Copyright (c) 1999 Association of Mizar Users
environ vocabulary WAYBEL_0, WAYBEL11, WAYBEL_9, WAYBEL17, LATTICES, ORDINAL2, PRE_TOPC, YELLOW_1, ORDERS_1, RELAT_2, SEQM_3, FUNCT_1, RELAT_1, FUNCOP_1, LATTICE3, BOOLE, GROUP_1, BHSP_3, YELLOW_0, FUNCT_2, BORSUK_1, FUNCT_5, MCART_1, QUANTAL1, CAT_1, MONOID_0, WAYBEL_3, PBOOLE, CARD_3, FUNCT_4, WAYBEL24; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, STRUCT_0, RELAT_1, FUNCT_1, FUNCT_2, FUNCT_5, SEQM_3, MONOID_0, CARD_3, FUNCOP_1, BORSUK_1, PRE_TOPC, ORDERS_1, LATTICE3, PBOOLE, FUNCT_7, YELLOW_0, ORDERS_3, WAYBEL_0, YELLOW_1, YELLOW_2, WAYBEL_3, WAYBEL_9, WAYBEL11, WAYBEL17, YELLOW_3; constructors FUNCT_7, SEQM_3, ORDERS_3, WAYBEL_3, WAYBEL_5, WAYBEL_1, WAYBEL17, MONOID_0, BORSUK_1; clusters STRUCT_0, WAYBEL_0, WAYBEL_3, RELSET_1, YELLOW_1, LATTICE3, BORSUK_2, WAYBEL10, WAYBEL11, WAYBEL17, YELLOW_3, YELLOW_0, MONOID_0, SUBSET_1, FUNCT_2, XBOOLE_0; requirements SUBSET, BOOLE; definitions TARSKI, WAYBEL_0, LATTICE3, ORDERS_1, MONOID_0, XBOOLE_0; theorems WAYBEL_0, TARSKI, FUNCT_1, FUNCT_2, YELLOW_0, STRUCT_0, YELLOW_2, WAYBEL_1, ZFMISC_1, LATTICE3, WAYBEL10, CAT_2, WAYBEL17, YELLOW_3, MCART_1, FUNCT_5, YELLOW_5, YELLOW10, WAYBEL_9, FUNCOP_1, BORSUK_1, YELLOW_1, SEQM_3, ORDERS_1, WAYBEL_3, PBOOLE, CARD_3, FUNCT_7, RELAT_1, WAYBEL15, RELSET_1, XBOOLE_0, XBOOLE_1; schemes XBOOLE_0; begin :: Preliminaries theorem for S, T being up-complete Scott TopLattice for M being Subset of SCMaps (S,T) holds "\/" (M, SCMaps (S,T)) is continuous map of S, T proof let S, T be up-complete Scott TopLattice; let M be Subset of SCMaps (S,T); A1: the carrier of SCMaps (S,T) c= the carrier of MonMaps (S,T) by YELLOW_0:def 13; "\/" (M, SCMaps (S,T)) in the carrier of MonMaps (S,T) by A1,TARSKI:def 3; then "\/" (M, SCMaps (S,T)) is Element of MonMaps (S,T); hence thesis by WAYBEL10:10,WAYBEL17:def 2; end; definition let S be non empty RelStr, T be non empty reflexive RelStr; cluster constant -> monotone map of S, T; coherence proof let f be map of S, T; assume A1: f is constant; for x,y being Element of S st x <= y holds f.x <= f.y proof let x,y be Element of S; assume x <= y; x in the carrier of S & y in the carrier of S; then x in dom f & y in dom f by FUNCT_2:def 1; hence thesis by A1,SEQM_3:def 5; end; hence f is monotone by WAYBEL_1:def 2; end; end; definition let S be non empty RelStr, T be reflexive non empty RelStr, a be Element of T; cluster S --> a -> monotone; coherence proof set f = S --> a; for x, y being Element of S st x <= y holds f.x <= f.y proof let x, y be Element of S; assume x <= y; f.x = ((the carrier of S) --> a). x by BORSUK_1:def 3 .= a by FUNCOP_1:13 .= ((the carrier of S) --> a). y by FUNCOP_1:13 .= f.y by BORSUK_1:def 3; hence thesis; end; hence thesis by WAYBEL_1:def 2; end; end; theorem for S being non empty RelStr, T being lower-bounded antisymmetric reflexive non empty RelStr holds Bottom MonMaps(S, T) = S --> Bottom T proof let S be non empty RelStr, T be lower-bounded antisymmetric reflexive non empty RelStr; set L = MonMaps(S, T); reconsider f = S --> Bottom T as Element of L by WAYBEL10:10; A1: f is_>=_than {} by YELLOW_0:6; reconsider f' = f as map of S, T; for b being Element of L st b is_>=_than {} holds f <= b proof let b be Element of L; assume b is_>=_than {}; reconsider b' = b as map of S, T by WAYBEL10:10; reconsider b'' = b', f'' = f as Element of T|^ the carrier of S by YELLOW_0:59; for x being Element of S holds f'.x <= b'.x proof let x be Element of S; f'. x = ((the carrier of S) --> Bottom T). x by BORSUK_1:def 3 .= Bottom T by FUNCOP_1:13; hence thesis by YELLOW_0:44; end; then f' <= b' by YELLOW_2:10; then f'' <= b'' by WAYBEL10:12; hence thesis by YELLOW_0:61; end; then f = "\/"({}, L) by A1,YELLOW_0:30; hence thesis by YELLOW_0:def 11; end; theorem for S being non empty RelStr, T being upper-bounded antisymmetric reflexive non empty RelStr holds Top MonMaps(S, T) = S --> Top T proof let S be non empty RelStr, T be upper-bounded antisymmetric reflexive non empty RelStr; set L = MonMaps(S, T); reconsider f = S --> Top T as Element of L by WAYBEL10:10; A1: f is_<=_than {} by YELLOW_0:6; reconsider f' = f as map of S, T; for b being Element of L st b is_<=_than {} holds f >= b proof let b be Element of L; assume b is_<=_than {}; reconsider b' = b as map of S, T by WAYBEL10:10; reconsider b'' = b', f'' = f as Element of T|^ the carrier of S by YELLOW_0:59; for x being Element of S holds f'.x >= b'.x proof let x be Element of S; f'. x = ((the carrier of S) --> Top T). x by BORSUK_1:def 3 .= Top T by FUNCOP_1:13; hence thesis by YELLOW_0:45; end; then f' >= b' by YELLOW_2:10; then f'' >= b'' by WAYBEL10:12; hence thesis by YELLOW_0:61; end; then f = "/\"({}, L) by A1,YELLOW_0:31; hence thesis by YELLOW_0:def 12; end; FuncFraenkelSL{ B, C() -> non empty RelStr, A(set) -> Element of C(), f() -> Function, P[set]}: f().:{A(x) where x is Element of B(): P[x]} = {f().A(x) where x is Element of B(): P[x]} provided A1: the carrier of C() c= dom f() proof set f = f(); thus f.:{A(x) where x is Element of B(): P[x]} c= {f.A(x) where x is Element of B(): P[x]} proof let y be set; assume y in f.:{A(x) where x is Element of B(): P[x]}; then consider z being set such that A2: z in dom f & z in {A(x) where x is Element of B(): P[x]} & y = f.z by FUNCT_1:def 12; ex x being Element of B() st z = A(x) & P[x] by A2; hence thesis by A2; end; let y be set; assume y in {f.A(x) where x is Element of B(): P[x]}; then consider x being Element of B() such that A3: y = f.A(x) & P[x]; A(x) in the carrier of C(); then A(x) in dom f & A(x) in {A(z) where z is Element of B(): P[z]} by A1,A3; hence thesis by A3,FUNCT_1:def 12; end; Fraenkel6A{ B() -> LATTICE, F(set) -> set, P[set], Q[set] } : { F(v1) where v1 is Element of B() : P[v1] } = { F(v2) where v2 is Element of B() : Q[v2] } provided A1: for v being Element of B() holds P[v] iff Q[v] proof thus { F(v1) where v1 is Element of B() : P[v1] } c= { F(v2) where v2 is Element of B() : Q[v2] } proof let a be set; assume a in { F(v1) where v1 is Element of B() : P[v1] }; then consider V1 being Element of B() such that A2: a = F(V1) & P[V1]; Q[V1] by A1,A2; hence thesis by A2; end; thus { F(v2) where v2 is Element of B() : Q[v2] } c= { F(v1) where v1 is Element of B() : P[v1] } proof let a be set; assume a in { F(v1) where v1 is Element of B() : Q[v1] }; then consider V1 being Element of B() such that A3: a = F(V1) & Q[V1]; P[V1] by A1,A3; hence thesis by A3; end; end; theorem Th4: for S, T being complete LATTICE, f being monotone map of S, T holds for x being Element of S holds f.x = sup (f.:downarrow x) proof let S, T be complete LATTICE, f be monotone map of S, T; let x be Element of S; A1: ex_sup_of downarrow x, S by WAYBEL_0:34; sup downarrow x = x by WAYBEL_0:34; then downarrow x is_<=_than x by A1,YELLOW_0:30; then A2: f.:downarrow x is_<=_than f.x by YELLOW_2:16; for b being Element of T st b is_>=_than f.:downarrow x holds f.x <= b proof let b be Element of T; assume A3: b is_>=_than f.:downarrow x; A4: dom f = the carrier of S by FUNCT_2:def 1; x <= x; then x in downarrow x by WAYBEL_0:17; then f.x in f.:downarrow x by A4,FUNCT_1:def 12; hence thesis by A3,LATTICE3:def 9; end; hence thesis by A2,YELLOW_0:30; end; theorem for S, T being complete lower-bounded LATTICE, f being monotone map of S, T holds ( for x being Element of S holds f.x = "\/"({ f.w where w is Element of S : w <= x },T) ) proof let S, T be complete lower-bounded LATTICE; let f be monotone map of S, T; let x be Element of S; A1: sup (f.:downarrow x) = f. x by Th4 .