5,CAT_1:41; end; theorem Th7: for C being Category, I being Indexing of the Cod of C, the Dom of C holds I is coIndexing of C iff (for a being Object of C holds I`2.id a = id (I`1.a)) & for m1, m2 being Morphism of C st dom m2 = cod m1 holds I`2.(m2*m1) = (I`2.m1)*(I`2.m2) proof let C be Category; A1: C opp = CatStr(# the Objects of C, the Morphisms of C, the Cod of C, the Dom of C, ~the Comp of C, the Id of C#) by OPPCAT_1:def 1; let I be Indexing of the Cod of C, the Dom of C; hereby assume A2: I is coIndexing of C; hereby let a be Object of C; thus I`2.id a = I`2.((the Id of C).a) by CAT_1:def 5 .= id (I`1.a) by A1,A2,Def10; end; let m1, m2 be Morphism of C; assume A3: dom m2 = cod m1; dom m2 = (the Dom of C).m2 & cod m1 = (the Cod of C).m1 by CAT_1:def 2,def 3; then I`2.((~the Comp of C).[m1,m2]) = (I`2.m1)*(I`2.m2) & [m2,m1] in dom the Comp of C by A1,A2,A3,Def10,CAT_1:40; then I`2.((the Comp of C).[m2,m1]) = (I`2.m1)*(I`2.m2) by FUNCT_4:def 2; hence I`2.(m2*m1) = (I`2.m1)*(I`2.m2) by A3,CAT_1:41; end; assume A4: (for a being Object of C holds I`2.id a = id (I`1.a)) & for m1, m2 being Morphism of C st dom m2 = cod m1 holds I`2.(m2*m1) = (I`2.m1)*(I`2.m2); thus ex D being Category st D = CatStr(# the Objects of C, the Morphisms of C, the Cod of C, the Dom of C, ~the Comp of C, the Id of C#) by A1; hereby let a be Object of C; thus I`2.((the Id of C).a) = I`2.id a by CAT_1:def 5 .= id (I`1.a) by A4; end; let m1, m2 be Morphism of C; assume (the Cod of C).m2 = (the Dom of C).m1; then (the Dom of C).m1 = cod m2 by CAT_1:def 3; then A5: dom m1 = cod m2 by CAT_1:def 2; then I`2.(m1*m2) = (I`2.m2)*(I`2.m1) by A4; then I`2.((the Comp of C).[m1,m2]) = (I`2.m2)*(I`2.m1) & [m1,m2] in dom the Comp of C by A5,CAT_1:40,41; hence I`2.((~the Comp of C).[m2,m1]) = (I`2.m2)*(I`2.m1) by FUNCT_4:def 2; end; theorem for C being Category, x be set holds x is coIndexing of C iff x is Indexing of C opp proof let C be Category, x be set; C opp = CatStr(# the Objects of C, the Morphisms of C, the Cod of C, the Dom of C, ~the Comp of C, the Id of C#) by OPPCAT_1:def 1; hence thesis; end; theorem for C being Category, I being Indexing of C for c1,c2 being Object of C st Hom(c1,c2) is non empty for m being Morphism of c1,c2 holds I`2.m is Functor of I`1.c1, I`1.c2 proof let C be Category, I be Indexing of C; let c1,c2 be Object of C such that A1: Hom(c1,c2) is non empty; let m be Morphism of c1,c2; dom (I`1*(the Dom of C)) = the Morphisms of C & dom (I`1*(the Cod of C)) = the Morphisms of C & dom m = c1 & cod m = c2 by A1,CAT_1:23,PBOOLE:def 3; then (I`1*(the Dom of C)).m = I`1.((the Dom of C).m) & (I`1*(the Cod of C)).m = I`1.((the Cod of C).m) & (the Dom of C).m = c1 & (the Cod of C).m = c2 by CAT_1:def 2,def 3,FUNCT_1:22; hence I`2.m is Functor of I`1.c1, I`1.c2; end; theorem for C being Category, I being coIndexing of C for c1,c2 being Object of C st Hom(c1,c2) is non empty for m being Morphism of c1,c2 holds I`2.m is Functor of I`1.c2, I`1.c1 proof let C be Category, I be coIndexing of C; let c1,c2 be Object of C such that A1: Hom(c1,c2) is non empty; let m be Morphism of c1,c2; dom (I`1*(the Dom of C)) = the Morphisms of C & dom (I`1*(the Cod of C)) = the Morphisms of C & dom m = c1 & cod m = c2 by A1,CAT_1:23,PBOOLE:def 3; then (I`1*(the Dom of C)).m = I`1.((the Dom of C).m) & (I`1*(the Cod of C)).m = I`1.((the Cod of C).m) & (the Dom of C).m = c1 & (the Cod of C).m = c2 by CAT_1:def 2,def 3,FUNCT_1:22; hence I`2.m is Functor of I`1.c2, I`1.c1; end; definition let C be Category; let I be Indexing of C; let T be TargetCat of I; func I-functor(C,T) -> Functor of C,T means: Def11: for f being Morphism of C holds it.f = [[I`1.dom f, I`1.cod f], I`2.f]; existence proof A1: dom I`1 = the Objects of C by PBOOLE:def 3; rng I`1 c= the Objects of T proof let x be set; assume x in rng I`1; then consider a being set such that A2: a in dom I`1 & x = I`1.a by FUNCT_1:def 5; reconsider a as Object of C by A2,PBOOLE:def 3; I`1.a is Object of T by Def9; hence thesis by A2; end; then reconsider I1 = I`1 as Function of the Objects of C, the Objects of T by A1,FUNCT_2:def 1,RELSET_1:11; deffunc F(Morphism of C) = [[I`1.dom $1, I`1.cod $1], I`2.