Copyright (c) 2000 Association of Mizar Users
environ vocabulary INT_1, RAT_1, ARYTM_1, TARSKI, FUNCT_1, FUNCT_2, RELAT_1, BOOLE, ARYTM_3, CARD_4, CARD_1, FINSET_1, PARTFUN1, SUPINF_1, SEQ_1, MEASURE6, RLVECT_1, GROUP_1, COMPLEX1, MEASURE1, SETFAM_1, MESFUNC1, ARYTM; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, MEASURE6, REAL_1, RELAT_1, FUNCT_1, NAT_1, INT_1, RAT_1, FINSET_1, CARD_1, CARD_4, SUPINF_1, SUPINF_2, MEASURE1, MEASURE2, MEASURE3, FUNCT_2, PARTFUN1, EXTREAL1; constructors NAT_1, MEASURE3, MEASURE6, RAT_1, WELLORD2, EXTREAL1, REAL_1, SUPINF_2, MEMBERED; clusters SUBSET_1, SUPINF_1, MEASURE1, XREAL_0, RELSET_1, INT_1, CARD_1, NAT_1, MEMBERED, NUMBERS, ORDINAL2; requirements NUMERALS, BOOLE, SUBSET, REAL, ARITHM; definitions TARSKI; theorems MEASURE1, TARSKI, SUBSET_1, SUPINF_1, PARTFUN1, FUNCT_1, FUNCT_2, MEASURE5, MEASURE6, SETFAM_1, NAT_1, REAL_1, REAL_2, SUPINF_2, INT_1, AXIOMS, CARD_1, CARD_4, WELLORD2, RAT_1, RELSET_1, PRALG_3, XREAL_0, EXTREAL1, HAHNBAN, XBOOLE_0, XBOOLE_1, SQUARE_1, XCMPLX_1; schemes SETWISEO, FUNCT_2, SEQ_1; begin :: Cardinal numbers of INT and RAT reserve k for Nat; reserve r for Real; reserve i for Integer; reserve q for Rational; definition func INT- -> Subset of REAL means :Def1: r in it iff ex k st r = - k; existence proof defpred P[Element of REAL] means ex k st $1 = -k; consider IT being Subset of REAL such that A1:for r being Element of REAL holds r in IT iff P[r] from SubsetEx; take IT; thus thesis by A1; end; uniqueness proof let D1,D2 be Subset of REAL such that A2:r in D1 iff ex k st r = - k and A3:r in D2 iff ex k st r = - k; for r holds r in D1 iff r in D2 proof let r; thus r in D1 implies r in D2 proof assume r in D1; then ex k st r = - k by A2; hence thesis by A3; end; assume r in D2; then ex k st r = - k by A3; hence thesis by A2; end; hence D1 = D2 by SUBSET_1:8; end; correctness; end; definition cluster INT- -> non empty; coherence by Def1,REAL_1:26; end; theorem Th1: NAT,INT- are_equipotent proof defpred P[Element of NAT,Element of INT-] means $2=-$1; A1:for x being Element of NAT ex y being Element of INT- st P[x,y] proof let x be Element of NAT; -x in INT- by Def1; hence thesis; end; consider f being Function of NAT,INT- such that A2:for x being Element of NAT holds P[x,f.x] from FuncExD(A1); A3:f in Funcs(NAT,INT-) by FUNCT_2:11; then A4:dom f = NAT & rng f c= INT- by PRALG_3:4; f is one-to-one & rng f = INT- proof A5: for x1,x2 being set st x1 in dom f & x2 in dom f & f.x1=f.x2 holds x1=x2 proof let x1,x2 be set; assume A6: x1 in dom f & x2 in dom f & f.x1=f.x2; then reconsider x1 as Nat by A3,PRALG_3:4; reconsider x2 as Nat by A3,A6,PRALG_3:4; f.x1 = -x1 & f.x2 = -x2 by A2; then --x1=x2 by A6; hence thesis; end; for y being set st y in INT- holds f"{y} <> {} proof let y being set; assume A7:y in INT-; then reconsider y as Real; consider k being Nat such that A8: y = -k by A7,Def1; f.k = -k by A2; then f.k in {y} by A8,TARSKI:def 1; hence thesis by A4,FUNCT_1:def 13; end; hence thesis by A5,FUNCT_1:def 8,FUNCT_2:49; end; hence thesis by A4,WELLORD2:def 4; end; theorem Th2: INT=INT- \/ NAT proof for x being set st x in INT holds x in INT- \/ NAT proof let x be set; assume x in INT; then consider k being Nat such that A1: x = k or x = - k by INT_1:def 1; now per cases by A1; suppose x = k; hence x in INT- \/ NAT by XBOOLE_0:def 2; suppose x = -k; then x in INT- by Def1; hence x