= f.sup downarrow x by WAYBEL_0:34; A2: the carrier of S c= dom f by FUNCT_2:def 1; deffunc A(Element of S) = $1; defpred P[Element of S] means $1 <= x; A3:f.:{ A(w) where w is Element of S: P[w]} = {f.A(w) where w is Element of S: P[w]} from FuncFraenkelSL(A2); defpred R[Element of S] means ex y1 being Element of S st $1 <= y1 & y1 in {x}; A4: for x2 be Element of S holds P[x2] iff R[x2] :: (ex y1 being Element of S st x2 <= y1 & y1 in {x}) proof let x2 be Element of S; hereby assume A5: x2 <= x; x in {x} by TARSKI:def 1; hence ex y1 being Element of S st x2 <= y1 & y1 in {x} by A5; end; given y1 being Element of S such that A6: x2 <= y1 & y1 in {x}; thus thesis by A6,TARSKI:def 1; end; A7: { A(w) where w is Element of S : P[w]} = {A(x1) where x1 is Element of S : R[x1]} from Fraenkel6A (A4); downarrow x = downarrow {x} by WAYBEL_0:def 17 .= { w where w is Element of S : w <= x } by A7,WAYBEL_0:14; hence thesis by A1,A3,WAYBEL_0:34; end; theorem Th6: for S being RelStr, T being non empty RelStr, F being Subset of (T |^ the carrier of S) holds sup F is map of S, T proof let S be RelStr, T be non empty RelStr; let F be Subset of (T |^ the carrier of S); set f = sup F; f in the carrier of (T |^ the carrier of S); then f in Funcs (the carrier of S, the carrier of T) by YELLOW_1:28; then consider f' being Function such that A1: f' = f & dom f' = the carrier of S & rng f' c= the carrier of T by FUNCT_2:def 2; f' is Function of the carrier of S, the carrier of T by A1,FUNCT_2:def 1,RELSET_1:11; hence thesis by A1; end; begin :: On the Scott continuity of maps definition let X1, X2, Y be non empty RelStr; let f be map of [:X1, X2:], Y; let x be Element of X1; func Proj (f, x) -> map of X2, Y equals :Def1: (curry f).x; correctness proof [:the carrier of X1, the carrier of X2:] = the carrier of [:X1, X2:] by YELLOW_3:def 2; then curry f is Function of the carrier of X1, Funcs(the carrier of X2, the carrier of Y) by CAT_2:1; then (curry f).x in Funcs(the carrier of X2, the carrier of Y) by FUNCT_2:7; then (curry f).x is Function of the carrier of X2, the carrier of Y by FUNCT_2:121; hence thesis; end; end; reserve X1, X2, Y for non empty RelStr, f for map of [:X1, X2:], Y, x for Element of X1, y for Element of X2; theorem Th7: for y being Element of X2 holds (Proj (f, x)). y = f. [x, y] proof let y be Element of X2; A1:Proj (f, x) = (curry f). x by Def1; dom f = the carrier of [:X1, X2:] by FUNCT_2:def 1 .= [:the carrier of X1, the carrier of X2:] by YELLOW_3:def 2; then [x,y] in dom f by ZFMISC_1:106; hence thesis by A1,FUNCT_5:27; end; definition let X1, X2, Y be non empty RelStr; let f be map of [:X1, X2:], Y; let y be Element of X2; func Proj (f, y) -> map of X1, Y equals :Def2: (curry' f).y; correctness proof [:the carrier of X1, the carrier of X2:] = the carrier of [:X1, X2:] by YELLOW_3:def 2; then curry' f is Function of the carrier of X2, Funcs(the carrier of X1, the carrier of Y) by CAT_2:2; then (curry' f).y in Funcs(the carrier of X1, the carrier of Y) by FUNCT_2:7; then (curry' f).y is Function of the carrier of X1, the carrier of Y by FUNCT_2:121; hence thesis; end; end; theorem Th8: for x being Element of X1 holds (Proj (f, y)). x = f. [x, y] proof let x be Element of X1; A1:Proj (f, y) = (curry' f). y by Def2; dom f = the carrier of [:X1, X2:] by FUNCT_2:def 1 .= [:the carrier of X1, the carrier of X2:] by YELLOW_3:def 2; then [x,y] in dom f by ZFMISC_1:106; hence thesis by A1,FUNCT_5:29; end; theorem Th9: for R, S, T being non empty RelStr, f being map of [:R,S:], T, a being Element of R, b being Element of S holds Proj (f, a). b = Proj (f, b). a proof let R, S, T be non empty RelStr, f be map of [:R,S:], T, a be Element of R, b be Element of S; Proj (f, a). b = f. [a, b] by Th7 .= Proj (f, b). a by Th8; hence thesis; end; definition let S be non empty RelStr, T be non empty reflexive RelStr; cluster antitone map of S, T; existence proof consider c being Element of T; take f = S --> c; let x, y be Element of S; assume [x, y] in the InternalRel of S; A1: f = (the carrier of S) --> c by BORSUK_1:def 3; let a, b be Element of T; assume a = f.x & b = f.y; then a = c & b = c by A1,FUNCOP_1:13; hence b <= a; end; end; theorem Th10: for R, S, T being non empty reflexive RelStr, f being map of [:R,S:], T, a being Element of R, b being Element of S st f is monotone holds Proj (f, a) is monotone & Proj (f, b) is monotone proof let R, S, T be non empty reflexive RelStr, f be map of [:R,S:], T; let a be Element of R, b be Element of S; reconsider a as Element of R; reconsider b as Element of S; set g = Proj (f, b), h = Proj (f, a); assume A1: f is monotone; A2:now let x, y be Element of S; assume A3: x <= y; A4: h. x = f. [a, x] by Th7; A5: h. y = f. [a, y] by Th7; a <= a; then [a, x] <= [a, y] by A3,YELLOW_3:11; hence h.x <= h.y by A1,A4,A5,WAYBEL_1:def 2; end; now let x, y be Element of R; assume A6: x <= y; A7: g. x = f. [x, b] by Th8; A8: g. y = f. [y, b] by Th8; b <= b; then [x, b] <= [y, b] by A6,YELLOW_3:11; hence g.x <= g.y by A1,A7,A8,WAYBEL_1:def 2; end; hence thesis by A2,WAYBEL_1:def 2; end; theorem Th11: for R, S, T being non empty reflexive RelStr, f being map of [:R,S:], T, a being Element of R, b being Element of S st f is antitone holds Proj (f, a) is antitone & Proj (f, b) is antitone proof let R, S, T be non empty reflexive RelStr, f be map of [:R,S:], T; let a be Element of R, b be Element of S; reconsider a' = a as Element of R; set g = Proj (f, b), h = Proj (f, a); assume A1: f is antitone; A2:now let x, y be Element of S; assume A3: x <= y; A4: h. x = f. [a, x] by Th7; A5: h. y = f. [a, y] by Th7; a' <= a'; then [a', x] <= [a', y] by A3,YELLOW_3:11; hence h.x >= h.y by A1,A4,A5,WAYBEL_0:def 5; end; now let x, y be Element of R; assume A6: x <= y; A7: g. x = f. [x, b] by Th8; A8: g. y = f. [y, b] by Th8; reconsider b' = b as Element of S; b' <= b'; then [x, b'] <= [y, b'] by A6,YELLOW_3:11; hence g.x >= g.y by A1,A7,A8,WAYBEL_0:def 5; end; hence thesis by A2,WAYBEL_9:def 1; end; definition let R, S, T be non empty reflexive RelStr; let f be monotone map of [:R, S:], T; let a be Element of R; cluster Proj (f, a) -> monotone; coherence by Th10; end; definition let R, S, T be non empty reflexive RelStr; let f be monotone map of [:R, S:], T; let b be Element of S; cluster Proj (f, b) -> monotone; coherence by Th10; end; definition let R, S, T be non empty reflexive RelStr; let f be antitone map of [:R, S:], T; let a be Element of R; cluster Proj (f, a) -> antitone; coherence by Th11; end; definition let R, S, T be non empty reflexive RelStr; let f be antitone map of [:R, S:], T; let b be Element of S; cluster Proj (f, b) -> antitone; coherence by Th11; end; theorem Th12: for R, S, T being LATTICE, f being map of [:R,S:], T st ( for a being Element of R, b being Element of S holds Proj (f, a) is monotone & Proj (f, b) is monotone) holds f is monotone proof let R, S, T be LATTICE, f be map of [:R,S:], T; assume A1: for a being Element of R, b being Element of S holds Proj (f, a) is monotone & Proj (f, b) is monotone; now let x, y be Element of [:R, S:]; assume x <= y; then A2: x`1 <= y`1 & x`2 <= y`2 by YELLOW_3:12; A3: Proj (f, y`2) is monotone by A1; A4: f. [x`1, y`2] = Proj (f, y`2). x`1 by Th8; f. [y`1, y`2] = Proj (f, y`2). y`1 by Th8; then A5: f. [x`1, y`2] <= f. [y`1, y`2] by A2,A3,A4,WAYBEL_1:def 2; A6: Proj (f, x`1) is monotone by A1; A7: f. [x`1, x`2] = Proj (f, x`1). x`2 by Th7; f. [x`1, y`2] = Proj (f, x`1). y`2 by Th7; then f. [x`1, x`2] <= f. [x`1, y`2] by A2,A6,A7,WAYBEL_1:def 2; then A8: f. [x`1, x`2] <= f. [y`1, y`2] by A5,YELLOW_0:def 2; A9: [:the carrier of R, the carrier of S:] = the carrier of [:R, S:] by YELLOW_3:def 2; then f. [y`1, y`2] = f. y by MCART_1:23; hence f.x <= f.y by A8,A9,MCART_1:23; end; hence thesis by WAYBEL_1:def 2; end; theorem for R, S, T being LATTICE, f being map of [:R,S:], T st ( for a being Element of R, b being Element of S holds Proj (f, a) is antitone & Proj (f, b) is antitone) holds f is antitone proof let R, S, T be LATTICE, f be map of [:R,S:], T; assume A1: for a being Element of R, b being Element of S holds Proj (f, a) is antitone & Proj (f, b) is antitone; now let x, y be Element of [:R, S:]; assume x <= y; then A2: x`1 <= y`1 & x`2 <= y`2 by YELLOW_3:12; A3: Proj (f, y`2) is antitone by A1; A4: f. [x`1, y`2] = Proj (f, y`2). x`1 by Th8; f. [y`1, y`2] = Proj (f, y`2). y`1 by Th8; then A5: f. [x`1, y`2] >= f. [y`1, y`2] by A2,A3,A4,WAYBEL_9:def 1; A6: Proj (f, x`1) is antitone by A1; A7: f. [x`1, x`2] = Proj (f, x`1). x`2 by Th7; f. [x`1, y`2] = Proj (f, x`1). y`2 by Th7; then f. [x`1, x`2] >= f. [x`1, y`2] by A2,A6,A7,WAYBEL_9:def 1; then A8: f. [x`1, x`2] >= f. [y`1, y`2] by A5,YELLOW_0:def 2; A9: [:the carrier of R, the carrier of S:] = the carrier of [:R, S:] by YELLOW_3:def 2; then f. [y`1, y`2] = f. y by MCART_1:23; hence f.x >= f.y by A8,A9,MCART_1:23; end; hence thesis by WAYBEL_9:def 1; end; theorem Th14: for R, S, T being LATTICE, f being map of [:R,S:], T, b being Element of S, X being Subset of R holds Proj (f, b).:X = f.:[:X, {b}:] proof let R, S, T be LATTICE, f be map of [:R,S:], T, b be Element of S, X be Subset of R; Proj (f, b).:X = f.:[:X, {b}:] proof thus Proj (f, b).:X c= f.:[:X, {b}:] proof let c be set; assume c in Proj (f, b).:X; then consider k be set such that A1: k in dom Proj (f, b) & k in X & c = Proj (f, b).k by FUNCT_1:def 12; A2: c = f. [k, b] by A1,Th8; b in {b} by TARSKI:def 1; then A3: [k, b] in [:X, {b}:] by A1,ZFMISC_1:106; [:the carrier of R, the carrier of S:] = the carrier of [:R, S:] by YELLOW_3:def 2; then dom f = [:the carrier of R, the carrier of S:] by FUNCT_2:def 1; then [k, b] in dom f by A1,ZFMISC_1:106; hence thesis by A2,A3,FUNCT_1:def 12; end; thus f.:[:X, {b}:] c= Proj (f, b).:X proof let c be set; assume c in f.:[:X, {b}:]; then consider k be set such that A4: k in dom f & k in [:X, {b}:] & c = f.k by FUNCT_1:def 12; consider k1, k2 be set such that A5: k1 in X & k2 in {b} & k = [k1, k2] by A4,ZFMISC_1:def 2; A6: dom Proj (f, b) = the carrier of R by FUNCT_2:def 1; k2 = b by A5,TARSKI:def 1; then c = Proj (f, b). k1 by A4,A5,Th8; hence thesis by A5,A6,FUNCT_1:def 12; end; end; hence thesis; end; theorem Th15: for R, S, T being LATTICE, f being map of [:R,S:], T, b being Element of R, X being Subset of S holds Proj (f, b).:X = f.:[:{b}, X:] proof let R, S, T be LATTICE, f be map of [:R,S:], T, b be Element of R, X be Subset of S; Proj (f, b).:X = f.:[:{b}, X:] proof thus Proj (f, b).:X c= f.:[:{b}, X:] proof let c be set; assume c in Proj (f, b).:X; then consider k be set such that A1: k in dom Proj (f, b) & k in X & c = Proj (f, b).k by FUNCT_1:def 12; A2: c = f. [b, k] by A1,Th7; b in {b} by TARSKI:def 1; then A3: [b, k] in [:{b}, X:] by A1,ZFMISC_1:106; [:the carrier of R, the carrier of S:] = the carrier of [:R, S:] by YELLOW_3:def 2; then dom f = [:the carrier of R, the carrier of S:] by FUNCT_2:def 1; then [b, k] in dom f by A1,ZFMISC_1:106; hence thesis by A2,A3,FUNCT_1:def 12; end; thus f.:[:{b}, X:] c= Proj (f, b).:X proof let c be set; assume c in f.:[:{b}, X:]; then consider k be set such that A4: k in dom f & k in [:{b}, X:] & c = f.k by FUNCT_1:def 12; consider k1, k2 be set such that A5: k1 in {b} & k2 in X & k = [k1, k2] by A4,ZFMISC_1:def 2; A6: dom Proj (f, b) = the carrier of S by FUNCT_2:def 1; k1 = b by A5,TARSKI:def 1; then c = Proj (f, b). k2 by A4,A5,Th7; hence thesis by A5,A6,FUNCT_1:def 12; end; end; hence thesis; end; theorem :: Lemma 2.9 p. 116 (1) => (2) for R, S, T being LATTICE, f being map of [:R,S:], T, a being Element of R, b being Element of S st f is directed-sups-preserving holds Proj (f, a) is directed-sups-preserving & Proj (f, b) is directed-sups-preserving proof let R, S, T be LATTICE, f be map of [:R,S:], T, a be Element of R, b be Element of S; assume A1: f is directed-sups-preserving; A2:for X being Subset of R st X is non empty directed holds Proj (f, b) preserves_sup_of X proof let X be Subset of R; assume X is non empty directed; then reconsider X' = X as non empty directed Subset of R; reconsider Y' = {b} as non empty directed Subset of S by WAYBEL_0:5; Proj (f, b) preserves_sup_of X proof assume A3: ex_sup_of X, R; A4: ex_sup_of Y', S by YELLOW_0:38; then A5: ex_sup_of [:X', Y':], [:R, S:] by A3,YELLOW_3:39; A6: f preserves_sup_of [:X', Y':] by A1,WAYBEL_0:def 37; A7: Proj (f, b).:X = f.:[:X', Y':] by Th14; A8: sup Y' = b by YELLOW_0:39; sup (Proj (f, b).:X) = sup (f.:[:X', Y':]) by Th14 .= f.(sup [:X', Y':]) by A5,A6,WAYBEL_0:def 31 .= f. [sup X', sup Y'] by A3,A4,YELLOW_3:43 .= Proj (f, b).sup X by A8,Th8; hence thesis by A5,A6,A7,WAYBEL_0:def 31; end; hence thesis; end; for X being Subset of S st X is non empty directed holds Proj (f, a) preserves_sup_of X proof let X be Subset of S; assume X is non empty directed; then reconsider X' = X as non empty directed Subset of S; reconsider Y' = {a} as non empty directed Subset of R by WAYBEL_0:5; Proj (f, a) preserves_sup_of X proof assume A9: ex_sup_of X, S; A10: ex_sup_of Y', R by YELLOW_0:38; then A11: ex_sup_of [:Y', X':], [:R, S:] by A9,YELLOW_3:39; A12: f preserves_sup_of [:Y', X':] by A1,WAYBEL_0:def 37; A13: Proj (f, a).:X = f.:[:Y', X':] by Th15; A14: sup Y' = a by YELLOW_0:39; sup (Proj (f, a).:X) = sup (f.:[:Y', X':]) by Th15 .= f.(sup [:Y', X':]) by A11,A12,WAYBEL_0:def 31 .= f. [sup Y', sup X'] by A9,A10,YELLOW_3:43 .= Proj (f, a).sup X by A14,Th7; hence thesis by A11,A12,A13,WAYBEL_0:def 31; end; hence thesis; end; hence thesis by A2,WAYBEL_0:def 37; end; theorem Th17: for R, S, T being LATTICE, f being monotone map of [:R, S:], T, a being Element of R, b being Element of S, X being directed Subset of [:R, S:] st ex_sup_of f.:X, T & a in proj1 X & b in proj2 X holds f. [a, b] <= sup (f.:X) proof let R, S, T be LATTICE, f be monotone map of [:R, S:], T, a be Element of R, b be Element of S, X be directed Subset of [:R, S:]; assume that A1: ex_sup_of f.:X, T and A2: a in proj1 X and A3: b in proj2 X; consider c being set such that A4: [a, c] in X by A2,FUNCT_5:def 1; c in proj2 X by A4,FUNCT_5:def 2; then reconsider c as Element of S; consider d being set such that A5: [d, b] in X by A3,FUNCT_5:def 2; d in proj1 X by A5,FUNCT_5:def 1; then reconsider d as Element of R; consider z being Element of [:R, S:] such that A6: z in X & [a, c] <= z & [d, b] <= z by A4,A5,WAYBEL_0:def 1; [a, c] "\/" [d, b] <= z "\/" z by A6,YELLOW_3:3; then [a, c] "\/" [d, b] <= z by YELLOW_5:1; then A7: [a "\/" d, c "\/" b] <= z by YELLOW10:16; a <= a "\/" d & b <= c "\/" b by YELLOW_0:22; then [a, b] <= [a "\/" d, c "\/" b] by YELLOW_3:11; then A8: f. [a, b] <= f. [a "\/" d, c "\/" b] by WAYBEL_1:def 2; f. [a "\/" d, c "\/" b] <= f. z by A7,WAYBEL_1:def 2; then A9: f. [a, b] <= f. z by A8,YELLOW_0:def 2; dom f = the carrier of [:R, S:] by FUNCT_2:def 1; then A10: f.z in f.:X by A6,FUNCT_1:def 12; f.:X is_<=_than sup (f.:X) by