$1]; deffunc O(Object of C) = I1.$1; A3: now let f be Morphism of C; dom f = (the Dom of C).f & cod f = (the Cod of C).f by CAT_1:def 2,def 3; hence F(f) is Morphism of T by Def9; let g be Morphism of T; assume A4: g = F(f); hence dom g = F(f)`11 by CAT_5:13 .= O(dom f) by COMMACAT:1; thus cod g = F(f)`12 by A4,CAT_5:13 .= O(cod f) by COMMACAT:1; end; A5: now let a be Object of C; dom id a = a & cod id a = a by CAT_1:44; hence F(id a) = [[I1.a, I1.a], id (I`1.a)] by Th6 .= id O(a) by CAT_5:def 6; end; A6: now let f1,f2 be Morphism of C; let g1,g2 be Morphism of T; assume A7: g1 = F(f1) & g2 = F(f2) & dom f2 = cod f1; A8: I`2.f1 is Functor of I`1.dom f1, I`1.cod f1 & I`2.f2 is Functor of I`1.dom f2, I`1.cod f2 by Th5; dom (f2*f1) = dom f1 & cod (f2*f1) = cod f2 & I`2.(f2*f1) = (I`2.f2)*(I`2.f1) by A7,Th6,CAT_1:42; hence F(f2*f1) = g2*g1 by A7,A8,CAT_5:def 6; end; thus ex F being Functor of C,T st for f being Morphism of C holds F.f = F(f) from FunctorEx(A3,A5,A6); end; uniqueness proof let F1, F2 be Functor of C,T such that A9: for f being Morphism of C holds F1.f = [[I`1.dom f, I`1.cod f], I`2.f] and A10: for f being Morphism of C holds F2.f = [[I`1.dom f, I`1.cod f], I`2.f]; now let f be Morphism of C; thus F1.f = [[I`1.dom f, I`1.cod f], I`2.f] by A9 .= F2.f by A10; end; hence thesis by FUNCT_2:113; end; end; Lm3: for C being Category, I being Indexing of C for T being TargetCat of I holds Obj (I-functor(C,T)) = I`1 proof let C be Category, I be Indexing of C; let T be TargetCat of I; A1: dom Obj (I-functor(C,T)) = the Objects of C & dom I`1 = the Objects of C by FUNCT_2:def 1,PBOOLE:def 3; now let x be set; assume x in the Objects of C; then reconsider a = x as Object of C; A2: dom id a = a & cod id a = a by CAT_1:44; (Obj (I-functor(C,T))).a = dom id ((Obj (I-functor(C,T))).a) by CAT_1: 44 .= dom ((I-functor(C,T)).(id a qua Morphism of C)) by CAT_1:104 .= ((I-functor(C,T) qua Function).id a)`11 by CAT_5:2 .= [[I`1.a, I`1.a], I`2.id a]`11 by A2,Def11; hence (Obj (I-functor(C,T))).x = I`1.x by COMMACAT:1; end; hence Obj (I-functor(C,T)) = I`1 by A1,FUNCT_1:9; end; theorem Th11: for C being Category, I being Indexing of C for T1,T2 being TargetCat of I holds I-functor(C,T1) = I-functor(C,T2) & Obj (I-functor(C,T1)) = Obj (I-functor(C,T2)) proof let C be Category, I be Indexing of C; let T1,T2 be TargetCat of I; A1: dom (I-functor(C,T1)) = the Morphisms of C & dom (I-functor(C,T2)) = the Morphisms of C by FUNCT_2:def 1; now let x be set; assume x in the Morphisms of C; then reconsider f = x as Morphism of C; thus (I-functor(C,T1)).x = [[I`1.dom f, I`1.cod f], I`2.f] by Def11 .= (I-functor(C,T2)).x by Def11; end; hence I-functor(C,T1) = I-functor(C,T2) by A1,FUNCT_1:9; A2: dom Obj (I-functor(C,T1)) = the Objects of C & dom Obj (I-functor(C,T2)) = the Objects of C by FUNCT_2:def 1; now let x be set; assume x in the Objects of C; then reconsider a = x as Object of C; thus (Obj (I-functor(C,T1))).x = I`1.a by Lm3 .= (Obj (I-functor(C,T2))).x by Lm3; end; hence thesis by A2,FUNCT_1:9; end; theorem for C being Category, I being Indexing of C for T being TargetCat of I holds Obj (I-functor(C,T)) = I`1 by Lm3; theorem for C being Category, I being Indexing of C for T being TargetCat of I, x being Object of C holds (I-functor(C,T)).x = I`1.x proof let C be Category, I be Indexing of C; let T be TargetCat of I, x be Object of C; thus (I-functor(C,T)).x = (Obj (I-functor(C,T))).x by CAT_1:def 20 .= I`1.x by Lm3; end; definition let C be Category; let I be Indexing of C; func rng I -> strict TargetCat of I means: Def12: for T being TargetCat of I holds it = Image (I-functor(C,T)); uniqueness proof let T1,T2 be strict TargetCat of I such that A1: for T being TargetCat of I holds T1 = Image (I-functor(C,T)) and A2: for T being TargetCat of I holds T2 = Image (I-functor(C,T)); consider T being TargetCat of I; thus T1 = Image (I-functor(C,T)) by A1 .