in INT- \/ NAT by XBOOLE_0:def 2; end; hence thesis; end; then A2:INT c= INT- \/ NAT by TARSKI:def 3; for x being set st x in INT- \/ NAT holds x in INT proof let x be set; assume A3:x in INT- \/ NAT; now per cases by A3,XBOOLE_0:def 2; suppose x in INT-; then consider k being Nat such that A4: x = -k by Def1; thus thesis by A4,INT_1:def 1; suppose x in NAT; hence thesis by INT_1:def 1; end; hence thesis; end; then INT- \/ NAT c= INT by TARSKI:def 3; hence thesis by A2,XBOOLE_0:def 10; end; theorem Th3: NAT,INT are_equipotent by Th1,Th2,CARD_4:15,35; definition redefine func INT -> Subset of REAL; correctness by INT_1:11,TARSKI:def 3; end; definition let n be Nat; func RAT_with_denominator n -> Subset of RAT means :Def2: q in it iff ex i st q = i/n; existence proof defpred P[Element of RAT] means ex i st $1 = i/n; consider X being Subset of RAT such that A1:for r being Element of RAT holds r in X iff P[r] from SubsetEx; A2:for q being Rational holds q in X iff ex i st q = i/n proof let q be Rational; reconsider q as Element of RAT by RAT_1:def 2; q in X iff ex i st q = i/n by A1; hence thesis; end; take X; thus thesis by A2; end; uniqueness proof let D1,D2 be Subset of RAT such that A3:q in D1 iff ex i st q = i/n and A4:q in D2 iff ex i st q = i/n; for q being Element of RAT holds q in D1 iff q in D2 proof let q be Element of RAT; thus q in D1 implies q in D2 proof assume A5:q in D1; reconsider q as Rational; ex i st q = i/n by A3,A5; hence thesis by A4; end; assume A6:q in D2; reconsider q as Rational; ex i st q = i/n by A4,A6; hence thesis by A3; end; hence D1 = D2 by SUBSET_1:8; end; end; definition let n be Nat; cluster RAT_with_denominator(n+1) -> non empty; coherence proof consider i being Integer; reconsider i1=n+1 as Integer; i/i1 in RAT by RAT_1:def 1; then reconsider q=i/i1 as Rational by RAT_1:def 2; q in RAT_with_denominator(n+1) by Def2; hence thesis; end; end; theorem Th4: for n being Nat holds INT,RAT_with_denominator (n+1) are_equipotent proof let n be Nat; defpred P[Element of INT,Element of RAT_with_denominator(n+1)] means $2=$1/(n+1); A1:for x being Element of INT ex y being Element of RAT_with_denominator(n+1) st P[x,y] proof let x be Element of INT; reconsider x as Integer; reconsider i1=n+1 as Integer; set y=x/i1; y in RAT by RAT_1:def 1; then reconsider y as Rational by RAT_1:def 2; y in RAT_with_denominator(n+1) by Def2; hence thesis; end; consider f being Function of INT,RAT_with_denominator(n+1) such that A2:for x being Element of INT holds P[x,f.x] from FuncExD(A1); A3:f in Funcs(INT,RAT_with_denominator(n+1)) by FUNCT_2:11; then A4:dom f = INT & rng f c= RAT_with_denominator(n+1) by PRALG_3:4; f is one-to-one & rng f = RAT_with_denominator(n+1) proof A5: for x1,x2 being set st x1 in dom f & x2 in dom f & f.x1=f.x2 holds x1=x2 proof let x1,x2 be set; assume A6:x1 in dom f & x2 in dom f & f.x1=f.x2; then reconsider x1 as Element of INT by A3,PRALG_3:4; reconsider x2 as Element of INT by A3,A6,PRALG_3:4; reconsider x1 as Element of REAL; reconsider x2 as Element of REAL; f.x1 = x1/(n+1) & f.x2 = x2/(n+1) & n+1 <> 0 by A2; hence thesis by A6,XCMPLX_1:53; end; for y being set st y in RAT_with_denominator(n+1) holds f"{y} <> {} proof let y being set; assume A7:y in RAT_with_denominator(n+1); then reconsider y as Rational by RAT_1:def 2; consider i being Integer such that A8: y = i/(n+1) by A7,Def2; reconsider i as Element