A1,YELLOW_0:30; then f. z <= sup (f.:X) by A10,LATTICE3:def 9; hence thesis by A9,YELLOW_0:def 2; end; theorem :: Lemma 2.9 p. 116 (2) => (1) for R, S, T being complete LATTICE, f being map of [:R, S:], T st ( for a being Element of R, b being Element of S holds Proj (f, a) is directed-sups-preserving & Proj (f, b) is directed-sups-preserving ) holds f is directed-sups-preserving proof let R, S, T be complete LATTICE, f be map of [:R,S:], T; assume A1: for a being Element of R, b being Element of S holds Proj (f, a) is directed-sups-preserving & Proj (f, b) is directed-sups-preserving; for X being Subset of [:R, S:] st X is non empty directed holds f preserves_sup_of X proof let X be Subset of [:R, S:]; assume X is non empty directed; then reconsider X as non empty directed Subset of [:R, S:]; now let a be Element of R, b be Element of S; Proj (f, a) is directed-sups-preserving by A1; hence Proj (f, a) is monotone by WAYBEL17:3; Proj (f, b) is directed-sups-preserving by A1; hence Proj (f, b) is monotone by WAYBEL17:3; end; then A2: f is monotone by Th12; f preserves_sup_of X proof assume A3: ex_sup_of X, [:R, S:]; A4: ex_sup_of f.:X, T by YELLOW_0:17; A5: proj1 X is directed non empty by YELLOW_3:21,22; A6: ex_sup_of proj1 X, R by A3,YELLOW_3:41; A7: ex_sup_of proj2 X, S by A3,YELLOW_3:41; Proj (f, "\/"(proj2 X, S)) is directed-sups-preserving by A1; then A8: Proj (f, "\/"(proj2 X, S)) preserves_sup_of proj1 X by A5,WAYBEL_0:def 37; A9: f.sup X = f. [sup proj1 X, sup proj2 X] by YELLOW_3:46 .= Proj (f, sup proj2 X). sup proj1 X by Th8 .= sup (Proj (f, sup proj2 X).:proj1 X) by A6,A8,WAYBEL_0:def 31; for b being Element of T st b in Proj (f, sup proj2 X).:proj1 X holds b <= sup (f.:X) proof let b be Element of T; assume b in Proj (f, sup proj2 X).:proj1 X; then consider b1 being set such that A10: b1 in dom Proj (f, sup proj2 X) & b1 in proj1 X & b = (Proj (f, sup proj2 X)).b1 by FUNCT_1:def 12; reconsider b1 as Element of R by A10; A11: proj2 X is non empty directed by YELLOW_3:21,22; Proj (f, b1) is directed-sups-preserving by A1; then A12: Proj (f, b1) preserves_sup_of proj2 X by A11,WAYBEL_0:def 37; A13: b = (Proj (f, b1)). sup proj2 X by A10,Th9 .= sup (Proj (f, b1).:proj2 X) by A7,A12,WAYBEL_0:def 31; b <= sup (f.:X) proof A14: ex_sup_of (Proj (f, b1).:proj2 X), T by YELLOW_0:17; (Proj (f, b1).:proj2 X) is_<=_than sup (f.:X) proof let k be Element of T; assume k in (Proj (f, b1).:proj2 X); then consider k1 being set such that A15: k1 in dom Proj (f, b1) & k1 in proj2 X & k = (Proj (f, b1)).k1 by FUNCT_1:def 12; reconsider k1 as Element of S by A15; k = f. [b1, k1] by A15,Th7; hence thesis by A2,A4,A10,A15,Th17; end; hence thesis by A13,A14,YELLOW_0:def 9; end; hence thesis; end; then A16: Proj (f, sup proj2 X).:proj1 X is_<=_than sup (f.:X) by LATTICE3: def 9; ex_sup_of (Proj (f, sup proj2 X).:proj1 X), T by YELLOW_0:17; then A17: f.sup X <= sup (f.:X) by A9,A16,YELLOW_0:def 9; sup (f.:X) <= f.sup X by A2,WAYBEL17:16; hence thesis by A17,YELLOW_0:17,def 3; end; hence thesis; end; hence thesis by WAYBEL_0:def 37; end; theorem Th19: for S being non empty 1-sorted, T being non empty RelStr, f be set holds f is Element of (T |^ the carrier of S) iff f is map of S, T proof let S be non empty 1-sorted, T be non empty RelStr, f be set; hereby assume f is Element of (T |^ the carrier of S); then f in the carrier of (T |^ the carrier of S); then f in Funcs (the carrier of S, the carrier of T) by YELLOW_1:28; then consider a being Function such that A1: a = f & dom a = the carrier of S & rng a c= the carrier of T by FUNCT_2:def 2; f is Function of the carrier of S, the carrier of T by A1,FUNCT_2:def 1,RELSET_1:11; hence f is map of S, T; end; assume f is map of S, T; then f in Funcs (the carrier of S, the carrier of T) by FUNCT_2:11; then f in the carrier of (T |^ the carrier of S) by YELLOW_1:28; hence thesis; end; begin :: The poset of continuous maps definition let S be TopStruct, T be non empty TopRelStr; func ContMaps (S, T) -> strict RelStr means :Def3: it is full SubRelStr of T |^ the carrier of S & for x being set holds x in the carrier of it iff ex f being map of S, T st x = f & f is continuous; existence proof defpred P[set] means ex f be map of S, T st $1 = f & f is continuous; consider X be set such that A1: for a be set holds a in X iff a in the carrier of (T |^ the carrier of S) & P[a] from Separation; X c= the carrier of T |^ the carrier of S proof let a be set; assume a in X; hence thesis by A1; end; then reconsider X as Subset of (T |^ the carrier of S); take SX = subrelstr X; thus SX is full SubRelStr of T |^ the carrier of S; let f be set; hereby assume f in the carrier of SX; then f in X by YELLOW_0:def 15; then consider f' be map of S, T such that A2: f' = f & f' is continuous by A1; take f'; thus f' = f & f' is continuous by A2; end; given f' being map of S, T such that A3: f = f' & f' is continuous; f in Funcs (the carrier of S, the carrier of T) by A3,FUNCT_2:11; then f in the carrier of (T |^ the carrier of S) by YELLOW_1:28; then f in X by A1,A3; hence f in the carrier of SX by YELLOW_0:def 15; end; uniqueness proof let A1, A2 be strict RelStr; assume that A4: A1 is full SubRelStr of T |^ the carrier of S & for x being set holds x in the carrier of A1 iff ex f being map of S, T st x = f & f is continuous and A5: A2 is full SubRelStr of T |^ the carrier of S & for x being set holds x in the carrier of A2 iff ex f being map of S, T st x = f & f is continuous; the carrier of A1 = the carrier of A2 proof hereby let x be set; assume x in the carrier of A1; then consider f being map of S, T such that A6: x = f & f is continuous by A4; thus x in the carrier of A2 by A5,A6; end; let x be set; assume x in the carrier of A2; then consider f being map of S, T such that A7: x = f & f is continuous by A5; thus x in the carrier of A1 by A4,A7; end; hence thesis by A4,A5,YELLOW_0:58; end; end; definition let S be non empty TopSpace, T be non empty TopSpace-like TopRelStr; cluster ContMaps (S, T) -> non empty; coherence proof consider f being continuous map of S, T; f in the carrier of ContMaps(S,T) by Def3; hence thesis by STRUCT_0:def 1; end; end; definition let S be non empty TopSpace, T be non empty TopSpace-like TopRelStr; cluster ContMaps (S, T) -> constituted-Functions; coherence proof let a be Element of ContMaps (S, T); consider a' being map of S, T such that A1: a' = a & a' is continuous by Def3; thus thesis by A1; end; end; theorem Th20: for S being non empty TopSpace, T being non empty reflexive TopSpace-like TopRelStr, x, y being Element of ContMaps (S, T) holds x <= y iff for i being Element of S holds [x.i, y.i] in the InternalRel of T proof let S be non empty TopSpace, T be non empty reflexive TopSpace-like TopRelStr, x, y be Element of ContMaps (S, T); A1:ContMaps (S, T) is full SubRelStr of T |^ the carrier of S by Def3; then reconsider x' = x, y' = y as Element of (T |^ the carrier of S) by YELLOW_0:59; reconsider xa = x', ya = y' as map of S, T by Th19; hereby assume x <= y; then x' <= y' by A1,YELLOW_0:60; then A2: xa <= ya by WAYBEL10:12; let i be Element of S; consider a, b being Element of T such that A3: a = xa.i & b = ya.i & a <= b by A2,YELLOW_2:def 1; thus [x.i, y.i] in the InternalRel of T by A3,ORDERS_1:def 9; end; assume A4: for i being Element of S holds [x.i, y.i] in the InternalRel of T; now let j be set; assume j in the carrier of S; then reconsider i = j as Element of S; reconsider a = xa.i, b = ya.i as Element of T; take