= T2 by A2; end; existence proof consider S being TargetCat of I; reconsider T = Image (I-functor(C,S)) as strict Subcategory of S; reconsider F = I-functor(C,S) as Functor of C,T by CAT_5:8; T is TargetCat of I proof the Objects of T = rng Obj (I-functor(C,S)) by CAT_5:def 3; then the Objects of T = rng I`1 & dom I`1 = the Objects of C by Lm3,PBOOLE:def 3; hence for a being Object of C holds I`1.a is Object of T by FUNCT_1:def 5; let b be Morphism of C; F.b = [[I`1.dom b, I`1.cod b], I`2.b] & dom b = (the Dom of C).b & cod b = (the Cod of C).b by Def11,CAT_1:def 2,def 3; hence [[I`1.((the Dom of C).b),I`1.((the Cod of C).b)],I`2.b] is Morphism of T; end; then reconsider T as strict TargetCat of I; take T; let T' be TargetCat of I; I-functor(C,S) = I-functor(C,T') by Th11; hence thesis by CAT_5:22; end; end; theorem Th14: for C being Category, I be Indexing of C for D being Categorial Category holds rng I is Subcategory of D iff D is TargetCat of I proof let C be Category, I be Indexing of C; let D be Categorial Category; hereby assume A1: rng I is Subcategory of D; thus D is TargetCat of I proof hereby let a be Object of C; I`1.a is Object of rng I by Def9; hence I`1.a is Object of D by A1,CAT_2:12; end; let b be Morphism of C; [[I`1.((the Dom of C).b),I`1.((the Cod of C).b)],I`2.b] is Morphism of rng I by Def9; hence [[I`1.((the Dom of C).b),I`1.((the Cod of C).b)],I`2.b] is Morphism of D by A1,CAT_2:14; end; end; assume D is TargetCat of I; then reconsider T = D as TargetCat of I; rng I = Image (I-functor(C,T)) by Def12; hence thesis; end; definition let C be Category; let I be Indexing of C; let m be Morphism of C; func I.m -> Functor of I`1.dom m, I`1.cod m equals I`2.m; coherence proof dom (I`1*(the Dom of C)) = the Morphisms of C & dom (I`1*(the Cod of C)) = the Morphisms of C by PBOOLE:def 3; then A1: (I`1*(the Dom of C)).m = I`1.((the Dom of C).m) & (I`1*(the Cod of C)).m = I`1.((the Cod of C).m) by FUNCT_1:22; dom m = (the Dom of C).m & cod m = (the Cod of C).m by CAT_1:def 2,def 3; hence thesis by A1; end; end; definition let C be Category; let I be coIndexing of C; let m be Morphism of C; func I.m -> Functor of I`1.cod m, I`1.dom m equals I`2.m; coherence proof dom (I`1*(the Dom of C)) = the Morphisms of C & dom (I`1*(the Cod of C)) = the Morphisms of C by PBOOLE:def 3; then A1: (I`1*(the Dom of C)).m = I`1.((the Dom of C).m) & (I`1*(the Cod of C)).m = I`1.((the Cod of C).m) by FUNCT_1:22; dom m = (the Dom of C).m & cod m = (the Cod of C).m by CAT_1:def 2,def 3; hence thesis by A1; end; end; Lm4: now let C,D be Category; set F = (the Objects of C) --> D, G = (the Morphisms of C) --> id D; set H = [F,G]; let m be Morphism of C; dom (H`1*(the Dom of C)) = the Morphisms of C by PBOOLE:def 3; then A1: (H`1*(the Dom of C)).m = H`1.((the Dom of C).m) by FUNCT_1:22 .= F.((the Dom of C).m) by MCART_1:7 .= D by FUNCOP_1:13; dom (H`1*(the Cod of C)) = the Morphisms of C by PBOOLE:def 3; then A2: (H`1*(the Cod of C)).m = H`1.((the Cod of C).m) by FUNCT_1:22 .= F.((the Cod of C).m) by MCART_1:7 .= D by FUNCOP_1:13; H`2.m = G.m by MCART_1:7 .= id D by FUNCOP_1:13; hence H`2.m is Functor of (H`1*(the Dom of C)).m, (H`1*(the Cod of C)).m by A1,A2; end; Lm5: now let C,D be Category; set F = (the Objects of C) --> D, G = (the Morphisms of C) --> id D; set H = [F,G]; let m be Morphism of C; dom (H`1*(the Dom of C)) = the Morphisms of C by PBOOLE:def 3; then A1: (H`1*(the Dom of C)).m = H`1.((the Dom of C).m) by FUNCT_1:22 .= F.((the Dom of C).m) by MCART_1:7 .= D by FUNCOP_1:13; dom (H`1*(the Cod of C)) = the Morphisms of C by PBOOLE:def 3; then A2: (H`1*(the Cod of C)).m = H`1.((the Cod of C).m) by FUNCT_1:22 .= F.((the Cod of C).m) by MCART_1:7 .= D by FUNCOP_1:13; H`2.m = G.m by MCART_1:7 .= id D by FUNCOP_1:13; hence H`2.m is Functor of (H`1*(the Cod of C)).m, (H`1*(the Dom of C)).m by A1,A2; end; theorem for C,D being Category holds [(the Objects of C) --> D, (the Morphisms of C) --> id D] is Indexing of C & [(the Objects of C) --> D, (the Morphisms of C) --> id D] is coIndexing of C proof let C,D be Category; set H = [(the Objects of C) --> D, (the Morphisms of C) --> id D], I = H; A1: H`1 = (the Objects of C) --> D & H`2 = (the Morphisms of C) --> id D by MCART_1:7; A2: now let a be Object of C; thus H`2.id a = id D by A1,FUNCOP_1:13 .