Copyright (c) 2003 Association of Mizar Users
environ vocabulary LFUZZY_0, ORDERS_1, ARYTM, RELAT_2, LATTICES, RCOMP_1, FUNCT_1, RELAT_1, QC_LANG1, FUZZY_3, SEQ_1, TARSKI, FUZZY_1, GROUP_1, LATTICE2, LATTICE3, WAYBEL_0, MEASURE5, SQUARE_1, BOOLE, YELLOW_0, SEQ_2, INTEGRA1, SEQ_4, PRE_TOPC, BHSP_3, ORDINAL2, YELLOW_1, PBOOLE, FUNCT_2, PARTFUN1, MONOID_0, CARD_3, WAYBEL_3, WAYBEL_1, FUNCOP_1; notation TARSKI, XBOOLE_0, ZFMISC_1, RELAT_1, FUNCT_1, FUNCT_2, DOMAIN_1, ORDINAL1, PARTFUN1, SUBSET_1, NUMBERS, XREAL_0, REAL_1, SQUARE_1, SEQ_1, SEQ_4, RCOMP_1, RELSET_1, STRUCT_0, ORDERS_1, PRE_TOPC, PBOOLE, MONOID_0, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, WAYBEL_1, WAYBEL_3, FUZZY_1, FUZZY_3, FUZZY_4, INTEGRA1, PRE_CIRC; constructors XCMPLX_0, SQUARE_1, INTEGRA1, WAYBEL_2, YELLOW10, PSCOMP_1, WAYBEL_3, FUZZY_4, DOMAIN_1, ORDERS_3, PRE_CIRC, MONOID_0; clusters SUBSET_1, LATTICE3, XREAL_0, YELLOW14, RELSET_1, ORDERS_4, STRUCT_0, MONOID_0, WAYBEL_2, WAYBEL10, WAYBEL17, WAYBEL_3, MEMBERED; requirements BOOLE, SUBSET, NUMERALS, REAL; definitions TARSKI, XBOOLE_0, RELAT_1, STRUCT_0, LATTICE3, YELLOW_0, WAYBEL_0, WAYBEL_1; theorems ZFMISC_1, ORDERS_1, STRUCT_0, WAYBEL_1, AXIOMS, RCOMP_1, FUNCT_1, XBOOLE_1, LATTICE3, SQUARE_1, YELLOW_0, TOPREAL5, INTEGRA1, XREAL_0, XBOOLE_0, TARSKI, SEQ_4, TOPMETR3, JORDAN5A, PSCOMP_1, FUNCT_2, YELLOW_2, WAYBEL_3, YELLOW_3, PBOOLE, CARD_3, YELLOW_1, FUNCOP_1, FUZZY_2, FUZZY_4, FUNCT_3, FUZZY_1, PARTFUN1, MONOID_0, WAYBEL_0; schemes YELLOW_0, PRALG_2, YELLOW_3, FRAENKEL; begin :: Posets of Real Numbers definition let R be RelStr; attr R is real means: Def1: the carrier of R c= REAL & for x,y being real number st x in the carrier of R & y in the carrier of R holds [x,y] in the InternalRel of R iff x <= y; end; definition let R be RelStr; attr R is interval means: Def2: R is real & ex a,b being real number st a <= b & the carrier of R = [.a,b.]; end; definition cluster interval -> real non empty RelStr; coherence proof let R being RelStr; assume A1:R is real; given a,b being real number such that A2:a <= b & the carrier of R = [.a,b.]; thus R is real by A1; thus the carrier of R is non empty by A2,RCOMP_1:15; end; end; definition cluster empty -> real RelStr; coherence proof let R being RelStr; assume A1:the carrier of R is empty; hence the carrier of R c= REAL by XBOOLE_1:2; thus thesis by A1; end; end; theorem Th1: for X being Subset of REAL ex R being strict RelStr st the carrier of R = X & R is real proof let X be Subset of REAL; per cases; suppose A1: X is empty; consider E being empty strict RelStr; take E; thus thesis by A1,STRUCT_0:def 1; suppose X is non empty; then reconsider Y = X as non empty set; defpred X[set,set] means ex x,y being real number st $1=x & $2=y & x<=y; consider L being non empty strict RelStr such that A2: the carrier of L = Y and A3: for a,b being Element of L holds a <= b iff X[a,b] from RelStrEx; take L; thus the carrier of L = X by A2; thus the carrier of L c= REAL by A2; let x,y be real number; assume x in the carrier of L & y in the carrier of L; then reconsider a = x, b = y as Element of L; a <= b iff [x,y] in the InternalRel of L by ORDERS_1:def 9; then [x,y] in the InternalRel of L iff ex x,y being real number st a=x & b=y & x<=y by A3; hence [x,y] in the InternalRel of L iff x <= y; end; definition cluster interval strict RelStr; existence proof set X = [. 0,1 qua real number .]; consider R being strict RelStr such that A1: the carrier of R = X & R is real by Th1; take R; thus thesis by A1,Def2; end; end; theorem Th2: for R1,R2 being real RelStr st the carrier of R1 = the carrier of R2 holds the RelStr of R1 = the RelStr of R2 proof let R1, R2 be real RelStr such that A1: the carrier of R1 = the carrier of R2; set X = the carrier of R1; the InternalRel of R1 = the InternalRel of R2 proof let a,b be set; A2: X c= REAL by Def1; hereby assume A3: [a,b] in the InternalRel of R1; then A4: a in X & b in X by ZFMISC_1:106; then reconsider a' = a, b' = b as Element of REAL by A2; a' <= b' by A3,A4,Def1; hence [a,b] in the InternalRel of R2 by A1,A4,Def1; end; assume A5: [a,b] in the InternalRel of R2; then A6: a in X & b in X by A1,ZFMISC_1:106; then reconsider a' = a, b' = b as Element of REAL by A2; A7: a' <= b' by A1,A5,A6,Def1; thus [a,b] in the InternalRel of R1 by A6,A7,Def1; end; hence thesis by A1; end; definition let R be non empty real RelStr; cluster -> real Element of R; coherence proof let x be Element of R; A1: the carrier of R c= REAL by Def1; x in the carrier of R;then x is Element of REAL by A1; hence x is real; end; end; definition let X be Subset of REAL; func RealPoset X -> real strict RelStr means: Def3: the carrier of it = X; existence proof ex R being strict RelStr st the carrier of R = X & R is real by Th1; hence thesis; end; uniqueness by Th2; end; definition let X be non empty Subset of REAL; cluster RealPoset X -> non empty; coherence proof thus the carrier of RealPoset X is non empty by Def3; end; end; definition let R be RelStr; let x,y be Element of R; redefine pred x <= y; synonym x <<= y; synonym y >>= x; antonym x ~<= y; antonym y ~>= x; end; definition let x,y be real number; redefine pred x <= y; synonym x <R= y; antonym y <R x; synonym y >R= x; antonym x >R y; end; theorem Th3: for R being non empty real RelStr for x,y being Element of R holds x <R= y iff x <<= y proof let R be non empty real RelStr; let x,y be Element of R; [x,y] in the InternalRel of R iff x <R= y by Def1; hence thesis by ORDERS_1:def 9; end; definition cluster real -> reflexive antisymmetric transitive RelStr; coherence proof let R be RelStr such that A1: R is real; per cases; suppose R is empty; then R is empty RelStr; hence thesis; suppose R is non empty; then reconsider R' = R as non empty real RelStr by A1; A2: R' is reflexive proof let x be Element of R'; thus thesis by Th3; end; A3: R' is antisymmetric proof let x,y be Element of R'; assume x <<= y & x >>= y; then x <R= y & x >R= y by Th3; hence x = y by AXIOMS:21; end; R' is transitive proof let x,y,z be Element of R'; assume x <<= y & y <<= z; then x <R= y & y <R= z by Th3; then x <R= z by AXIOMS:22; hence thesis by Th3; end; hence thesis by A2,A3; end; end; definition cluster -> connected (real non empty RelStr); coherence proof let R be non empty real RelStr; let x,y be Element of R; x <R= y or x >R= y; hence thesis by Th3; end; end; definition let R be non empty real RelStr; let x,y be Element of R; redefine func max(x,y) -> Element of R; coherence by SQUARE_1:49; end; definition let R be non empty real RelStr; let x,y be Element of R; redefine func min(x,y) -> Element of R; coherence by SQUARE_1:38; end; definition cluster -> with_suprema with_infima (real non empty