a, b; thus a = xa.j & b = ya.j; [a, b] in the InternalRel of T by A4; hence a <= b by ORDERS_1:def 9; end; then xa <= ya by YELLOW_2:def 1; then x' <= y' by WAYBEL10:12; hence thesis by A1,YELLOW_0:61; end; theorem Th21: for S being non empty TopSpace, T being non empty reflexive TopSpace-like TopRelStr, x being set holds x is continuous map of S, T iff x is Element of ContMaps (S, T) proof let S be non empty TopSpace, T be non empty reflexive TopSpace-like TopRelStr, x be set; thus x is continuous map of S, T implies x is Element of ContMaps (S, T) by Def3; assume x is Element of ContMaps (S, T); then x in the carrier of ContMaps (S, T); then consider f being map of S, T such that A1: x = f & f is continuous by Def3; thus thesis by A1; end; definition let S be non empty TopSpace, T be non empty reflexive TopSpace-like TopRelStr; cluster ContMaps (S, T) -> reflexive; coherence proof reconsider CS = ContMaps (S, T) as full SubRelStr of T |^ the carrier of S by Def3; CS is reflexive; hence thesis; end; end; definition let S be non empty TopSpace, T be non empty transitive TopSpace-like TopRelStr; cluster ContMaps (S, T) -> transitive; coherence proof reconsider CS = ContMaps (S, T) as full SubRelStr of T |^ the carrier of S by Def3; CS is transitive; hence thesis; end; end; definition let S be non empty TopSpace, T be non empty antisymmetric TopSpace-like TopRelStr; cluster ContMaps (S, T) -> antisymmetric; coherence proof reconsider CS = ContMaps (S, T) as full SubRelStr of T |^ the carrier of S by Def3; CS is antisymmetric; hence thesis; end; end; definition let S be non empty 1-sorted, T be non empty TopSpace-like TopRelStr; cluster T |^ the carrier of S -> constituted-Functions; coherence proof let a be Element of (T |^ the carrier of S); thus thesis by Th19; end; end; theorem for S being non empty 1-sorted, T being complete LATTICE for f, g, h being map of S, T, i being Element of S st h = "\/" ({f, g}, T |^ the carrier of S) holds h.i = sup {f.i, g.i} proof let S be non empty 1-sorted, T be complete LATTICE; let f, g, h be map of S, T, i be Element of S; assume A1: h = "\/" ({f, g}, T |^ the carrier of S); dom ((the carrier of S) --> T) = the carrier of S by FUNCOP_1:19; then reconsider SYT = (the carrier of S) --> T as non-Empty RelStr-yielding ManySortedSet of the carrier of S by PBOOLE:def 3; reconsider SYT as non-Empty reflexive-yielding RelStr-yielding ManySortedSet of the carrier of S; A2:for i being Element of S holds SYT.i is complete LATTICE by FUNCOP_1:13; reconsider f' = f, g' = g as Element of (T |^ the carrier of S) by Th19; reconsider f', g' as Element of product SYT by YELLOW_1:def 5; reconsider DU = {f', g'} as Subset of product SYT; h.i = (sup DU).i by A1,YELLOW_1:def 5 .= "\/" (pi({f,g},i), SYT.i) by A2,WAYBEL_3:32 .= "\/" (pi({f,g},i), T) by FUNCOP_1:13 .= sup {f.i, g.i} by CARD_3:26; hence thesis; end; theorem Th23: for I being non empty set for J being RelStr-yielding non-Empty reflexive-yielding ManySortedSet of I st for i being Element of I holds J.i is complete LATTICE for X being Subset of product J, i being Element of I holds (inf X).i = inf pi(X,i) proof let I be non empty set; let J be RelStr-yielding non-Empty reflexive-yielding ManySortedSet of I; assume A1: for i being Element of I holds J.i is complete LATTICE; then reconsider L = product J as complete LATTICE by WAYBEL_3:31; A2: L is complete; let X be Subset of product J, i be Element of I; A3: inf X is_<=_than X by A2,YELLOW_0:33; A4: (inf X).i is_<=_than pi(X,i) proof let a be Element of J.i; assume a in pi(X,i); then consider f being Function such that A5: f in X & a = f.i by CARD_3:def 6; reconsider f as Element of product J by A5; inf X <= f by A3,A5,LATTICE3:def 8; hence thesis by A5,WAYBEL_3:28; end; A6: J.i is complete LATTICE by A1; now let a be Element of J.i; assume A7: a is_<=_than pi(X,i); set f = (inf X)+*(i,a); A8: dom f = dom inf X & dom inf X = I by FUNCT_7:32,WAYBEL_3:27; now let j be Element of I; j = i or j <> i; then f.j = (inf X).j or f.j = a & j = i by A8,FUNCT_7:33,34; hence f.j is Element of J.j; end; then reconsider f as Element of product J by A8,WAYBEL_3:27; f is_<=_than X proof let g be Element of product J; assume g in X; then A9: g >= inf X & g.i in pi(X,i) by A2,CARD_3:def 6,YELLOW_2:24; now let j be Element of I; j = i or j <> i; then f.j = (inf X).j or f.j = a & j = i by A8,FUNCT_7:33,34; hence f.j <= g.j by A7,A9,LATTICE3:def 8,WAYBEL_3:28; end; hence f <= g by WAYBEL_3:28; end; then f <= inf X & f.i = a by A2,A8,FUNCT_7:33,YELLOW_0:33; hence (inf X).i >= a by WAYBEL_3:28; end; hence thesis by A4,A6,YELLOW_0:33; end; theorem for S being non empty 1-sorted, T being complete LATTICE for f, g, h being map of S, T, i being Element of S st h = "/\" ({f, g}, T |^ the carrier of S) holds h.i = inf {f.i, g.i} proof let S be non empty 1-sorted, T be complete LATTICE; let f, g, h be map of S, T, i be Element of S; assume A1: h = "/\" ({f, g}, T |^ the carrier of S); dom ((the carrier of S) --> T) = the carrier of S by FUNCOP_1:19; then reconsider SYT = (the carrier of S) --> T as non-Empty RelStr-yielding ManySortedSet of the carrier of S by PBOOLE:def 3; reconsider SYT as non-Empty reflexive-yielding RelStr-yielding ManySortedSet of the carrier of S; A2:for i being Element of S holds SYT.i is complete LATTICE by FUNCOP_1:13; reconsider f' = f, g' = g as Element of (T |^ the carrier of S) by Th19; reconsider f', g' as Element of product SYT by YELLOW_1:def 5; reconsider DU = {f', g'} as Subset of product SYT; h.i = (inf DU).i by A1,YELLOW_1:def 5 .= "/\" (pi({f,g},i), SYT.i) by A2,Th23 .= "/\" (pi({f,g},i), T) by FUNCOP_1:13 .= inf {f.i, g.i} by CARD_3:26; hence thesis; end; definition let S be non empty 1-sorted, T be LATTICE; cluster -> Function-like Relation-like Element of (T |^ the carrier of S); coherence by Th19; end; definition let S, T be TopLattice; cluster -> Function-like Relation-like Element of ContMaps (S, T); coherence; end; theorem Th25: for S being non empty RelStr, T being complete LATTICE for F being non empty Subset of (T |^ the carrier of S), i being Element of S holds (sup F).i = "\/" ({ f.i where f is Element of (T |^ the carrier of S) : f in F }, T ) proof let S be non empty RelStr, T be complete LATTICE; let F be non empty Subset of (T |^ the carrier of S), i be Element of S; dom ((the carrier of S) --> T) = the carrier of S by FUNCOP_1:19; then reconsider SYT = (the carrier of S) --> T as non-Empty RelStr-yielding ManySortedSet of the carrier of S by PBOOLE:def 3; reconsider SYT as non-Empty reflexive-yielding RelStr-yielding ManySortedSet of the carrier of S; A1:T |^ the carrier of S = product SYT by YELLOW_1:def 5; then reconsider X = F as Subset of product SYT; A2:pi(X,i) = { f.i where f is Element of (T |^ the carrier of S) : f in F } proof thus pi(X,i) c= { f.i where f is Element of (T |^ the carrier of S) : f in F } proof let a be set; assume a in pi(X,i); then consider g being Function such that A3: g in X & a = g.i by CARD_3:def 6; thus thesis by A3; end; thus { f.i where f is Element of (T |^ the carrier of S) : f in F } c= pi(X,i) proof let a be set; assume a in { f.i where f is Element of (T |^ the carrier of S) : f in F }; then consider g being Element of (T |^ the carrier of S) such that A4: a = g.i & g in F; thus thesis by A4,CARD_3:def 6; end; end; for i being Element of S holds SYT.i is complete LATTICE by FUNCOP_1:13; then (sup F).i = "\/" (pi(X,i), SYT.i) by A1,WAYBEL_3:32 .