= id (H`1.a) by A1,FUNCOP_1:13; end; H is Indexing of the Dom of C, the Cod of C proof thus H`2 is ManySortedFunctor of H`1*(the Dom of C), H`1*(the Cod of C) proof thus for m being Morphism of C holds H`2.m is Functor of (H`1*(the Dom of C)).m, (H`1*(the Cod of C)).m by Lm4; end; end; then reconsider H as Indexing of the Dom of C, the Cod of C; now let m1, m2 be Morphism of C; assume dom m2 = cod m1; H`2.(m2*m1) = id D & H`2.m2 = id D & H`2.m1 = id D by A1,FUNCOP_1:13; hence H`2.(m2*m1) = (H`2.m2)*(H`2.m1) by ISOCAT_1:4; end; hence I is Indexing of C by A2,Th6; H is Indexing of the Cod of C, the Dom of C proof thus H`2 is ManySortedFunctor of H`1*(the Cod of C), H`1*(the Dom of C) proof thus for m being Morphism of C holds H`2.m is Functor of (H`1*(the Cod of C)).m, (H`1*(the Dom of C)).m by Lm5; end; end; then reconsider H as Indexing of the Cod of C, the Dom of C; now let m1, m2 be Morphism of C; assume dom m2 = cod m1; H`2.(m2*m1) = id D & H`2.m2 = id D & H`2.m1 = id D by A1,FUNCOP_1:13; hence H`2.(m2*m1) = (H`2.m1)*(H`2.m2) by ISOCAT_1:4; end; hence thesis by A2,Th7; end; begin :: Indexing vs Functor into Categorial Categories definition let C be Category, D be Categorial Category; let F be Functor of C,D; cluster Obj F -> Category-yielding; coherence proof A1: rng Obj F c= the Objects of D by RELSET_1:12; let x be set; assume x in dom Obj F; then (Obj F).x in rng Obj F by FUNCT_1:def 5; hence thesis by A1,CAT_5:12; end; end; theorem Th16: for C being Category, D being Categorial Category, F being Functor of C,D holds [Obj F, pr2 F] is Indexing of C proof let C be Category, D be Categorial Category, F be Functor of C,D; set I = [Obj F, pr2 F]; A1: I`1 = Obj F & I`2 = pr2 F by MCART_1:7; dom Obj F = the Objects of C by FUNCT_2:def 1; then A2: Obj F is ManySortedSet of the Objects of C by PBOOLE:def 3; A3: dom pr2 F = dom F & dom F = the Morphisms of C by DTCONSTR:def 3,FUNCT_2:def 1; then pr2 F is ManySortedSet of the Morphisms of C by PBOOLE:def 3; then reconsider I as ManySortedSet of the Objects of C, the Morphisms of C by A2,Def4; I`1 is Category-yielding by MCART_1:7; then reconsider I as Category-yielding_on_first ManySortedSet of the Objects of C, the Morphisms of C by Def5; pr2 F is Function-yielding proof let x be set; assume x in dom pr2 F; then reconsider x as Morphism of C by A3; reconsider m = F.x as Morphism of D; (pr2 F).x = m`2 by A3,DTCONSTR:def 3; hence thesis; end; then reconsider FF = pr2 F as ManySortedFunction of the Morphisms of C by A3,PBOOLE:def 3; FF is ManySortedFunctor of I`1*the Dom of C, I`1*the Cod of C proof let m be Morphism of C; reconsider x = F.m as Morphism of D; A4: x`11 = dom (F.m) & x`12 = cod (F.m) by CAT_5:13; then consider f being Functor of x`11, x`12 such that A5: F.m = [[x`11, x`12], f] by CAT_5:def 6; A6: FF.m = x`2 by A3,DTCONSTR:def 3 .= f by A5,MCART_1:7; (the Dom of C).m = dom m by CAT_1:def 2; then A7: ((Obj F)*the Dom of C).m = (Obj F).dom m by FUNCT_2:21 .= dom (F.m) by CAT_1:105; (the Cod of C).m = cod m by CAT_1:def 3; then ((Obj F)*the Cod of C).m = (Obj F).cod m by FUNCT_2:21 .= cod (F.m) by CAT_1:105; hence thesis by A1,A4,A6,A7; end; then reconsider I as Indexing of the Dom of C, the Cod of C by A1,Def8; A8: dom F = the Morphisms of C by FUNCT_2:def 1; A9: now let a be Object of C; reconsider i = (Obj F).a as Object of D; thus I`2.id a = (F.(id a qua Morphism of C))`2 by A1,A8,DTCONSTR:def 3 .= (id i qua Morphism of D)`2 by CAT_1:104 .= [[I`1.a, I`1.a], id (I`1.a)]`2 by A1,CAT_5:def 6 .= id (I`1.a) by MCART_1:7; end; now let m1, m2 be Morphism of C; assume A10: dom m2 = cod m1; set h1 = F.m1, h2 = F.m2; A11: dom h2 = h2`11 & cod h2 = h2`12 & dom h1 = h1`11 & cod h1 = h1`12 by CAT_5:13; then consider f2 being Functor of h2`11, h2`12 such that A12: h2 = [[h2`11, h2`12], f2] by CAT_5:def 6; consider f1 being Functor of h1`11, h1`12 such that A13: h1 = [[h1`11, h1`12], f1] by A11,CAT_5:def 6; A14: dom h2 = F.dom m2 by CAT_1:109 .