RelStr); coherence; end; reserve X for non empty Subset of REAL, x,y,z,w for real number, R for real non empty RelStr, a,b,c for Element of R; theorem Th4: a"\/"b = max(a,b) proof max(a,b) >R= a & max(a,b) >R= b by SQUARE_1:46;then A1: max(a,b) >>= a & max(a,b) >>= b by Th3; for d being Element of R st d >>= a & d >>= b holds max(a,b) <<= d proof let d be Element of R; assume d >>= a & d >>= b;then d >R= a & d >R= b by Th3;then max(a,b) <R= d by SQUARE_1:50; hence max(a,b) <<= d by Th3; end; hence thesis by A1,YELLOW_0:22; end; theorem Th5: a"/\"b = min(a,b) proof min(a,b) <= a & min(a,b) <= b by SQUARE_1:35;then A1: min(a,b) <<= a & min(a,b) <<= b by Th3; for d being Element of R st d <<= a & d <<= b holds min(a,b) >>= d proof let d being Element of R; assume d <<= a & d <<= b; then d <= a & d <= b by Th3; then d <= min(a,b) by SQUARE_1:39; hence thesis by Th3; end; hence thesis by A1,YELLOW_0:23; end; theorem Th6: (ex x st x in the carrier of R & for y st y in the carrier of R holds x <= y) iff R is lower-bounded proof hereby given x such that A1: x in the carrier of R & for y st y in the carrier of R holds x <R= y; reconsider a = x as Element of R by A1; now let b;assume b in the carrier of R; b >R= a by A1; hence a <<= b by Th3; end; then a is_<=_than the carrier of R by LATTICE3:def 8; hence R is lower-bounded by YELLOW_0:def 4; end; given x being Element of R such that A2: x is_<=_than the carrier of R; take x; thus x in the carrier of R; let y; assume y in the carrier of R; then reconsider b = y as Element of R; x <<= b by A2,LATTICE3:def 8; hence x <= y by Th3; end; theorem Th7: (ex x st x in the carrier of R & for y st y in the carrier of R holds x >= y) iff R is upper-bounded proof hereby given x such that A1: x in the carrier of R & for y st y in the carrier of R holds x >R= y; reconsider a = x as Element of R by A1; now let b;assume b in the carrier of R; a >R= b by A1; hence a >>= b by Th3; end; then a is_>=_than the carrier of R by LATTICE3:def 9; hence R is upper-bounded by YELLOW_0:def 5; end; given x being Element of R such that A2: x is_>=_than the carrier of R; take x; thus x in the carrier of R; let y; assume y in the carrier of R; then reconsider b = y as Element of R; x >>= b by A2,LATTICE3:def 9; hence x >= y by Th3; end; definition cluster interval -> bounded (non empty RelStr); coherence proof let R being non empty RelStr; assume A1: R is real; given x,y being real number such that A2: x <= y & the carrier of R = [.x,y.]; ex x st x in the carrier of R & for y st y in the carrier of R holds x >= y proof take y; thus y in the carrier of R by A2,RCOMP_1:15; let z; assume z in the carrier of R; hence thesis by A2,TOPREAL5:1; end; then A3: R is upper-bounded by A1,Th7; ex x st x in the carrier of R & for y st y in the carrier of R holds x <= y proof take x; thus x in the carrier of R by A2,RCOMP_1:15; let z; assume z in the carrier of R; hence thesis by A2,TOPREAL5:1; end; then R is lower-bounded by A1,Th6; hence R is bounded by A3,YELLOW_0:def 6; end; end; theorem Th8: for R being interval non empty RelStr, X being set holds ex_sup_of X,R proof let R being interval non empty RelStr; let X being set; consider a,b being real number such that A1:a <= b & the carrier of R = [.a,b.] by Def2; reconsider a,b as Real by XREAL_0:def 1; A2:X /\ [.a,b.] c= [.a,b.] by XBOOLE_1:17;then reconsider Y = X /\ [.a,b.] as Subset of REAL by XBOOLE_1:1; [.a,b.] is closed-interval by A1,INTEGRA1:def 1;then A3:[.a,b.] is bounded_above by INTEGRA1:3;then A4:Y is bounded_above by A2,RCOMP_1:4; ex a being Element of R st Y is_<=_than a & for b being Element of R st Y is_<=_than b holds a <<= b proof per cases; suppose A5: Y is empty; a in [.a,b.] by A1,RCOMP_1:15;then reconsider x = a as Element of R by A1; take x; thus Y is_<=_than x by A5,YELLOW_0:6; let y be Element of R; x <= y by A1,TOPREAL5:1; hence thesis by Th3; suppose A6:Y is non empty; upper_bound [.a,b.] = b by A1,JORDAN5A:20;then A7: upper_bound Y <= b by A2,A3,A6,PSCOMP_1:13; consider c being Element of Y; A8: c in Y by A6;then reconsider c as Real; A9: c <= upper_bound Y by A4,A6,SEQ_4:def 4; a<=c by A2,A8,TOPREAL5:1;then a <= upper_bound Y by A9,AXIOMS:22;then upper_bound Y in [.a,b.] by A7,TOPREAL5:1; then reconsider x = upper_bound Y as Element of R by A1; take x; A10: for b being Element of R st b in Y holds b <<= x proof let b be Element of R; assume b in Y; then b <= upper_bound Y by A4,SEQ_4:def 4; hence thesis by Th3; end; for y being Element of R st Y is_<=_than y holds x <<= y proof let y be Element of R; assume A11:Y is_<=_than y; for z being real number st z in Y holds z <= y proof let z being real number; assume A12: z in Y; then z in [.a,b.] by XBOOLE_0:def 3;then reconsider z as Element of R by A1; z <<= y by A11,A12,LATTICE3:def 9; hence thesis by Th3; end; then upper_bound Y <= y by A6,TOPMETR3:1; hence thesis by Th3; end; hence thesis by A10,LATTICE3:def 9; end; then ex_sup_of Y,R by YELLOW_0:15; hence thesis by A1,YELLOW_0:50; end; definition cluster -> complete (interval non empty RelStr); coherence proof let R be interval non empty RelStr; for X being Subset of R holds ex_sup_of X,R by Th8; hence thesis by YELLOW_0:53; end; end; definition cluster -> distributive Chain; coherence proof let C be Chain; for x,y,z being Element of C holds x "/\" (y "\/" z) = (x "/\" y) "\/" (x "/\" z) proof let x,y,z be Element of C; A1: x <= x by YELLOW_0:def 1; per cases by WAYBEL_0:def 29; suppose A2:x <= y & x <= z & y <= z; A3: x "/\" (y "\/" z) = x "/\" z by A2,YELLOW_0:24 .= x by A2,YELLOW_0:25; (x "/\" y) "\/" (x "/\" z) = x "\/" (x "/\" z) by A2,YELLOW_0:25 .= x "\/" x by A2,YELLOW_0:25 .= x by A1,YELLOW_0:24; hence thesis by A3; suppose A4:x <= y & x <= z & y >= z; A5: x "/\" (y "\/" z) = x "/\" y by A4,YELLOW_0:24 .= x by A4,YELLOW_0:25; (x "/\" y) "\/" (x "/\" z) = x "\/" (x "/\" z) by A4,YELLOW_0:25 .= x "\/" x by A4,YELLOW_0:25 .= x by A1,YELLOW_0:24; hence thesis by A5; suppose A6:x <= y & x >= z & y <= z;then z <= y by YELLOW_0:def 2;then A7: z = y by A6,YELLOW_0:def 3; A8: x "/\" (y "\/" z) = x "/\" z by A6,YELLOW_0:24 .= z by A6,YELLOW_0:25; (x "/\" y) "\/" (x "/\" z) = x "\/" (x "/\" z) by A6,YELLOW_0:25 .= x "\/" z by A6,YELLOW_0:25 .= x by A6,YELLOW_0:24; hence thesis by A6,A7,A8,YELLOW_0:def 3; suppose A9:x <= y & x >= z & y >= z; A10: x "/\" (y "\/" z) = x "/\" y by A9,YELLOW_0:24 .= x by A9,YELLOW_0:25; (x "/\" y) "\/" (x "/\" z) = x "\/" (x "/\" z) by A9,YELLOW_0:25 .= x "\/" z by A9,YELLOW_0:25 .= x by A9,YELLOW_0:24; hence thesis by A10; suppose A11:x >= y & x <= z & y <= z; A12: x "/\" (y "\/" z) = x "/\" z by A11,YELLOW_0:24 .= x by A11,YELLOW_0:25; (x "/\" y) "\/" (x "/\" z) = y "\/" (x "/\" z) by A11,YELLOW_0:25 .