= "\/" ({ f.i where f is Element of (T |^ the carrier of S) : f in F }, T ) by A2,FUNCOP_1:13; hence thesis; end; theorem Th26: for S, T being complete TopLattice for F being non empty Subset of ContMaps (S, T), i being Element of S holds "\/" (F, (T |^ the carrier of S)).i = "\/" ({ f.i where f is Element of (T |^ the carrier of S) : f in F }, T ) proof let S, T be complete TopLattice; let F be non empty Subset of ContMaps (S, T), i be Element of S; dom ((the carrier of S) --> T) = the carrier of S by FUNCOP_1:19; then reconsider SYT = (the carrier of S) --> T as non-Empty RelStr-yielding ManySortedSet of the carrier of S by PBOOLE:def 3; reconsider SYT as non-Empty reflexive-yielding RelStr-yielding ManySortedSet of the carrier of S; ContMaps (S, T) is full SubRelStr of (T |^ the carrier of S) by Def3; then the carrier of ContMaps (S, T) c= the carrier of (T |^ the carrier of S ) by YELLOW_0:def 13; then A1:F c= the carrier of (T |^ the carrier of S) by XBOOLE_1:1; A2:T |^ the carrier of S = product SYT by YELLOW_1:def 5; then reconsider X = F as Subset of product SYT by A1; A3:pi(X,i) = { f.i where f is Element of (T |^ the carrier of S) : f in F } proof thus pi(X,i) c= { f.i where f is Element of (T |^ the carrier of S) : f in F } proof let a be set; assume a in pi(X,i); then consider g being Function such that A4: g in X & a = g.i by CARD_3:def 6; g is Element of (T |^ the carrier of S) by A1,A4; hence thesis by A4; end; thus { f.i where f is Element of (T |^ the carrier of S) : f in F } c= pi(X,i) proof let a be set; assume a in { f.i where f is Element of (T |^ the carrier of S) : f in F }; then consider g being Element of (T |^ the carrier of S) such that A5: a = g.i & g in F; thus thesis by A5,CARD_3:def 6; end; end; for i being Element of S holds SYT.i is complete LATTICE by FUNCOP_1:13; then "\/" (F, (T |^ the carrier of S)).i = "\/" (pi(X,i), SYT.i) by A2,WAYBEL_3:32 .= "\/" ({ f.i where f is Element of (T |^ the carrier of S) : f in F }, T ) by A3,FUNCOP_1:13; hence thesis; end; reserve S for non empty RelStr, T for complete LATTICE; theorem Th27: for F being non empty Subset of (T |^ the carrier of S), D being non empty Subset of S holds (sup F).:D = { "\/" ({ f.i where f is Element of (T |^ the carrier of S) : f in F }, T ) where i is Element of S : i in D } proof let F be non empty Subset of (T |^ the carrier of S), D be non empty Subset of S; thus (sup F).:D c= { "\/" ({ f.i where f is Element of (T |^ the carrier of S) : f in F }, T ) where i is Element of S: i in D } proof let a be set; assume a in (sup F).:D; then consider x being set such that A1: x in dom (sup F) & x in D & a = (sup F).x by FUNCT_1:def 12; reconsider x' = x as Element of S by A1; a = "\/" ({ f.x' where f is Element of (T |^ the carrier of S) : f in F }, T ) by A1,Th25; hence thesis by A1; end; thus { "\/" ({ f.i where f is Element of (T |^ the carrier of S) : f in F }, T ) where i is Element of S: i in D } c= (sup F).:D proof let a be set; assume a in { "\/" ({ f.i where f is Element of (T |^ the carrier of S) : f in F }, T ) where i is Element of S: i in D }; then consider i1 being Element of S such that A2: a = "\/" ({ f.i1 where f is Element of (T |^ the carrier of S) : f in F }, T ) & i1 in D; A3: a = (sup F).i1 by A2,Th25; sup F is map of S, T by Th6; then dom (sup F) = the carrier of S by FUNCT_2:def 1; hence thesis by A2,A3,FUNCT_1:def 12; end; end; theorem Th28: for S, T being complete Scott TopLattice, F being non empty Subset of ContMaps (S, T), D being non empty Subset of S holds ("\/" (F, (T |^ the carrier of S))).:D = { "\/" ({ f.i where f is Element of (T |^ the carrier of S) : f in F }, T ) where i is Element of S : i in D } proof let S, T be complete Scott TopLattice, F be non empty Subset of ContMaps (S, T), D be non empty Subset of S; thus ("\/" (F, (T |^ the carrier of S))).:D c= { "\/" ({ f.i where f is Element of (T |^ the carrier of S) : f in F }, T ) where i is Element of S: i in D } proof let a be set; assume a in ("\/" (F, (T |^ the carrier of S))).:D; then consider x being set such that A1: x in dom ("\/" (F, (T |^ the carrier of S))) & x in D & a = ("\/" (F, (T |^ the carrier of S))).x by FUNCT_1:def 12; reconsider x' = x as Element of S by A1; a = "\/" ({ f.x' where f is Element of (T |^ the carrier of S) : f in F }, T ) by A1,Th26; hence thesis by A1; end; thus { "\/" ({ f.i where f is Element of (T |^ the carrier of S) : f in F }, T ) where i is Element of S: i in D } c= ("\/" (F, (T |^ the carrier of S))).:D proof let a be set; assume a in { "\/" ({ f.i where f is Element of (T |^ the carrier of S) : f in F }, T ) where i is Element of S: i in D }; then consider i1 being Element of S such that A2: a = "\/" ({ f.i1 where f is Element of (T |^ the carrier of S) : f in F }, T ) & i1 in D; A3: a = ("\/" (F, (T |^ the carrier of S))).i1 by A2,Th26; ("\/" (F, (T |^ the carrier of S))) is map of S, T by Th19; then dom ("\/" (F, (T |^ the carrier of S))) = the carrier of S by FUNCT_2:def 1; hence thesis by A2,A3,FUNCT_1:def 12; end; end; FraenkelF'RS{ B() -> non empty TopRelStr, F(set) -> set, G(set) -> set, P[set] } : { F(v1) where v1 is Element of B() : P[v1] } = { G(v2) where v2 is Element of B() : P[v2] } provided A1: for v being Element of B() st P[v] holds F(v) = G(v) proof set X = { F(v1) where v1 is Element of B() : P[v1] }, Y = { G(v2) where v2 is Element of B() : P[v2] }; A2: X c= Y proof let x be set; assume x in X; then consider v1 being Element of B() such that A3: x = F(v1) & P[v1]; x = G(v1) by A1,A3; hence x in Y by A3; end; Y c= X proof let x be set; assume x in Y; then consider v1 being Element of B() such that A4: x = G(v1) & P[v1]; x = F(v1) by A1,A4; hence x in X by A4; end; hence thesis by A2,XBOOLE_0:def 10; end; scheme FraenkelF'RSS{ B() -> non empty RelStr, F(set) -> set, G(set) -> set, P[set] } : { F(v1) where v1 is Element of B() : P[v1] } = { G(v2) where v2 is Element of B() : P[v2] } provided A1: for v being Element of B() st P[v] holds F(v) = G(v) proof set X = { F(v1) where v1 is Element of B() : P[v1] }, Y = { G(v2) where v2 is Element of B() : P[v2] }; A2: X c= Y proof let x be set; assume x in X; then consider v1 being Element of B() such that A3: x = F(v1) & P[v1]; x = G(v1) by A1,A3; hence x in Y by A3; end; Y c= X proof let x be set; assume x in Y; then consider v1 being Element of B() such that A4: x = G(v1) & P[v1]; x = F(v1) by A1,A4; hence x in X by A4; end; hence thesis by A2,XBOOLE_0:def 10; end; FuncFraenkelS{ B, C() -> non empty TopRelStr, A(set) -> Element of C(), f() -> Function, P[set]}: f().:{A(x) where x is Element of B(): P[x]} = {f().A(x) where x is Element of B(): P[x]} provided A1: the carrier of C() c= dom f() proof set f = f(); thus f.:{A(x) where x is Element of B(): P[x]} c= {f.A(x) where x is Element of B(): P[x]} proof let y be set; assume y in f.:{A(x) where x is Element of B(): P[x]}; then consider z being set such that A2: z in dom f & z in {A(x) where x is Element of B(): P[x]} & y = f.z by FUNCT_1:def 12; ex x being Element of B() st z = A(x) & P[x] by A2; hence thesis by A2; end; let y be set; assume y in {f.A(x) where x is Element of B(): P[x]}; then consider x being Element of B() such that A3: y = f.A(x) & P[x]; A(x) in the carrier of C(); then A(x) in dom f & A(x) in {A(z) where z is Element of B(): P[z]} by A1,A3; hence thesis by A3,FUNCT_1:def 12; end; Lm1:for S being non empty RelStr, D being non empty Subset of S holds D = { i where i is Element of S: i in D } proof let S be non empty RelStr, D be