= cod h1 by A10,CAT_1:109; thus I`2.(m2*m1) = (F.(m2*m1))`2 by A1,A8,DTCONSTR:def 3 .= (h2*h1)`2 by A10,CAT_1:99 .= [[h1`11, h2`12], f2*f1]`2 by A11,A12,A13,A14,CAT_5:def 6 .= f2*f1 by MCART_1:7 .= h2`2*f1 by A12,MCART_1:7 .= h2`2*h1`2 by A13,MCART_1:7 .= (I`2.m2)*h1`2 by A1,A8,DTCONSTR:def 3 .= (I`2.m2)*(I`2.m1) by A1,A8,DTCONSTR:def 3; end; hence thesis by A9,Th6; end; definition let C be Category; let D be Categorial Category; let F be Functor of C,D; func F-indexing_of C -> Indexing of C equals: Def15: [Obj F, pr2 F]; coherence by Th16; end; theorem for C being Category, D being Categorial Category, F being Functor of C,D holds D is TargetCat of F-indexing_of C proof let C be Category, D be Categorial Category, F be Functor of C,D; set I = F-indexing_of C; I = [Obj F, pr2 F] by Def15; then A1: I`1 = Obj F & I`2 = pr2 F by MCART_1:7; A2: dom F = the Morphisms of C & dom Obj F = the Objects of C by FUNCT_2:def 1 ; thus for a being Object of C holds I`1.a is Object of D by A1,FUNCT_2:7; let b be Morphism of C; set h = F.b; A3: dom h = h`11 & cod h = h`12 by CAT_5:13; then consider f being Functor of h`11, h`12 such that A4: h = [[h`11, h`12], f] by CAT_5:def 6; A5: dom h = (Obj F).dom b by CAT_1:105 .= (Obj F).((the Dom of C).b) by CAT_1:def 2; A6: cod h = (Obj F).cod b by CAT_1:105 .= (Obj F).((the Cod of C).b) by CAT_1:def 3; I`2.b = h`2 by A1,A2,DTCONSTR:def 3 .= f by A4,MCART_1:7; hence thesis by A1,A3,A4,A5,A6; end; theorem Th18: for C being Category, D being Categorial Category, F being Functor of C,D for T being TargetCat of F-indexing_of C holds F = (F-indexing_of C)-functor(C,T) proof let C be Category, D be Categorial Category, F be Functor of C,D; set I = F-indexing_of C; let T be TargetCat of I; I = [Obj F, pr2 F] by Def15; then A1: I`1 = Obj F & I`2 = pr2 F by MCART_1:7; A2: dom F = the Morphisms of C & dom Obj F = the Objects of C by FUNCT_2:def 1 ; A3: dom (I-functor(C,T)) = the Morphisms of C by FUNCT_2:def 1; now let x be set; assume x in the Morphisms of C; then reconsider f = x as Morphism of C; set h = F.f; A4: dom h = h`11 & cod h = h`12 by CAT_5:13; then consider g being Functor of h`11, h`12 such that A5: h = [[h`11, h`12], g] by CAT_5:def 6; A6: dom h = (Obj F).dom f & cod h = (Obj F).cod f by CAT_1:105; I`2.f = h`2 by A1,A2,DTCONSTR:def 3 .= g by A5,MCART_1:7; hence F.x = (I-functor(C,T)).x by A1,A4,A5,A6,Def11; end; hence thesis by A2,A3,FUNCT_1:9; end; theorem Th19: for C being Category, D,E being Categorial Category for F being Functor of C,D for G being Functor of C,E st F = G holds F-indexing_of C = G-indexing_of C proof let C be Category, D,E be Categorial Category; let F be Functor of C,D; let G be Functor of C,E; assume A1: F = G; thus F-indexing_of C = [Obj F, pr2 F] by Def15 .= [Obj G, pr2 G] by A1,Th2 .= G-indexing_of C by Def15; end; theorem Th20: for C being Category, I being (Indexing of C), T being TargetCat of I holds pr2 (I-functor(C,T)) = I`2 proof let C be Category, I be Indexing of C; let T be TargetCat of I; A1: dom pr2 (I-functor(C,T)) = dom (I-functor(C,T)) by DTCONSTR:def 3; A2: dom (I-functor(C,T)) = the Morphisms of C by FUNCT_2:def 1; A3: dom I`2 = the Morphisms of C by PBOOLE:def 3; now let x be set; assume x in the Morphisms of C; then reconsider f = x as Morphism of C; (I-functor(C,T)).f = [[I`1.dom f, I`1.cod f], I`2.f] by Def11; then ((I-functor(C,T)).x)`2 = I`2.f by MCART_1:7; hence (pr2 (I-functor(C,T))).x = I`2.x by A2,DTCONSTR:def 3; end; hence thesis by A1,A2,A3,FUNCT_1:9; end; theorem for C being Category, I being (Indexing of C), T being TargetCat of I holds (I-functor(C,T))-indexing_of C = I proof let C be Category, I be Indexing of C; let T be TargetCat of I; set F = I-functor(C,T); A1: ex f being (ManySortedSet of the Objects of C), g being ManySortedSet of the Morphisms of C st I = [f,g] by Def4; thus F-indexing_of C = [Obj F, pr2 F] by Def15 .= [I`1, pr2 F] by Lm3 .