= y "\/" x by A11,YELLOW_0:25 .= x by A11,YELLOW_0:24; hence thesis by A12; suppose A13:x >= y & x <= z & y >= z;then z >= y by YELLOW_0:def 2;then A14: z = y by A13,YELLOW_0:def 3; A15: x "/\" (y "\/" z) = x "/\" y by A13,YELLOW_0:24 .= y by A13,YELLOW_0:25; (x "/\" y) "\/" (x "/\" z) = y "\/" (x "/\" z) by A13,YELLOW_0:25 .= y "\/" x by A13,YELLOW_0:25 .= x by A13,YELLOW_0:24; hence thesis by A13,A14,A15,YELLOW_0:def 3; suppose A16:x >= y & x >= z & y <= z; A17: x "/\" (y "\/" z) = x "/\" z by A16,YELLOW_0:24 .= z by A16,YELLOW_0:25; (x "/\" y) "\/" (x "/\" z) = y "\/" (x "/\" z) by A16,YELLOW_0:25 .= y "\/" z by A16,YELLOW_0:25 .= z by A16,YELLOW_0:24; hence thesis by A17; suppose A18:x >= y & x >= z & y >= z; A19: x "/\" (y "\/" z) = x "/\" y by A18,YELLOW_0:24 .= y by A18,YELLOW_0:25; (x "/\" y) "\/" (x "/\" z) = y "\/" (x "/\" z) by A18,YELLOW_0:25 .= y "\/" z by A18,YELLOW_0:25 .= y by A18,YELLOW_0:24; hence thesis by A19; end; hence thesis by WAYBEL_1:def 3; end; end; definition cluster -> Heyting (interval non empty RelStr); coherence proof let R be interval non empty RelStr; thus R is LATTICE; let x be Element of R; set f = x "/\"; f is sups-preserving proof let X be Subset of R; assume ex_sup_of X,R; thus ex_sup_of f.:X,R by Th8; A1: f.sup X is_>=_than f.:X proof let a be Element of R; assume a in f.:X; then consider y being set such that A2: y in the carrier of R & y in X & a = f.y by FUNCT_2:115; reconsider y as Element of R by A2; y <<= sup X & a = x "/\" y & f.sup X = x "/\" sup X & x <<= x by A2,WAYBEL_1:def 18,YELLOW_0:def 1,YELLOW_2:24; hence thesis by YELLOW_3:2; end; now let b being Element of R such that A3: b is_>=_than f.:X; per cases; suppose A4: x is_>=_than X; then x >>= sup X by YELLOW_0:32; then A5: x "/\" sup X = sup X by YELLOW_0:25; b is_>=_than X proof let a be Element of R; assume A6: a in X; then x >>= a by A4,LATTICE3:def 9;then x "/\" a = a by YELLOW_0:25;then A7: a = f.a by WAYBEL_1:def 18; f.a in f.:X by A6,FUNCT_2:43; hence thesis by A3,A7,LATTICE3:def 9; end; then b >>= sup X by YELLOW_0:32; hence f.sup X <<= b by A5,WAYBEL_1:def 18; suppose not x is_>=_than X; then consider a being Element of R such that A8: a in X & not x >>= a by LATTICE3:def 9; A9: x <<= a by A8,WAYBEL_0:def 29; A10: x = x "/\" a by A9,YELLOW_0:25 .= f.a by WAYBEL_1:def 18; a <<= sup X by A8,YELLOW_2:24;then x <<= sup X by A9,ORDERS_1:26;then x = x "/\"sup X by YELLOW_0:25;then A11: x = f.sup X by WAYBEL_1:def 18; f.a in f.:X by A8,FUNCT_2:43; hence f.sup X <<= b by A3,A10,A11,LATTICE3:def 9; end; hence sup (f.:X) = f.sup X by A1,YELLOW_0:30; end; hence thesis by WAYBEL_1:18; end; end; definition cluster [.0,1 .] -> non empty; coherence proof [. 0,1 .] is closed-interval by INTEGRA1:def 1; hence thesis by INTEGRA1:2; end; end; definition cluster RealPoset [.0,1 .] -> interval; coherence proof the carrier of RealPoset [. 0,1 .] = [. 0,1 .] by Def3; hence thesis by Def2; end; end; begin :: Product of Heyting Lattices theorem Th9: 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 sup-Semilattice holds product J is with_suprema proof let I being non empty set; let J being RelStr-yielding non-Empty reflexive-yielding ManySortedSet of I such that A1:for i being Element of I holds J.i is sup-Semilattice; set IT = product J; for x,y being Element of IT ex z being Element of IT st x <= z & y <= z & for z' being Element of IT st x <= z' & y <= z' holds z <= z' proof let x,y being Element of IT; deffunc U(Element of I) = x.$1 "\/" y.$1; consider z be ManySortedSet of I such that A2: for i be Element of I holds z.i = U(i) from LambdaDMS; A3: dom z = I by PBOOLE:def 3; for i being Element of I holds z.i is Element of J.i proof let i being Element of I; z.i = x.i "\/" y.i by A2; hence thesis; end; then reconsider z as Element of product J by A3,WAYBEL_3:27; take z; for i being Element of I holds x.i <= z.i & y.i <= z.i proof let i being Element of I; J.i is antisymmetric with_suprema RelStr & z.i = x.i "\/" y.i by A1,A2; hence thesis by YELLOW_0:22; end; hence x <= z & y <= z by WAYBEL_3:28; let z' be Element of IT; assume A4:x <= z' & y <= z'; for i being Element of I holds z.i <= z'.i proof let i being Element of I; J.i is antisymmetric with_suprema RelStr & z'.i >= x.i & z'.i >= y.i & z.i = x.i "\/" y.i by A1,A2,A4,WAYBEL_3:28; hence thesis by YELLOW_0:22; end; hence z <= z' by WAYBEL_3:28; end; hence thesis by LATTICE3:def 10; end; theorem Th10: 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 Semilattice holds product J is with_infima proof let I being non empty set; let J being RelStr-yielding non-Empty reflexive-yielding ManySortedSet of I such that A1:for i being Element of I holds J.i is Semilattice; set IT = product J; for x,y being Element of IT ex z being Element of IT st x >= z & y >= z & for z' being Element of IT st x >= z' & y >= z' holds z >= z' proof let x,y being Element of IT; deffunc U(Element of I) = x.$1 "/\" y.$1; consider z be ManySortedSet of I such that A2: for i be Element of I holds z.i = U(i) from LambdaDMS; A3: dom z = I by PBOOLE:def 3; for i being Element of I holds z.i is Element of J.i proof let i being Element of I; z.i = x.i "/\" y.i by A2; hence thesis; end; then reconsider z as Element of product J by A3,WAYBEL_3:27; take z; for i being Element of I holds x.i >= z.i & y.i >= z.i proof let i being Element of I; J.i is antisymmetric with_infima RelStr & z.i = x.i "/\" y.i by A1,A2; hence thesis by YELLOW_0:23; end; hence x >= z & y >= z by WAYBEL_3:28; let z' be Element of IT; assume A4:x >= z' & y >= z'; for i being Element of I holds z.i >= z'.i proof let i being Element of I; J.i is antisymmetric with_infima RelStr & x.i >= z'.i & z'.i <= y.i & z.i = x.i "/\" y.i by A1,A2,A4,WAYBEL_3:28; hence thesis by YELLOW_0:23; end; hence z >= z' by WAYBEL_3:28; end; hence thesis by LATTICE3:def 11; end; theorem Th11: 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 Semilattice for f,g being Element of product J, i being Element of I holds (f "/\" g).i = (f.i) "/\" (g.i) proof let I being non empty set; let J being RelStr-yielding non-Empty reflexive-yielding ManySortedSet of I such that A1: for i being Element of I holds J.i is Semilattice; let f,g being Element of product J; let i being Element of I; A2: for i being Element of I holds J.i is antisymmetric by A1; A3: J.i is Semilattice by A1; A4: product J is antisymmetric with_infima by A1,A2,Th10,WAYBEL_3:30; A5: (f "/\" g).i <= (f.i) "/\" (g.i) proof f >= f "/\" g & g >= f "/\" g by A4,YELLOW_0:23;then f.i >= (f "/\" g).i & g.i >= (f "/\" g).i by WAYBEL_3:28; hence thesis by A3,YELLOW_0:23; end; (f "/\" g).i >= (f.i) "/\" (g.i) proof deffunc U(Element of I) = f.$1 "/\" g.