non empty Subset of S; thus D c= { i where i is Element of S: i in D } proof let a be set; assume A1: a in D; then reconsider a' = a as Element of S; a' in { i where i is Element of S: i in D } by A1; hence thesis; end; thus { i where i is Element of S: i in D } c= D proof let a be set; assume a in { i where i is Element of S: i in D }; then consider j being Element of S such that A2: j = a & j in D; thus thesis by A2; end; end; theorem Th29: for S, T being complete Scott TopLattice, F being non empty Subset of ContMaps (S, T) holds "\/" (F, (T |^ the carrier of S)) is monotone map of S, T proof let S, T be complete Scott TopLattice, F be non empty Subset of ContMaps (S, T); ContMaps (S, T) is full SubRelStr of (T |^ the carrier of S) by Def3; then the carrier of ContMaps (S, T) c= the carrier of (T |^ the carrier of S ) by YELLOW_0:def 13; then F c= the carrier of (T |^ the carrier of S) by XBOOLE_1:1; then reconsider F' = F as Subset of (T |^ the carrier of S); reconsider sF = sup F' as map of S, T by Th6; now let x, y be Element of S; assume A1: x <= y; set G1 = { f.x where f is Element of (T |^ the carrier of S) : f in F' }; A2: sF.x = "\/" (G1, T) by Th25; G1 is_<=_than sF.y proof let a be Element of T; assume a in { f.x where f is Element of (T |^ the carrier of S) : f in F' }; then consider f' being Element of (T |^ the carrier of S) such that A3: a = f'.x & f' in F'; f' is Element of ContMaps (S, T) by A3; then reconsider f1 = f' as continuous map of S, T by Th21; reconsider f1 as monotone map of S, T; f' <= sup F' by A3,YELLOW_2:24; then f1 <= sF by WAYBEL10:12; then A4: f1.y <= sF.y by YELLOW_2:10; f1.x <= f1.y by A1,WAYBEL_1:def 2; hence thesis by A3,A4,YELLOW_0:def 2; end; hence sF.x <= sF.y by A2,YELLOW_0:32; end; hence thesis by WAYBEL_1:def 2; end; theorem Th30: for S, T being complete Scott TopLattice, F being non empty Subset of ContMaps (S, T), D being directed non empty Subset of S holds "\/"({ "\/"({g.i where i is Element of S : i in D }, T ) where g is Element of (T |^ the carrier of S) : g in F }, T ) = "\/"({ "\/" ({g'.i' where g' is Element of (T |^ the carrier of S) : g' in F }, T ) where i' is Element of S : i' in D }, T) proof let S, T be complete Scott TopLattice, F be non empty Subset of ContMaps (S, T), D be directed non empty Subset of S; set F' = F; set L = "\/"({ "\/"({g.i where i is Element of S : i in D }, T ) where g is Element of (T |^ the carrier of S) : g in F }, T ); set P = "\/"({ "\/" ({g'.i' where g' is Element of (T |^ the carrier of S) : g' in F }, T ) where i' is Element of S : i' in D }, T); set L1 = { "\/"({g.i where i is Element of S : i in D }, T) where g is Element of (T |^ the carrier of S) : g in F }; set P1 = { "\/"({g2.i3 where g2 is Element of (T |^ the carrier of S) : g2 in F }, T ) where i3 is Element of S : i3 in D }; reconsider L, P as Element of T; reconsider sF = "\/" (F, (T |^ the carrier of S)) as map of S, T by Th19; A1: P = sup (sF.:D) by Th28; deffunc A(Element of S) = $1; defpred P[set] means $1 in D; L1 is_<=_than P proof let b be Element of T; assume b in L1; then consider g' being Element of T |^ the carrier of S such that A2: b = "\/"({g'.i where i is Element of S : i in D }, T) & g' in F; g' is Element of ContMaps (S, T) by A2; then reconsider g1 = g' as continuous map of S, T by Th21; the carrier of S c= dom g1 by FUNCT_2:def 1; then A3: the carrier of S c= dom g'; g'.:{A(i2) where i2 is Element of S : P[i2]} = {g'.A(i) where i is Element of S : P[i]} from FuncFraenkelS (A3); then A4: b = "\/" (g'.:D, T) by A2,Lm1; g' <= "\/" (F, (T |^ the carrier of S)) by A2,YELLOW_2:24; then A5: g1 <= sF by WAYBEL10:12; g1.:D is_<=_than sup (sF.:D) proof let a be Element of T; assume a in g1.:D; then consider x be set such that A6: x in dom g1 & x in D & a = g1.x by FUNCT_1:def 12; A7: x in the carrier of S by A6; reconsider x' = x as Element of S by A6; A8: g1.x' <= sF.x' by A5,YELLOW_2:10; x' in dom sF by A7,FUNCT_2:def 1; then sF.x' in sF.:D by A6,FUNCT_1:def 12; then sF.x' <= sup (sF.:D) by YELLOW_2:24; hence thesis by A6,A8,YELLOW_0:def 2; end; hence thesis by A1,A4,YELLOW_0:32; end; then A9: L <= P by YELLOW_0:32; deffunc F(Element of (T |^ the carrier of S)) = "\/"({$1.i4 where i4 is Element of S : i4 in D }, T ); deffunc G(Element of (T |^ the carrier of S)) = $1.sup D; defpred Q[set] means $1 in F'; A10: for g8 being Element of (T |^ the carrier of S) st Q[g8] holds F(g8) = G(g8) proof let g1 be Element of (T |^ the carrier of S); assume g1 in F'; then g1 is Element of ContMaps (S, T); then reconsider g' = g1 as continuous map of S, T by Th21; A11: g' preserves_sup_of D by WAYBEL_0:def 37; A12: ex_sup_of D,S by YELLOW_0:17; the carrier of S c= dom g' by FUNCT_2:def 1; then A13: the carrier of S c= dom g1; g1.:{A(i2) where i2 is Element of S : P[i2]} = {g1.A(i) where i is Element of S : P[i]} from FuncFraenkelS (A13); then "\/"({g1.i where i is Element of S : i in D }, T ) = sup (g'.:D) by Lm1 .= g1.sup D by A11,A12,WAYBEL_0:def 31; hence thesis; end; {F(g3) where g3 is Element of (T |^ the carrier of S): Q[g3]} = {G(g4) where g4 is Element of (T |^ the carrier of S): Q[g4]} from FraenkelF'RSS (A10); then A14: L = sF.sup D by Th26; P1 is_<=_than L proof let b be Element of T; assume b in P1; then consider i' being Element of S such that A15: b = "\/"({g'.i' where g' is Element of (T |^ the carrier of S) : g' in F }, T ) & i' in D; A16: b = sF.i' by A15,Th26; sF is monotone by Th29; then A17:sup (sF.:D) <= L by A14,WAYBEL17:15; i' in the carrier of S; then i' in dom sF by FUNCT_2:def 1; then b in sF.:D by A15,A16,FUNCT_1:def 12; then b <= sup (sF.:D) by YELLOW_2:24; hence thesis by A17,YELLOW_0:def 2; end; then P <= L by YELLOW_0:32; hence thesis by A9,YELLOW_0:def 3; end; theorem Th31: for S, T being complete Scott TopLattice, F being non empty Subset of ContMaps (S, T), D being directed non empty Subset of S holds "\/" (("\/"(F, (T |^ the carrier of S))).:D, T) = "\/" (F, (T |^ the carrier of S)).sup D proof let S, T be complete Scott TopLattice, F be non empty Subset of ContMaps (S, T), D be directed non empty Subset of S; ContMaps (S, T) is full SubRelStr of (T |^ the carrier of S) by Def3; then the carrier of ContMaps (S, T) c= the carrier of (T |^ the carrier of S ) by YELLOW_0:def 13; then F c= the carrier of (T |^ the carrier of S) by XBOOLE_1:1; then reconsider F' = F as non empty Subset of (T |^ the carrier of S) ; reconsider sF = sup F' as map of S, T by Th19; set L = "\/"({ "\/"({g.i where i is Element of S : i in D }, T ) where g is Element of (T |^ the carrier of S) : g in F }, T ); set P = "\/"({ "\/" ({g'.i' where g' is Element of (T |^ the carrier of S) : g' in F }, T ) where i' is Element of S : i' in D }, T); deffunc F(Element of (T |^ the carrier of S)) = "\/"({$1.i4 where i4 is Element of S : i4 in D }, T ); deffunc G(Element of (T |^ the carrier of S)) = $1.sup D; defpred Q[set] means $1 in F'; A1: for g8 being Element of (T |^ the carrier of S) st Q[g8] holds F(g8) = G(g8) proof let g1 be Element of (T |^ the carrier of S); assume g1 in F'; then g1 is Element of ContMaps (S, T); then reconsider g' = g1 as continuous map of S, T by Th21; A2: g' preserves_sup_of D by WAYBEL_0:def 37; A3: ex_sup_of D,S by YELLOW_0:17; the carrier of S c= dom g' by FUNCT_2:def 1; then A4: the carrier of S c= dom g1; deffunc A(Element of S) = $1; defpred P[set] means $1 in D; g1.:{A(i2) where i2 is Element of S : P[i2]} = {g1.A(i) where i is Element of S : P[i]} from FuncFraenkelS (A4); then "\/"({g1.i where i is Element of S : i in D }, T ) = sup (g'.:D) by Lm1 .