= [I`1, I`2] by Th20 .= I by A1,MCART_1:8; end; begin :: Composing Indexings and Functors Lm6: now let c,d be Category, e be Subcategory of d; let f be Functor of c,e; (incl e)*f = (id e)*f by CAT_2:def 5 .= (id the Morphisms of e)*f by CAT_1:def 21 .= f by FUNCT_2:23; hence f is Functor of c,d; end; definition let C,D,E be Category; let F be Functor of C,D; let I be Indexing of E; assume Image F is Subcategory of E; then reconsider A = Image F as Subcategory of E; reconsider G = F as Functor of C, A by CAT_5:8; reconsider G as Functor of C, E by Lm6; func I*F -> Indexing of C means: Def16: for F' being Functor of C,E st F' = F holds it = (I-functor(E,rng I)*F')-indexing_of C; existence proof take (I-functor(E,rng I)*G)-indexing_of C; thus thesis; end; uniqueness proof let J1, J2 be Indexing of C such that A1: for F' being Functor of C,E st F' = F holds J1 = (I-functor(E,rng I)*F')-indexing_of C and A2: for F' being Functor of C,E st F' = F holds J2 = (I-functor(E,rng I)*F')-indexing_of C; thus J1 = (I-functor(E,rng I)*G)-indexing_of C by A1 .= J2 by A2; end; end; theorem Th22: for C,D1,D2,E being Category, I being Indexing of E for F being Functor of C,D1 for G being Functor of C,D2 st Image F is Subcategory of E & Image G is Subcategory of E & F = G holds I*F = I*G proof let C,D1,D2,E be Category, I be Indexing of E; let F be Functor of C,D1, G be Functor of C,D2; assume A1: Image F is Subcategory of E & Image G is Subcategory of E & F = G; reconsider F' = F as Functor of C, Image F by CAT_5:8; reconsider G' = G as Functor of C, Image G by CAT_5:8; reconsider F', G' as Functor of C,E by A1,Lm6; I*F = (I-functor(E,rng I)*F')-indexing_of C & I*G = (I-functor(E,rng I)*G')-indexing_of C by A1,Def16; hence I*F = I*G by A1; end; theorem Th23: for C,D being Category, F being Functor of C,D, I being Indexing of D for T being TargetCat of I holds I*F = ((I-functor(D,T))*F)-indexing_of C proof let C,D be Category; let F be Functor of C,D; let I be Indexing of D; let T be TargetCat of I; Image F is Subcategory of D & I-functor(D,rng I) = I-functor(D,T) by Th11; then (I-functor(D,rng I))*F = (I-functor(D,T))*F & I*F = (I-functor(D,rng I)*F)-indexing_of C by Def16; hence thesis by Th19; end; theorem Th24: for C,D being Category, F being Functor of C,D, I being Indexing of D for T being TargetCat of I holds T is TargetCat of I*F proof let C,D be Category; let F be Functor of C,D; let I be Indexing of D; let T be TargetCat of I; consider T' being TargetCat of I*F; A1: rng (I*F) = Image ((I*F)-functor(C,T')) by Def12; I*F = ((I-functor(D,T))*F)-indexing_of C by Th23; then (I*F)-functor(C,T') = (I-functor(D,T))*F by Th18; then rng (I*F) = Image ((I-functor(D,T))*F) by A1,CAT_5:22; hence thesis by Th14; end; theorem for C,D being Category, F being Functor of C,D, I being Indexing of D for T being TargetCat of I holds rng (I*F) is Subcategory of T proof let C,D be Category; let F be Functor of C,D; let I be Indexing of D; let T be TargetCat of I; T is TargetCat of I*F by Th24; hence thesis by Th14; end; theorem Th26: for C,D,E being Category, F being Functor of C,D for G being Functor of D,E, I being Indexing of E holds (I*G)*F = I*(G*F) proof let C,D,E be Category; let F be Functor of C,D; let G be Functor of D,E; let I be Indexing of E; set T = rng I; reconsider T' = T as TargetCat of I*G by Th24; I*G = ((I-functor(E,T))*G)-indexing_of D by Th23; then (I*G)-functor(D,T') = (I-functor(E,T))*G by Th18; hence (I*G)*F = ((I-functor(E,T))*G*F)-indexing_of C by Th23 .= ((I-functor(E,T))*(G*F))-indexing_of C by RELAT_1:55 .