$1; consider z be ManySortedSet of I such that A6: for i be Element of I holds z.i = U(i) from LambdaDMS; A7: dom z = I by PBOOLE:def 3; for i being Element of I holds z.i is Element of J.i proof let i being Element of I; z.i = f.i "/\" g.i by A6; hence thesis; end; then reconsider z as Element of product J by A7,WAYBEL_3:27; for i being Element of I holds z.i <= f.i & z.i <= g.i proof let i being Element of I; A8: J.i is antisymmetric with_infima RelStr by A1; z.i = f.i "/\" g.i by A6; hence thesis by A8,YELLOW_0:23; end; then z <= f & z <= g by WAYBEL_3:28;then A9: f "/\" g >= z by A4,YELLOW_0:23; for i being Element of I holds (f "/\" g).i >= f.i "/\" g.i proof let i being Element of I; f.i "/\" g.i = z.i & (f "/\" g).i >= z.i by A6,A9,WAYBEL_3:28; hence thesis; end; hence thesis; end; hence thesis by A3,A5,YELLOW_0:def 3; end; theorem Th12: 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 sup-Semilattice for f,g being Element of product J, i being Element of I holds (f "\/" g).i = (f.i) "\/" (g.i) proof let I being non empty set; let J being RelStr-yielding non-Empty reflexive-yielding ManySortedSet of I such that A1: for i being Element of I holds J.i is sup-Semilattice; let f,g being Element of product J; let i being Element of I; A2: for i being Element of I holds J.i is antisymmetric by A1; A3: J.i is sup-Semilattice by A1; A4: product J is antisymmetric with_suprema by A1,A2,Th9,WAYBEL_3:30; A5: (f "\/" g).i >= (f.i) "\/" (g.i) proof f <= f "\/" g & g <= f "\/" g by A4,YELLOW_0:22;then f.i <= (f "\/" g).i & g.i <= (f "\/" g).i by WAYBEL_3:28; hence thesis by A3,YELLOW_0:22; end; (f "\/" g).i <= (f.i) "\/" (g.i) proof deffunc U(Element of I) = f.$1 "\/" g.$1; consider z be ManySortedSet of I such that A6: for i be Element of I holds z.i = U(i) from LambdaDMS; A7: dom z = I by PBOOLE:def 3; for i being Element of I holds z.i is Element of J.i proof let i being Element of I; z.i = f.i "\/" g.i by A6; hence thesis; end; then reconsider z as Element of product J by A7,WAYBEL_3:27; for i being Element of I holds z.i >= f.i & z.i >= g.i proof let i being Element of I; A8: J.i is antisymmetric with_suprema RelStr by A1; z.i = f.i "\/" g.i by A6; hence thesis by A8,YELLOW_0:22; end; then z >= f & z >= g by WAYBEL_3:28;then A9: f "\/" g <= z by A4,YELLOW_0:22; for i being Element of I holds (f "\/" g).i <= f.i "\/" g.i proof let i being Element of I; f.i "\/" g.i = z.i & (f "\/" g).i <= z.i by A6,A9,WAYBEL_3:28; hence thesis; end; hence thesis; end; hence thesis by A3,A5,YELLOW_0:def 3; end; theorem Th13: 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 Heyting complete LATTICE holds product J is complete Heyting proof let I be non empty set; let J be RelStr-yielding non-Empty reflexive-yielding ManySortedSet of I such that A1: for i being Element of I holds J.i is Heyting complete LATTICE; set H = product J; A2: for i being Element of I holds J.i is Semilattice by A1; A3: for i being Element of I holds J.i is complete LATTICE by A1; then reconsider H as complete LATTICE by WAYBEL_3:31; H = H; hence product J is complete & product J is LATTICE; let f be Element of product J; reconsider g = f as Element of H; reconsider F = f "/\" as map of H,H; F is sups-preserving proof let X be Subset of H; reconsider Y = F.:X, X' = X as Subset of product J; reconsider sX = sup X as Element of product J; assume ex_sup_of X, H; thus ex_sup_of F.:X, H by YELLOW_0:17; reconsider f1 = sup (F.:X), f2 = F.sup X as Element of product J; A4: dom f1 = I & dom f2 = I by WAYBEL_3:27; now let x be set; assume x in dom f1; then reconsider i = x as Element of I by WAYBEL_3:27; A5: J.i is complete Heyting LATTICE by A1; then (f.i) "/\" is lower_adjoint by WAYBEL_1:def 19; then (f.i) "/\" is sups-preserving by A5,WAYBEL_1:14; then A6: (f.i) "/\" preserves_sup_of pi(X',i) & ex_sup_of pi(X',i), J.i by A5,WAYBEL_0:def 33,YELLOW_0:17; f2 = g "/\" sup X by WAYBEL_1:def 18; then A7: f2.i = (f.i) "/\" (sX.i) by A2,Th11 .= (f.i) "/\" sup pi(X', i) by A3,WAYBEL_3:32 .= ((f.i) "/\").sup pi(X',i) by WAYBEL_1:def 18 .= sup (((f.i) "/\").:pi(X',i)) by A6,WAYBEL_0:def 31; pi(Y,i) = ((f.i) "/\").:pi(X',i) proof thus pi(Y,i) c= ((f.i) "/\").:pi(X',i) proof let a be set; assume a in pi(Y,i);then consider f3 being Function such that A8: f3 in Y & a = f3.i by CARD_3:def 6; reconsider f3 as Element of product J by A8; consider h being set such that A9: h in dom F & h in X & f3=F.h by A8,FUNCT_1:def 12; reconsider h as Element of product J by A9; A10: f3.i = (f "/\" h).i by A9,WAYBEL_1:def 18 .= f.i "/\" h.i by A2,Th11 .= (f.i "/\").(h.i) by WAYBEL_1:def 18; (h.i) in pi(X',i) by A9,CARD_3:def 6; hence a in ((f.i) "/\").:pi(X',i) by A8,A10,FUNCT_2:43; end; thus ((f.i) "/\").:pi(X',i) c= pi(Y,i) proof let b be set; assume b in ((f.i) "/\").:pi(X',i);then consider f4 being set such that A11: f4 in dom ((f.i) "/\") & f4 in pi(X',i) & b = ((f.i) "/\").f4 by FUNCT_1:def 12; consider f5 being Function such that A12: f5 in X' & f4 = f5.i by A11,CARD_3:def 6; reconsider f5 as Element of product J by A12; A13: ((f.i) "/\").f4 = (f.i) "/\" (f5.i) by A12,WAYBEL_1:def 18 .= (f "/\" f5).i by A2,Th11; (f "/\" f5) = f "/\". f5 by WAYBEL_1:def 18;then (f "/\" f5) in f"/\" .: X by A12,FUNCT_2:43; hence b in pi(Y,i) by A11,A13,CARD_3:def 6; end; end; hence f1.x = f2.x by A3,A7,WAYBEL_3:32; end; hence sup (F.:X) = F.sup X by A4,FUNCT_1:9; end; hence thesis by WAYBEL_1:18; end; definition let A be non empty set; let R be complete Heyting LATTICE; cluster R |^ A -> Heyting; coherence proof for i being Element of A holds (A --> R).i is complete Heyting LATTICE by FUNCOP_1:13; then product (A --> R) is complete Heyting by Th13; hence thesis by YELLOW_1:def 5; end; end; begin :: Lattice of Fuzzy Sets definition let A be non empty set; func FuzzyLattice A -> Heyting complete LATTICE equals: Def4: (RealPoset [. 0,1 .]) |^ A; coherence; end; theorem Th14: for A being non empty set holds the carrier of FuzzyLattice A = Funcs(A, [. 0, 1 .]) proof let A be non empty set; thus the carrier of FuzzyLattice A = the carrier of (RealPoset [. 0,1 .]) |^ A by Def4 .= Funcs (A, the carrier of RealPoset [. 0,1 .]) by YELLOW_1:28 .= Funcs (A, [. 0,1 .]) by Def3; end; Lm1: for A being non empty set holds FuzzyLattice A is constituted-Functions proof let A be non empty set; for a being Element of FuzzyLattice A holds a is Function proof let a be Element of FuzzyLattice A; a in the carrier of FuzzyLattice A;then a in Funcs(A, [. 0, 1 .]) by Th14;then ex f being Function st a = f & dom f = A & rng f c= [. 