= g1.sup D by A2,A3,WAYBEL_0:def 31; hence thesis; end; {F(g3) where g3 is Element of (T |^ the carrier of S) : Q[g3]} = {G(g4) where g4 is Element of (T |^ the carrier of S): Q[g4]} from FraenkelF'RSS (A1); then A5: L = sF.sup D by Th25; P = sup (sF.:D) by Th27; hence thesis by A5,Th30; end; theorem Th32: for S, T being complete Scott TopLattice, F being non empty Subset of ContMaps (S, T) holds "\/"(F, (T |^ the carrier of S)) in the carrier of ContMaps (S, T) proof let S, T be complete Scott TopLattice, F be non empty Subset of ContMaps (S, T); reconsider Ex = "\/"(F, (T |^ the carrier of S)) as map of S, T by Th19; for X being Subset of S st X is non empty directed holds Ex preserves_sup_of X proof let X be Subset of S; assume X is non empty directed; then reconsider X' = X as directed non empty Subset of S; Ex preserves_sup_of X' proof assume ex_sup_of X',S; thus ex_sup_of Ex.:X',T by YELLOW_0:17; thus sup (Ex.:X') = Ex.sup X' by Th31; end; hence thesis; end; then Ex is directed-sups-preserving by WAYBEL_0:def 37; then Ex is continuous by WAYBEL17:22; hence thesis by Def3; end; theorem Th33: for S being non empty RelStr, T being lower-bounded antisymmetric non empty RelStr holds Bottom (T |^ the carrier of S) = S --> Bottom T proof let S be non empty RelStr, T be lower-bounded antisymmetric non empty RelStr; set L = (T |^ the carrier of S); reconsider f = S --> Bottom T as Element of L by Th19; A1: f is_>=_than {} by YELLOW_0:6; reconsider f' = f as map of S, T; for b being Element of L st b is_>=_than {} holds f <= b proof let b be Element of L; assume b is_>=_than {}; reconsider b' = b as map of S, T by Th19; for x being Element of S holds f'.x <= b'.x proof let x be Element of S; f'. x = ((the carrier of S) --> Bottom T). x by BORSUK_1:def 3 .= Bottom T by FUNCOP_1:13; hence thesis by YELLOW_0:44; end; then f' <= b' by YELLOW_2:10; hence thesis by WAYBEL10:12; end; then f = "\/"({}, L) by A1,YELLOW_0:30; hence thesis by YELLOW_0:def 11; end; theorem for S being non empty RelStr, T being upper-bounded antisymmetric non empty RelStr holds Top (T |^ the carrier of S) = S --> Top T proof let S be non empty RelStr, T be upper-bounded antisymmetric non empty RelStr; set L = (T |^ the carrier of S); reconsider f = S --> Top T as Element of L by Th19; A1: f is_<=_than {} by YELLOW_0:6; reconsider f' = f as map of S, T; for b being Element of L st b is_<=_than {} holds f >= b proof let b be Element of L; assume b is_<=_than {}; reconsider b' = b as map of S, T by Th19; for x being Element of S holds f'.x >= b'.x proof let x be Element of S; f'. x = ((the carrier of S) --> Top T). x by BORSUK_1:def 3 .= Top T by FUNCOP_1:13; hence thesis by YELLOW_0:45; end; then f' >= b' by YELLOW_2:10; hence thesis by WAYBEL10:12; end; then f = "/\"({}, L) by A1,YELLOW_0:31; hence thesis by YELLOW_0:def 12; end; definition let S be non empty reflexive RelStr, T be complete LATTICE, a be Element of T; cluster S --> a -> directed-sups-preserving; coherence proof set f = S --> a; for X being Subset of S st X is non empty directed holds f preserves_sup_of X proof let X be Subset of S; assume X is non empty directed; then reconsider X' = X as non empty directed Subset of S; f preserves_sup_of X proof assume ex_sup_of X,S; thus ex_sup_of f.:X, T by YELLOW_0:17; A1: f .: X c= rng f by RELAT_1:144; A2: f = (the carrier of S) --> a by BORSUK_1:def 3; then A3: f .: X c= {a} by A1,FUNCOP_1:14; {a} c= f .: X proof let b be set; assume b in {a}; then A4: b = a by TARSKI:def 1; consider n being Element of X'; A5: dom f = the carrier of S by A2,FUNCOP_1:19; a = ((the carrier of S) --> a).n by FUNCOP_1:13 .= f.n by BORSUK_1:def 3; hence thesis by A4,A5,FUNCT_1:def 12; end; hence sup (f.:X) = sup {a} by A3,XBOOLE_0:def 10 .= a by YELLOW_0:39 .= ((the carrier of S) --> a). sup X by FUNCOP_1:13 .= f.sup X by BORSUK_1:def 3; end; hence thesis; end; hence f is directed-sups-preserving by WAYBEL_0:def 37; end; end; theorem Th35: :: Lemma 2.4, p. 115 for S, T being complete Scott TopLattice holds ContMaps (S, T) is sups-inheriting SubRelStr of (T |^ the carrier of S) proof let S, T be complete Scott TopLattice; set L = T |^ the carrier of S; reconsider CS = ContMaps (S, T) as full SubRelStr of L by Def3; now let X be Subset of CS; assume ex_sup_of X,L; per cases; suppose X is non empty; hence "\/"(X,L) in the carrier of CS by Th32; suppose X is empty; then "\/"(X,L) = Bottom L by YELLOW_0:def 11; then "\/"(X,L) = S --> Bottom T by Th33; hence "\/"(X,L) in the carrier of CS by Def3; end; hence thesis by YELLOW_0:def 19; end; definition let S, T be complete Scott TopLattice; cluster ContMaps (S, T) -> complete; coherence proof ContMaps (S, T) is sups-inheriting non empty full SubRelStr of (T |^ the carrier of S) by Def3,Th35; hence thesis by YELLOW_2:33; end; end; theorem for S, T being non empty Scott complete TopLattice holds Bottom ContMaps (S, T) = S --> Bottom T proof let S, T be non empty Scott complete TopLattice; set L = ContMaps (S, T); reconsider f = S --> Bottom T as Element of L by Th21; A1: f is_>=_than {} by YELLOW_0:6; reconsider f' = f as map of S, T; for b being Element of L st b is_>=_than {} holds f <= b proof let b be Element of L; assume b is_>=_than {}; reconsider b' = b as map of S, T by Th21; for i being Element of S holds [f.i, b.i] in the InternalRel of T proof let i be Element of S; f. i = ((the carrier of S) --> Bottom T). i by BORSUK_1:def 3 .= Bottom T by FUNCOP_1:13; then f'.i <= b'.i by YELLOW_0:44; hence thesis by ORDERS_1:def 9; end; hence thesis by Th20; end; then f = "\/"({}, L) by A1,YELLOW_0:30; hence thesis by YELLOW_0:def 11; end; theorem for S, T being non empty Scott complete TopLattice holds Top ContMaps (S, T) = S --> Top T proof let S, T be non empty Scott complete TopLattice; set L = ContMaps (S, T); reconsider f = S --> Top T as Element of L by Th21; A1: f is_<=_than {} by YELLOW_0:6; reconsider f' = f as map of S, T; for b being Element of L st b is_<=_than {} holds f >= b proof let b be Element of L; assume b is_<=_than {}; reconsider b' = b as map of S, T by Th21; for i being Element of S holds [b.i, f.i] in the InternalRel of T proof let i be Element of S; f. i = ((the carrier of S) --> Top T). i by BORSUK_1:def 3 .= Top T by FUNCOP_1:13; then f'.i >= b'.i by YELLOW_0:45; hence thesis by ORDERS_1:def 9; end; hence thesis by Th20; end; then f = "/\"({}, L) by A1,YELLOW_0:31; hence thesis by YELLOW_0:def 12; end; theorem for S, T being Scott complete TopLattice holds SCMaps (S, T) = ContMaps (S, T) proof let S, T be Scott complete TopLattice; reconsider Sm = ContMaps (S, T) as full non empty SubRelStr of (T |^ the carrier of S) by Def3; reconsider SC = SCMaps (S, T) as full non empty SubRelStr of (T |^ the carrier of S) by WAYBEL15:1; A1:the carrier of SC c= the carrier of MonMaps (S, T) by YELLOW_0:def 13; the carrier of SC = the carrier of Sm proof thus the carrier of SC c= the carrier of Sm proof let a be set; assume A2: a in the carrier of SC; then a is Element of MonMaps (S, T) by A1; then reconsider f = a as map of S, T by WAYBEL10:10; f is continuous map of S, T by A2,WAYBEL17:def 2; then a is Element of Sm by Th21; hence thesis; end; thus the carrier of Sm c= the carrier of SC proof let a be set; assume a in the carrier of Sm; then a is Element of Sm; then a is continuous map of S, T by Th21; hence thesis by WAYBEL17:def 2; end; end; hence thesis by YELLOW_0:58; end;