= I*(G*F) by Th23; end; definition let C be Category; let I be Indexing of C; let D be Categorial Category such that A1: D is TargetCat of I; let E be Categorial Category; let F be Functor of D,E; reconsider T = D as TargetCat of I by A1; reconsider G = F as Functor of T,E; func F*I -> Indexing of C means: Def17: for T being TargetCat of I, G being Functor of T,E st T = D & G = F holds it = (G*(I-functor(C,T)))-indexing_of C; existence proof take (G*(I-functor(C,T)))-indexing_of C; thus thesis; end; uniqueness proof let J1, J2 be Indexing of C; assume A2: not thesis; then J1 = (G*(I-functor(C,T)))-indexing_of C; hence thesis by A2; end; end; theorem Th27: for C being Category, I being Indexing of C for T being TargetCat of I, D,E being Categorial Category for F being Functor of T,D for G being Functor of T,E st F = G holds F*I = G*I proof let C be Category; let I be Indexing of C; let T be TargetCat of I, D,E be Categorial Category; let F be Functor of T,D; let G be Functor of T,E; assume A1: F = G; thus F*I = (F*(I-functor(C,T)))-indexing_of C by Def17 .= (G*(I-functor(C,T)))-indexing_of C by A1,Th19 .= G*I by Def17; end; theorem Th28: for C being Category, I being Indexing of C for T being TargetCat of I, D being Categorial Category for F being Functor of T,D holds Image F is TargetCat of F*I proof let C be Category; let I be Indexing of C; let T be TargetCat of I, D be Categorial Category; let F be Functor of T,D; reconsider F' = F as Functor of T, Image F by CAT_5:8; consider T' being TargetCat of F*I; A1: rng (F*I) = Image ((F*I)-functor(C,T')) by Def12; F*I = F'*I by Th27 .= (F'*(I-functor(C,T)))-indexing_of C by Def17; then (F*I)-functor(C,T') = F'*(I-functor(C,T)) by Th18; then rng (F*I) = Image (F'*(I-functor(C,T))) by A1,CAT_5:22; hence thesis by Th14; end; theorem Th29: for C being Category, I being Indexing of C for T being TargetCat of I, D being Categorial Category for F being Functor of T,D holds D is TargetCat of F*I proof let C be Category; let I be Indexing of C; let T be TargetCat of I, D be Categorial Category; let F be Functor of T,D; Image F is TargetCat of F*I by Th28; then rng (F*I) is Subcategory of Image F by Th14; then rng (F*I) is Subcategory of D by CAT_5:4; hence thesis by Th14; end; theorem for C being Category, I being Indexing of C for T being TargetCat of I, D being Categorial Category for F being Functor of T,D holds rng (F*I) is Subcategory of Image F proof let C be Category; let I be Indexing of C; let T be TargetCat of I, D be Categorial Category; let F be Functor of T,D; Image F is TargetCat of F*I by Th28; hence thesis by Th14; end; theorem for C be Category, I being Indexing of C for T being TargetCat of I for D,E being Categorial Category, F being Functor of T,D for G being Functor of D,E holds (G*F)*I = G*(F*I) proof let C be Category; let I be Indexing of C; let T be TargetCat of I; let D,E be Categorial Category; let F be Functor of T,D; let G be Functor of D,E; reconsider D' = D as TargetCat of F*I by Th29; reconsider G' = G as Functor of D', E; F*I = (F*(I-functor(C,T)))-indexing_of C by Def17; then A1: (F*I)-functor(C,D') = F*(I-functor(C,T)) by Th18; thus (G*F)*I = ((G*F)*(I-functor(C,T)))-indexing_of C by Def17 .= (G'*((F*I)-functor(C,D')))-indexing_of C by A1,RELAT_1:55 .= G*(F*I) by Def17; end; definition let C,D be Category; let I1 be Indexing of C; let I2 be Indexing of D; func I2*I1 -> Indexing of C equals: Def18: I2*(I1-functor(C,rng I1)); correctness; end; theorem Th32: for C being Category, D being Categorial Category, I1 being Indexing of C for I2 being Indexing of D for T being TargetCat of I1 st D is TargetCat of I1 holds I2*I1 = I2*(I1-functor(C,T)) proof let C be Category, D be Categorial Category; let I1 be Indexing of C; let I2 be Indexing of D; let T be TargetCat of I1; assume D is TargetCat of I1; then reconsider D' = D as TargetCat of I1; A1: Image (I1-functor(C,D')) = rng I1 & Image (I1-functor(C,T)) = rng I1 & Image (I1-functor(C,rng I1)) = rng I1 by Def12; A2: I1-functor(C,rng I1) = I1-functor(C,T) by Th11; thus I2*I1 = I2*(I1-functor(C,rng I1)) by Def18 .= I2*(I1-functor(C,T)) by A1,A2,Th22; end; theorem for C being Category, D being Categorial Category, I1 being Indexing of C for I2 being Indexing of D for T being TargetCat of I2 st D is TargetCat of I1 holds I2*I1 = (I2-functor(D,T))*I1 proof let C be Category, D be Categorial Category; let I1 be Indexing of C; let I2 be Indexing of D; let T be TargetCat of I2; assume D is TargetCat of I1; then reconsider D' = D as TargetCat of I1; reconsider I2' = I2 as Indexing of D'; reconsider T' = T as TargetCat of I2'; A1: I1-functor(C,D') = I1-functor(C,rng I1) by Th11; A2: Image (I1-functor(C,D')) = rng I1 & Image (I1-functor(C,rng I1)) = rng I1 by Def12; thus I2*I1 = I2*(I1-functor(C,rng I1)) by Def18 .