0, 1 .] by FUNCT_2:def 2; hence thesis; end; hence thesis by MONOID_0:def 1; end; definition let A be non empty set; cluster FuzzyLattice A -> constituted-Functions; coherence by Lm1; end; theorem Th15: for R being complete Heyting LATTICE, X being Subset of R, y be Element of R holds "\/"(X,R) "/\" y = "\/"({x "/\" y where x is Element of R: x in X},R) proof let R be complete Heyting LATTICE, X be Subset of R, y be Element of R; A1:for z being Element of R holds z "/\" has_an_upper_adjoint & ex_sup_of X,R by WAYBEL_1:def 19,YELLOW_0:17; set Z = {y "/\" x where x is Element of R: x in X}, W = {x "/\" y where x is Element of R: x in X}; Z = W proof thus Z c= W proof let z be set; assume z in Z;then consider x being Element of R such that A2: z = y "/\" x and A3: x in X; thus thesis by A2,A3; end; thus W c= Z proof let w be set; assume w in W;then consider x being Element of R such that A4: w = x "/\" y and A5: x in X; thus thesis by A4,A5; end; end; hence thesis by A1,WAYBEL_1:66; end; Lm2: for X being non empty set for a being Element of FuzzyLattice X holds a is Membership_Func of X proof let X be non empty set; let a be Element of FuzzyLattice X; a in the carrier of FuzzyLattice X;then a in Funcs(X, [. 0,1 .]) by Th14;then consider f being Function such that A1: a = f & dom f = X & rng f c= [. 0,1 .] by FUNCT_2:def 2; reconsider a as PartFunc of X, [. 0,1 .] by A1,PARTFUN1:25; reconsider a as PartFunc of X, REAL by PARTFUN1:31; dom a = X & rng a c= [. 0,1 .] by A1; hence thesis by FUZZY_1:def 1; end; definition let X be non empty set; let a be Element of FuzzyLattice X; func @a -> Membership_Func of X equals :Def5: a; coherence by Lm2; end; Lm3: for X be non empty set for f be Membership_Func of X holds f is Element of FuzzyLattice X proof let X be non empty set; let f be Membership_Func of X; dom f = X & rng f c= [. 0,1 .] by FUZZY_1:def 1;then f in Funcs(X, [. 0,1 .]) by FUNCT_2:def 2;then f in the carrier of FuzzyLattice X by Th14; hence thesis; end; definition let X be non empty set; let f be Membership_Func of X; func (X,f)@ -> Element of FuzzyLattice X equals :Def6: f; coherence by Lm3; end; definition let X be non empty set; let f be Membership_Func of X; let x be Element of X; redefine func f.x -> Element of RealPoset [. 0,1 .]; coherence proof 0 <= f.x & f.x <= 1 by FUZZY_2:1;then A1: f.x in [. 0,1 .] by TOPREAL5:1; [. 0,1 .] = the carrier of RealPoset [. 0,1 .] by Def3; hence thesis by A1; end; end; definition let X be non empty set; let f be Element of FuzzyLattice X; let x be Element of X; redefine func f.x -> Element of RealPoset [. 0,1 .]; coherence proof 0 <= @f.x & @f.x <= 1 by FUZZY_2:1; hence thesis by Def5; end; end; reserve C for non empty set, c for Element of C, f,g for Membership_Func of C, s,t for Element of FuzzyLattice C; theorem Th16: (for c holds f.c <R= g.c) iff (C,f)@ <<= (C,g)@ proof A1: FuzzyLattice C = (RealPoset [. 0,1 .]) |^ C by Def4; A2: (RealPoset [. 0,1 .]) |^ C = product (C --> RealPoset [. 0,1 .]) by YELLOW_1:def 5; A3: (for c holds f.c <R= g.c) implies (C,f)@ <<= (C,g)@ proof assume A4: for c holds f.c <R= g.c; set f' = (C,f)@, g' = (C,g)@; reconsider f',g' as Element of product (C --> RealPoset [. 0,1 .]) by A2,Def4; for c holds f'.c <<= g'.c proof let c be Element of C; A5: (C --> RealPoset [. 0,1 .]).c = RealPoset [. 0,1 .] by FUNCOP_1:13; set f1 = f.c, g1=g.c; f1 <R= g1 & f = f' & g = g' by A4,Def6; hence thesis by A5,Th3; end; hence (C,f)@ <<= (C,g)@ by A1,A2,WAYBEL_3:28; end; (C,f)@ <<= (C,g)@ implies (for c holds f.c <R= g.c) proof assume A6: (C,f)@ <<= (C,g)@; reconsider ff = (C,f)@, gg = (C,g)@ as Element of product (C --> RealPoset [. 0,1 .]) by A2,Def4; let c; A7: (C --> RealPoset [. 0,1 .]).c = RealPoset [. 0,1 .] by FUNCOP_1:13; ff.c <<= gg.c & (C,f)@ = f & (C,g)@ = g by A1,A2,A6,Def6,WAYBEL_3:28; hence thesis by A7,Th3; end; hence thesis by A3; end; theorem s <<= t iff for c holds @s.c <R= @t.c proof @s = s & @t = t by Def5; then (C,@s)@ = s & (C,@t)@ = t by Def6; hence thesis by Th16; end; theorem Th18: max(f,g) = (C,f)@ "\/" (C,g)@ proof A1: FuzzyLattice C = (RealPoset [. 0,1 .]) |^ C by Def4; A2: (RealPoset [. 0,1 .]) |^ C = product (C --> RealPoset [. 0,1 .]) by YELLOW_1:def 5; set fg = (C,f)@ "\/" (C,g)@, R = RealPoset [. 0,1 .], J = C --> R; now let c; A3: for c being Element of C holds J.c is sup-Semilattice by FUNCOP_1:13; reconsider f' = (C,f)@ , g' = (C,g)@ as Element of product J by A2,Def4; A4: J.c = RealPoset [. 0,1 .] by FUNCOP_1:13; thus (@fg).c = (f' "\/" g').c by A1,A2,Def5 .= ((C,f)@.c) "\/" ((C,g)@.c) by A3,A4,Th12 .= max((C,f)@.c, (C,g)@.c) by Th4 .= max (f.c, (C,g)@.c) by Def6 .= max(f.c, g.c) by Def6; end; then @fg = max(f,g) by FUZZY_1:def 6; hence thesis by Def5; end; theorem s "\/" t = max(@s, @t) proof @s = s & @t = t by Def5; then (C,@s)@ = s & (C,@t)@ = t by Def6; hence thesis by Th18; end; theorem Th20: min(f,g) = (C,f)@ "/\" (C,g)@ proof A1: FuzzyLattice C = (RealPoset [. 0,1 .]) |^ C by Def4; A2: (RealPoset [. 0,1 .]) |^ C = product (C --> RealPoset [. 0,1 .]) by YELLOW_1:def 5; set fg = (C,f)@ "/\" (C,g)@, R = RealPoset [. 0,1 .], J = C --> R; now let c; A3: for c being Element of C holds J.c is Semilattice by FUNCOP_1:13; A4: J.c = RealPoset [. 0,1 .] by FUNCOP_1:13; thus (@fg).c = fg.c by Def5 .= ((C,f)@.c) "/\" ((C,g)@.c) by A1,A2,A3,A4,Th11 .= min((C,f)@ .c, (C,g)@.c) by Th5 .= min (f.c, (C,g)@.c) by Def6 .= min(f.c, g.c) by Def6; end; then @fg = min(f,g) by FUZZY_1:def 5; hence thesis by Def5; end; theorem s "/\" t = min(@s, @t) proof @s = s & @t = t by Def5; then (C,@s)@ = s & (C,@t)@ = t by Def6; hence thesis by Th20; end; begin :: Associativity of composition of fuzzy relations scheme SupDistributivity { L() -> complete LATTICE, X, Y() -> non empty set, F(set,set) -> Element of L(), P,Q[set]}: "\/"({ "\/"({F(x,y) where y is Element of Y(): Q[y]}, L()) where x is Element of X(): P[x] }, L()) = "\/"({F(x,y) where x is Element of X(), y is Element of Y(): P[x] & Q[y]}, L()) proof defpred R[set] means ex x being Element of X() st P[x] & for a being set holds a in $1 iff ex y being Element of Y() st a = F(x,y) & Q[y]; A1: "\/" ({ "\/"(A,L()) where A is Subset of L(): R[A] },L()) = "\/" (union {A where A is Subset of L(): R[A]}, L()) from Sup_of_Sups; A2: union {A where A is Subset of L(): R[A]} = {F(x,y) where x is Element of X(), y is Element of Y(): P[x] & Q[y]} proof thus union {A where A is Subset of L(): R[A]} c= {F(x,y) where x is Element of X(), y is Element of Y(): P[x] & Q[y]} proof let a be set; assume a in union {A where A is Subset of L(): R[A]};then consider A1 be set such that A3: a in A1 & A1 in {A where A is Subset of L(): R[A]} by TARSKI:def 4; consider A2 being Subset of L() such that A4: A1 = A2 & R[A2] by A3; consider x being Element of X() such that A5: P[x] and A6: for a being set holds a in A2 iff ex y being Element of Y() st a = F(x,y) & Q[y] by A4; ex y being Element of Y() st a = F(x,y) & Q[y] by A3,A4,A6; hence thesis by A5; end; thus {F(x,y) where x is Element of X(), y is Element of Y(): P[x] & Q[y]} c= union {A where A is Subset of L(): R[A]} proof let a be set; assume a in {F(x,y) where x is Element of X(), y is Element of Y() : P[x] & Q[y]}; then consider x being Element of X(), y being Element of Y() such that A7: a = F(x,y) & P[x] & Q[y]; set A1 = {F(x,y') where y' is Element of Y(): Q[y']}; A1 c= the carrier of L() proof let b be set; assume b in A1;then ex y' being Element of Y() st b = F(x,y') & Q[y']; hence thesis; end; then reconsider A1 as Subset of L(); A8: a in A1 by A7; R[A1] proof take x; thus P[x] by A7; let a be set; thus thesis; end; then A1 in {A where A is Subset of L(): R[A]}; hence thesis by A8,TARSKI:def 4; end; end; { "\/"(A,L()) where A is Subset of L(): R[A] } = { "\/"({F(x,y) where y is Element of Y(): Q[y]}, L()) where x is Element of X(): P[x] } proof thus { "\/"(A,L()) where A is Subset of L(): R[A] } c= { "\/"({F(x,y) where y is Element of Y(): Q[y]}, L()) where x is Element of X(): P[x] } proof let a be set; assume a in { "\/"(A,L()) where A is Subset of L(): R[A] };then consider A1 being Subset of L() such that A9: a = "\/"(A1,L()) & R[A1]; consider x being Element of X() such that A10: P[x] & for a being set holds a in A1 iff ex y being Element of Y() st a = F(x,y) & Q[y] by A9; now let a be set; a in {F(x,y) where y is Element of Y(): Q[y]} iff ex y being Element of Y() st a = F(x,y) & Q[y]; hence a in A1 iff a in {F(x,y) where y is Element of Y(): Q[y]} by A10; end; then A1 = {F(x,y) where y is Element of Y(): Q[y]} by TARSKI:2; hence thesis by A9,A10; end; thus { "\/"({F(x,y) where y is Element of Y(): Q[y]}, L()) where x is Element of X(): P[x] } c= { "\/"(A,L()) where A is Subset of L(): R[A] } proof let a be set; assume a in { "\/"({F(x,y) where y is Element of Y(): Q[y]}, L()) where x is Element of X(): P[x] }; then consider x being Element of X() such that A11: a = "\/"({F(x,y) where y is Element of Y(): Q[y]}, L()) & P[x]; ex A1 being Subset of L() st a = "\/"(A1,L()) & R[A1] proof consider A1 being Subset of L(); set A2 = {F(x,y) where y is Element of Y(): Q[y]}; A2 c= the carrier of L() proof let b be set; assume b in A2;then ex y being Element of Y() st b = F(x,y) & Q[y]; hence thesis; end; then reconsider A1 = A2 as Subset of L(); R[A1] proof take x; thus P[x] by A11; thus for a being set holds a in A1 iff ex y being Element of Y() st a = F(x,y) & Q[y]; end; hence thesis by A11; end; hence thesis; end; end; hence thesis by A1,A2; end; scheme SupDistributivity' { L() -> complete LATTICE, X, Y() -> non empty set, F(set,set) -> Element of L(), P,Q[set]}: "\/"({ "\/"({F(x,y) where x is Element of X(): P[x]},L()) where y is Element of Y(): Q[y] }, L()) = "\/"({F(x,y) where x is Element of X(), y is Element of Y(): P[x] & Q[y]}, L()) proof defpred R[set] means ex y being Element of Y() st Q[y] & for a being set holds a in $1 iff ex x being Element of X() st a = F(x,y) & P[x]; A1: "\/" ({ "\/"(A,L()) where A is Subset of L(): R[A] },L()) = "\/" (union {A where A is Subset of L(): R[A]}, L()) from Sup_of_Sups; A2: union {A where A is Subset of L(): R[A]} = {F(x,y) where x is Element of X(), y is Element of Y(): P[x] & Q[y]} proof thus union {A where A is Subset of L(): R[A]} c= {F(x,y) where x is Element of X(), y is Element of Y(): P[x] & Q[y]} proof let a be set; assume a in union {A where A is Subset of L(): R[A]};then consider A1 be set such that A3: a in A1 & A1 in {A where A is Subset of L(): R[A]} by TARSKI:def 4; consider A2 being Subset of L() such that A4: A1 = A2 & R[A2] by A3; consider y being Element of Y() such that A5: Q[y] and A6: for a being set holds a in A2 iff ex x being Element of X() st a = F(x,y) & P[x] by A4; ex x being Element of X() st a = F(x,y) & P[x] by A3,A4,A6; hence thesis by A5; end; thus {F(x,y) where x is Element of X(), y is Element of Y(): P[x] & Q[y]} c= union {A where A is Subset of L(): R[A]} proof let a be set; assume a in {F(x,y) where x is Element of X(), y is Element of Y() : P[x] & Q[y]}; then consider x being Element of X(), y being Element of Y() such that A7: a = F(x,y) & P[x] & Q[y]; set A1 = {F(x',y) where x' is Element of X(): P[x']}; A1 c= the carrier of L() proof let b be set; assume b in A1;then ex x' being Element of X() st b = F(x',y) & P[x']; hence thesis; end; then reconsider A1 as Subset of L(); A8: a in A1 by A7; R[A1] proof take y; thus Q[y] by A7; let a be set; thus thesis; end; then A1 in {A where A is Subset of L(): R[A]}; hence thesis by A8,TARSKI:def 4; end; end; { "\/"(A,L()) where A is Subset of L(): R[A] } = { "\/"({F(x,y) where x is Element of X(): P[x]}, L()) where y is Element of Y(): Q[y] } proof thus { "\/"(A,L()) where A is Subset of L(): R[A] } c= { "\/"({F(x,y) where x is Element of X(): P[x]}, L()) where y is Element of Y(): Q[y] } proof let a be set; assume a in { "\/"(A,L()) where A is Subset of L(): R[A] };then consider A1 being Subset of L() such that A9: a = "\/"(A1,L()) & R[A1]; consider y being Element of Y() such that A10: Q[y] & for a being set holds a in A1 iff ex x being Element of X() st a = F(x,y) & P[x] by A9; now let a be set; a in {F(x,y) where x is Element of X(): P[x]} iff ex x being Element of X() st a = F(x,y) & P[x]; hence a in A1 iff a in {F(x,y) where x is Element of X(): P[x]} by A10; end; then A1 = {F(x,y) where x is Element of X(): P[x]} by TARSKI:2; hence thesis by A9,A10; end; thus { "\/"({F(x,y) where x is Element of X(): P[x]}, L()) where y is Element of Y(): Q[y] } c= { "\/"(A,L()) where A is Subset of L(): R[A] } proof let a be set; assume a in { "\/"({F(x,y) where x is Element of X(): P[x]}, L()) where y is Element of Y(): Q[y] }; then consider y being Element of Y() such that A11: a = "\/"({F(x,y) where x is Element of X(): P[x]}, L()) & Q[y]; ex A1 being Subset of L() st a = "\/"(A1,L()) & R[A1] proof consider A1 being Subset of L(); set A2 = {F(x,y) where x is Element of X(): P[x]}; A2 c= the carrier of L() proof let b be set; assume b in A2;then ex x being Element of X() st b = F(x,y) & P[x]; hence thesis; end; then reconsider A1 = A2 as Subset of L(); R[A1] proof take y; thus Q[y] by A11; thus for a being set holds a in A1 iff ex x being Element of X() st a = F(x,y) & P[x]; end; hence thesis by A11; end; hence thesis; end; end; hence thesis by A1,A2; end; scheme FraenkelF'R'{ A() -> non empty set,B() -> non empty set, F1, F2(set,set) -> set, P[set,set] } : { F1(u1,v1) where u1 is (Element of A()), v1 is Element of B() : P[u1,v1] } = { F2(u2,v2) where u2 is (Element of A()), v2 is Element of B() : P[u2,v2] } provided A1:for u being (Element of A()), v being Element of B() st P[u,v] holds F1(u,v) = F2(u,v) proof set A = { F1(u1,v1) where u1 is (Element of A()), v1 is Element of B() : P[u1,v1] }, C = { F2(u3,v3) where u3 is (Element of A()), v3 is Element of B() : P[u3,v3] }; for a be set holds a in A iff a in C proof let a be set; hereby assume a in A;then consider u being (Element of A()), v being Element of B() such that A2: a = F1(u,v) & P[u,v]; a = F2(u,v) by A1,A2; hence a in C by A2; end; assume a in C; then consider u being (Element of A()), v being Element of B() such that A3: a = F2(u,v) & P[u,v]; a = F1(u,v) by A1,A3; hence a in A by A3; end; hence thesis by TARSKI:2; end; scheme FraenkelF6''R { A() -> non empty set, B() -> non empty set, F1, F2(set,set) -> set, P[set,set], Q[set,set] } : { F1(u1,v1) where u1 is (Element of A()), v1 is Element of B() : P[u1,v1] } = { F2(u2,v2) where u2 is (Element of A()), v2 is Element of B() : Q[u2,v2] } provided A1:for u being (Element of A()), v being Element of B() holds P[u,v] iff Q[u,v] and A2:for u being (Element of A()), v being Element of B() st P[u,v] holds F1(u,v) = F2(u,v) proof deffunc U(set,set) = F1($1,$2); deffunc V(set,set) = F2($1,$2); defpred X[set,set] means P[$1,$2]; defpred Y[set,set] means Q[$1,$2]; A3:for u being (Element of A()), v being Element of B() holds X[u,v] iff Y[u,v] by A1; A4:for u being (Element of A()), v being Element of B() st X[u,v] holds U(u,v) = V(u,v) by A2; set A = { U(u1,v1) where u1 is (Element of A()), v1 is Element of B() : X[u1,v1] }, B = { V(u2,v2) where u2 is (Element of A()), v2 is Element of B() : Y[u2,v2] }, C = { V(u3,v3) where u3 is (Element of A()), v3 is Element of B() : X[u3,v3] }; A5:C = B from Fraenkel6''(A3); A = C from FraenkelF'R'(A4); hence thesis by A5; end; scheme SupCommutativity { L() -> complete LATTICE, X, Y() -> non empty set, F1, F2(set,set) -> Element of L(), P,Q[set]}: "\/"({ "\/"({F1(x,y) where y is Element of Y(): Q[y]}, L()) where x is Element of X(): P[x] }, L()) = "\/"({ "\/"({F2(x',y') where x' is Element of X(): P[x']}, L()) where y' is Element of Y(): Q[y'] }, L()) provided A1: for x being Element of X(), y being Element of Y() st P[x] & Q[y] holds F1(x,y) = F2(x,y) proof deffunc U(set,set) = F1($1,$2); deffunc V(set,set) = F2($1,$2); defpred X[set,set] means P[$1] & Q[$2]; defpred Y[set,set] means P[$1] & Q[$2]; A2: for x being Element of X(), y being Element of Y() st X[x,y] holds U(x,y) = V(x,y) by A1; A3: for x being Element of X(), y being Element of Y() holds X[x,y] iff Y[x,y]; A4: {U(x',y') where x' is Element of X(), y' is Element of Y(): X[x',y']} = {V(x,y) where x is Element of X(), y is Element of Y(): Y[x,y]} from FraenkelF6''R(A3,A2); defpred P'[set] means P[$1]; defpred Q'[set] means Q[$1]; A5: "\/"({ "\/"({U(x,y) where y is Element of Y(): Q'[y]}, L()) where x is Element of X(): P'[x] }, L()) ="\/"({U(x,y) where x is Element of X(), y is Element of Y(): P'[x] & Q'[y]}, L()) from SupDistributivity; "\/"({ "\/"({V(x',y') where x' is Element of X(): P'[x']}, L()) where y' is Element of Y(): Q'[y'] }, L()) ="\/"({V(x',y') where x' is Element of X(), y' is Element of Y(): P'[x'] & Q'[y']}, L()) from SupDistributivity' .="\/"({U(x',y') where x' is Element of X(), y' is Element of Y(): P'[x'] & Q'[y']}, L()) by A4; hence thesis by A5; end; Lm4: for x being Element of RealPoset [. 0,1 .] holds x is Real proof let x be Element of RealPoset [. 0,1 .]; x in the carrier of RealPoset [. 0,1 .] & the carrier of RealPoset [. 0,1 .] = [. 0,1 .] by Def3; hence thesis; end; Lm5: for x being Element of [. 0,1 .] holds x is Element of RealPoset [. 0,1 .] by Def3; Lm6: for X,Y,Z being non empty set for R being RMembership_Func of X,Y for S being RMembership_Func of Y,Z for x being Element of X, z being Element of Z holds rng min(R,S,x,z) = {R. [x,y] "/\" S. [y,z] where y is Element of Y: not contradiction} & rng min(R,S,x,z) <> {} proof let X,Y,Z being non empty set; let R being RMembership_Func of X,Y; let S being RMembership_Func of Y,Z; let x being Element of X, z being Element of Z; A1: Y = dom min(R,S,x,z) & min(R,S,x,z) is PartFunc of dom min(R,S,x,z), rng min(R,S,x,z) by FUZZY_1:def 1,PARTFUN1:24; set L = {R. [x,y] "/\" S. [y,z] where y is Element of Y: not contradiction}; for c be set holds c in L iff c in rng min(R,S,x,z) proof let c be set; hereby assume c in L;then consider y being Element of Y such that A2: c = R. [x,y] "/\" S. [y,z]; c = min(R. [x,y], S. [y,z] qua real number) by A2,Th5 .= min(R,S,x,z).y by FUZZY_4:def 3; hence c in rng min(R,S,x,z) by A1,PARTFUN1:27; end; assume c in rng min(R,S,x,z);then consider y being Element of Y such that A3: y in dom min(R,S,x,z) & c = min(R,S,x,z).y by PARTFUN1:26; c = min(R. [x,y], S. [y,z] qua real number) by A3,FUZZY_4:def 3 .= R. [x,y] "/\" S. [y,z] by Th5; hence c in L; end; hence rng min(R,S,x,z) = L by TARSKI:2; thus rng min(R,S,x,z) <> {} proof ex d be set st d in rng min(R,S,x,z) proof consider y being Element of Y; take d = min(R,S,x,z).y; thus d in rng min(R,S,x,z) by A1,PARTFUN1:27; end; hence thesis; end; end; theorem Th22: for X,Y,Z being non empty set for R being RMembership_Func of X,Y for S being RMembership_Func of Y,Z for x being Element of X, z being Element of Z holds (R (#) S). [x,z] = "\/"({R. [x,y] "/\" S. [y,z] where y is Element of Y: not contradiction}, RealPoset [. 0,1 .]) proof let X,Y,Z being non empty set; let R being RMembership_Func of X,Y; let S being RMembership_Func of Y,Z; let x being Element of X, z being Element of Z; A1: (R (#) S). [x,z] = upper_bound(rng(min(R,S,x,z))) by FUZZY_4:def 4; set L = {R. [x,y] "/\" S. [y,z] where y is Element of Y: not contradiction}; A2: (R (#) S). [x,z] is_>=_than L proof for b being Element of RealPoset [. 0,1 .] st b in L holds (R (#) S). [x,z] >>= b proof let b be Element of RealPoset [. 0,1 .]; assume b in L;then consider y being Element of Y such that A3: b = R. [x,y] "/\" S. [y,z] & not contradiction; reconsider b as Real by Lm4; A4: dom min(R,S,x,z) = Y by FUZZY_1:def 1; b = min(R. [x,y], S. [y,z] qua real number) by A3,Th5 .= min(R,S,x,z).y by FUZZY_4:def 3; then b <= upper_bound rng min(R,S,x,z) by A4,FUZZY_4:1; hence thesis by A1,Th3; end; hence thesis by LATTICE3:def 9; end; for b being Element of RealPoset [. 0,1 .] st b is_>=_than L holds (R (#) S). [x,z] <<= b proof let b be Element of RealPoset [. 0,1 .]; assume A5: b is_>=_than {R. [x,y] "/\" S. [y,z] where y is Element of Y: not contradiction}; A6: Y = dom min(R,S,x,z) & rng min(R,S,x,z) c= [. 0,1 .] & min(R,S,x,z) is PartFunc of dom min(R,S,x,z), rng min(R,S,x,z) by FUZZY_1:def 1,PARTFUN1:24; A7: L = rng min(R,S,x,z) by Lm6; A8: rng min(R,S,x,z) <> {} by Lm6; for r being real number st r in rng min(R,S,x,z) holds r <= b proof let r be real number; assume A9: r in rng min(R,S,x,z);then reconsider r as Element of RealPoset [. 0,1 .] by A6,Lm5; r <<= b by A5,A7,A9,LATTICE3:def 9; hence thesis by Th3; end; then upper_bound rng min(R,S,x,z) <= b by A8,TOPMETR3:1; hence thesis by A1,Th3; end; hence thesis by A2,YELLOW_0:32; end; Lm7: for X,Y,Z being non empty set for R being RMembership_Func of X,Y for S being RMembership_Func of Y,Z for x being Element of X, z being Element of Z for a being Element of RealPoset [. 0,1 .] holds (R (#) S). [x,z] "/\" a = "\/"({R. [x,y] "/\" S. [y,z] "/\" a where y is Element of Y: not contradiction}, RealPoset [. 0,1 .]) proof let X,Y,Z being non empty set; let R being RMembership_Func of X,Y; let S being RMembership_Func of Y,Z; let x being Element of X, z being Element of Z; let a being Element of RealPoset [. 0,1 .]; {R. [x,y] "/\" S. [y,z] where y is Element of Y: not contradiction} c= the carrier of RealPoset [. 0,1 .] proof let d be set;assume d in {R. [x,y] "/\" S. [y,z] where y is Element of Y: not contradiction}; then consider t being Element of Y such that A1: d = R. [x,t] "/\" S. [t,z] & not contradiction; thus thesis by A1; end; then A2: {R. [x,y] "/\" S. [y,z] where y is Element of Y: not contradiction} is Subset of RealPoset [. 0,1 .]; A3: (R (#) S). [x,z] "/\" a = "\/"({R. [x,y] "/\" S. [y,z] where y is Element of Y: not contradiction}, RealPoset [. 0,1 .]) "/\" a by Th22 .= "\/"({b "/\" a where b is Element of RealPoset [. 0,1 .]: b in {R. [x,y] "/\" S. [y,z] where y is Element of Y: not contradiction}}, RealPoset [. 0,1 .]) by A2,Th15; set A = {b "/\" a where b is Element of RealPoset [. 0,1 .]: b in {R. [x,y] "/\" S. [y,z] where y is Element of Y: not contradiction}}, B = {R. [x,y] "/\" S. [y,z] "/\" a where y is Element of Y: not contradiction}; for c be set holds c in A iff c in B proof let c be set; hereby assume c in A;then consider b being Element of RealPoset [. 0,1 .] such that A4: c = b "/\" a & b in {R. [x,y] "/\" S. [y,z] where y is Element of Y:not contradiction}; consider y being Element of Y such that A5: b = R. [x,y] "/\" S. [y,z] & not contradiction by A4; thus c in B by A4,A5; end; assume c in B;then consider y being Element of Y such that A6: c = R. [x,y] "/\" S. [y,z] "/\" a & not contradiction; R. [x,y] "/\" S. [y,z] in {R. [x,y1] "/\" S. [y1,z] where y1 is Element of Y: not contradiction}; hence c in A by A6; end; hence thesis by A3,TARSKI:2; end; Lm8: for X,Y,Z being non empty set for R being RMembership_Func of X,Y for S being RMembership_Func of Y,Z for x being Element of X, z being Element of Z for a being Element of RealPoset [. 0,1 .] holds a "/\" (R (#) S). [x,z] = "\/"({a "/\" R. [x,y] "/\" S. [y,z] where y is Element of Y: not contradiction}, RealPoset [. 0,1 .]) proof let X,Y,Z being non empty set; let R being RMembership_Func of X,Y; let S being RMembership_Func of Y,Z; let x being Element of X, z being Element of Z; let a being Element of RealPoset [. 0,1 .]; set A = {a "/\" R. [x,y] "/\" S. [y,z] where y is Element of Y: not contradiction}, B = {R. [x,y] "/\" S. [y,z] "/\" a where y is Element of Y: not contradiction}; for b be set holds b in A iff b in B proof let b be set; hereby assume b in A; then consider y being Element of Y such that A1: b = a "/\" R. [x,y] "/\" S. [y,z]; b = R. [x,y] "/\" S. [y,z] "/\" a by A1,LATTICE3:16; hence b in B; end; assume b in B; then consider y being Element of Y such that A2: b = R. [x,y] "/\" S. [y,z] "/\" a; b = a "/\" R. [x,y] "/\" S. [y,z] by A2,LATTICE3:16; hence b in A; end; then A = B by TARSKI:2; hence thesis by Lm7; end; theorem for X,Y,Z,W being non empty set for R being RMembership_Func of X,Y for S being RMembership_Func of Y,Z for T being RMembership_Func of Z,W holds (R (#) S) (#) T = R (#) (S (#) T) proof let X,Y,Z,W be non empty set; let R be RMembership_Func of X,Y; let S be RMembership_Func of Y,Z; let T be RMembership_Func of Z,W; A1: dom ((R (#) S) (#) T) = [:X,W:] & dom (R (#) (S (#) T))= [:X,W:] by FUZZY_1:def 1; for x being set, w being set st x in X & w in W holds ((R (#) S) (#) T). [x,w] = (R (#) (S (#) T)). [x,w] proof let x being set, w being set; assume A2: x in X & w in W; then reconsider x as Element of X; reconsider w as Element of W by A2; A3: ((R (#) S) (#) T). [x,w] = "\/"({((R (#) S). [x,z]) "/\" T. [z,w] where z is Element of Z: not contradiction}, RealPoset [. 0,1 .]) & (R (#) (S (#) T)). [x,w] = "\/"({R. [x,y] "/\" ((S (#) T). [y,w]) where y is Element of Y: not contradiction}, RealPoset [. 0,1 .]) by Th22; set A = {"\/"({R. [x,y] "/\" S. [y,z] "/\" T. [z,w] where y is Element of Y: not contradiction}, RealPoset [. 0,1 .]) where z is Element of Z :not contradiction} , B = {(R (#) S). [x,z] "/\" T. [z,w] where z is Element of Z :not contradiction}; A4: for a be set holds a in A iff a in B proof let a be set; hereby assume a in A;then consider z being Element of Z such that A5: a = "\/"({R. [x,y] "/\" S. [y,z] "/\" T. [z,w] where y is Element of Y: not contradiction}, RealPoset [. 0,1 .]); a = (R (#) S). [x,z] "/\" T. [z,w] by A5,Lm7; hence a in B; end; assume a in B;then consider z being Element of Z such that A6: a = (R (#) S). [x,z] "/\" T. [z,w]; a = "\/"({R. [x,y] "/\" S. [y,z] "/\" T. [z,w] where y is Element of Y: not contradiction}, RealPoset [. 0,1 .]) by A6,Lm7; hence a in A; end; set C = {"\/"({R. [x,y] "/\" S. [y,z] "/\" T. [z,w] where z is Element of Z: not contradiction}, RealPoset [. 0,1 .]) where y is Element of Y :not contradiction} , D = {R. [x,y] "/\" ((S (#) T). [y,w]) where y is Element of Y : not contradiction}; for c be set holds c in C iff c in D proof let c be set; hereby assume c in C;then consider y being Element of Y such that A7: c = "\/"({R. [x,y] "/\" S. [y,z] "/\" T. [z,w] where z is Element of Z: not contradiction}, RealPoset [. 0,1 .]); c = R. [x,y] "/\" (S (#) T). [y,w] by A7,Lm8; hence c in D; end; assume c in D;then consider y being Element of Y such that A8: c = R. [x,y] "/\" ((S (#) T). [y,w]); c = "\/"({R. [x,y] "/\" S. [y,z] "/\" T. [z,w] where z is Element of Z: not contradiction}, RealPoset [. 0,1 .]) by A8,Lm8; hence c in C; end; then A9: C = D by TARSKI:2; defpred X[set] means not contradiction; deffunc U(Element of Y,Element of Z) = R. [x,$1] "/\" S. [$1,$2] "/\" T. [$2,w]; deffunc V(Element of Z,Element of Y) = R. [x,$2] "/\" S. [$2,$1] "/\" T. [$1,w]; A10: for y being Element of Y, z being Element of Z st X[y] & X[z] holds U(y,z) = U(y,z); "\/"({"\/"({U(y,z) where z is Element of Z: X[z]}, RealPoset [. 0,1 .]) where y is Element of Y :X[y]}, RealPoset [. 0,1 .]) ="\/"({"\/"({U(y1,z1) where y1 is Element of Y: X[y1]}, RealPoset [. 0,1 .]) where z1 is Element of Z :X[z1]}, RealPoset [. 0,1 .]) from SupCommutativity(A10); hence thesis by A3,A4,A9,TARSKI:2; end; hence thesis by A1,FUNCT_3:6; end;