= I2*(I1-functor(C,D')) by A1,A2,Th22 .= (I2'-functor(D',T')*(I1-functor(C,D')))-indexing_of C by Th23 .= (I2-functor(D,T))*I1 by Def17; end; theorem Th34: for C,D being Category, F being Functor of C,D, I being Indexing of D for T being TargetCat of I, E being Categorial Category for G being Functor of T,E holds (G*I)*F = G*(I*F) :: C --F-> D --I-> T --G-> E proof let C,D be Category, F be Functor of C,D, I be Indexing of D; let T be TargetCat of I; let E be Categorial Category, G be Functor of T,E; A1: G*I = (G*(I-functor(D,T)))-indexing_of D by Def17; then reconsider GI = rng (G*I) as TargetCat of (G*(I-functor(D,T)))-indexing_of D; A2: I*F = ((I-functor(D,T))*F)-indexing_of C by Th23; reconsider T' = T as TargetCat of I*F by Th24; reconsider G' = G as Functor of T', E; A3: ((G*(I-functor(D,T)))-indexing_of D)-functor(D,GI) = G*(I-functor(D,T)) by Th18; Image F is Subcategory of D; hence (G*I)*F = (((G*(I-functor(D,T)))-indexing_of D)-functor(D,GI)*F)-indexing_of C by A1,Def16 .= ((G*(I-functor(D,T))) * F) -indexing_of C by A3,Th19 .= (G * ((I-functor(D,T))*F)) -indexing_of C by RELAT_1:55 .= (G'*((I*F)-functor(C,T')))-indexing_of C by A2,Th18 .= G*(I*F) by Def17; end; theorem for C being Category, I being Indexing of C for T being TargetCat of I, D being Categorial Category for F being Functor of T,D, J being Indexing of D holds (J*F)*I = J*(F*I) :: C --I-> T --F-> D --J-> (?) CAT proof let C be Category, I be Indexing of C; let T be TargetCat of I; let D be Categorial Category, F be Functor of T,D; let J be Indexing of D; F*I = (F*(I-functor(C,T)))-indexing_of C by Def17; then A1: (F*I)-functor(C, rng (F*I)) = F*(I-functor(C,T)) by Th18; A2: Image (F*(I-functor(C,T))) is Subcategory of D; D is TargetCat of F*I by Th29; then rng (F*I) is Subcategory of D by Th14; then A3: Image ((F*I)-functor(C, rng (F*I))) is Subcategory of D by CAT_5:4; thus (J*F)*I = (J*F)*(I-functor(C,T)) by Th32 .= J*(F*(I-functor(C,T))) by Th26 .= J*((F*I)-functor(C, rng (F*I))) by A1,A2,A3,Th22 .= J*(F*I) by Def18; end; theorem for C being Category, I being Indexing of C for T1 being TargetCat of I, J being Indexing of T1 for T2 being TargetCat of J for D being Categorial Category, F being Functor of T2,D holds (F*J)*I = F*(J*I) :: C --I-> T1 --J-> T2 --F-> D proof let C be Category, I be Indexing of C; let T1 be TargetCat of I; let J be Indexing of T1; let T2 be TargetCat of J; let D be Categorial Category, F be Functor of T2,D; thus (F*J)*I = (F*J)*(I-functor(C,T1)) by Th32 .= F*(J*(I-functor(C,T1))) by Th34 .= F*(J*I) by Th32; end; theorem Th37: for C,D being Category, F being Functor of C,D, I being Indexing of D for T being TargetCat of I, J being Indexing of T holds (J*I)*F = J*(I*F) :: C --F-> D --I-> T --J-> (?) CAT proof let C,D be Category, F be Functor of C,D, I be Indexing of D; let T be TargetCat of I, J be Indexing of T; I*F = ((I-functor(D,T))*F)-indexing_of C by Th23; then A1: (I*F)-functor(C, rng (I*F)) = (I-functor(D,T))*F by Th18; A2: Image ((I-functor(D,T))*F) is Subcategory of T; T is TargetCat of I*F by Th24; then rng (I*F) is Subcategory of T by Th14; then A3: Image ((I*F)-functor(C, rng (I*F))) is Subcategory of T by CAT_5:4; thus (J*I)*F = (J*(I-functor(D,T)))*F by Th32 .= J*((I-functor(D,T))*F) by Th26 .= J*((I*F)-functor(C, rng (I*F))) by A1,A2,A3,Th22 .= J*(I*F) by Def18; end; theorem for C being Category, I being Indexing of C for D being TargetCat of I, J being Indexing of D for E being TargetCat of J, K being Indexing of E holds (K*J)*I = K*(J*I) :: C --I-> D --J-> E --K-> (?) CAT proof let C be Category, I be Indexing of C; let D be TargetCat of I, J be Indexing of D; let E be TargetCat of J, K be Indexing of E; thus (K*J)*I = (K*J)*(I-functor(C,D)) by Th32 .= K*(J*(I-functor(C,D))) by Th37 .= K*(J*I) by Th32; end;