of INT by INT_1:def 2; f.i = i/(n+1) by A2; then f.i in {y} by A8,TARSKI:def 1; hence thesis by A4,FUNCT_1:def 13; end; hence thesis by A5,FUNCT_1:def 8,FUNCT_2:49; end; hence thesis by A4,WELLORD2:def 4; end; theorem NAT,RAT are_equipotent proof assume A1:not NAT,RAT are_equipotent; defpred P[Element of NAT,Element of bool REAL] means $2 = RAT_with_denominator ($1+1); A2:for n being Element of NAT ex X being Element of bool REAL st P[n,X] proof let n be Element of NAT; for x being set st x in RAT_with_denominator (n+1) holds x in REAL by RAT_1:4,TARSKI:def 3; then reconsider X=RAT_with_denominator (n+1) as Element of bool REAL by TARSKI:def 3; take X; thus thesis; end; consider F being Function of NAT,bool REAL such that A3:for n being Element of NAT holds P[n,F.n] from FuncExD(A2); A4:union rng F = RAT proof for x being set st x in union rng F holds x in RAT proof let x being set; assume x in union rng F; then consider Y being set such that A5: x in Y & Y in rng F by TARSKI:def 4; dom F = NAT by FUNCT_2:def 1; then consider n being set such that A6: n in NAT & F.n = Y by A5,FUNCT_1:def 5; reconsider n as Nat by A6; Y = RAT_with_denominator (n+1) by A3,A6; hence thesis by A5; end; then A7: union rng F c= RAT by TARSKI:def 3; for x being set st x in RAT holds x in union rng F proof let x be set; assume x in RAT; then reconsider x as Rational by RAT_1:def 2; A8: dom F = NAT by FUNCT_2:def 1; denominator x <> 0 by RAT_1:def 3; then consider m being Nat such that A9: denominator x = m + 1 by NAT_1:22; reconsider n = denominator x - 1 as Nat by A9,XCMPLX_1:26; denominator x = n+1 by XCMPLX_1:27; then x = (numerator x)/(n+1) by RAT_1:37; then x in RAT_with_denominator (n+1) by Def2; then x in F.n & F.n in rng F by A3,A8,FUNCT_1:def 5; hence thesis by TARSKI:def 4; end; then RAT c= union rng F by TARSKI:def 3; hence thesis by A7,XBOOLE_0:def 10; end; dom F = NAT by FUNCT_2:def 1; then A10:rng F is countable by CARD_4:45; for Y being set st Y in rng F holds Y is countable proof let Y be set; assume Y in rng F; then consider n being set such that A11: n in dom F & F.n = Y by FUNCT_1:def 5; reconsider n as Nat by A11,FUNCT_2:def 1; Y = RAT_with_denominator (n+1) by A3,A11; then INT,Y are_equipotent by Th4; then NAT,Y are_equipotent by Th3,WELLORD2:22; then A12: Card NAT = Card Y by CARD_1:21; Card NAT <=` alef 0 by CARD_4:44,def 1; hence thesis by A12,CARD_4:def 1; end; then RAT is countable by A4,A10,CARD_4:62; then Card RAT <=` alef 0 by CARD_4:def 1; then not alef 0 <` Card RAT by CARD_1:14; then not alef 0 <=` Card RAT or not alef 0 <> Card RAT by CARD_1:13; then A13:Card RAT <` alef 0 by A1,CARD_1:14,21,38; not RAT is finite by CARD_1:38,CARD_4:14,RAT_1:12; hence contradiction by A13,CARD_4:2; end; begin :: Basic operations on R_EAL valued functions definition let C be non empty set, f be PartFunc of C, ExtREAL, x be set; redefine func f.x -> R_eal; coherence proof per cases; suppose x in dom f; hence thesis by PARTFUN1:27; suppose not x in dom f; hence thesis by FUNCT_1:def 4,SUPINF_2:def 1; end; end; definition let C be non empty set; let f1,f2 be PartFunc of C,ExtREAL; deffunc F(Element of C) = f1.$1 + f2.$1; func f1+f2 -> PartFunc of C,ExtREAL means :Def3: dom it = (dom f1 /\ dom f2)\((f1"{-infty} /\ f2"{+infty}) \/ (f1"{+infty} /\ f2"{-infty})) & for c being Element of C st c in dom it holds it.c = f1.c + f2.c; existence proof defpred P[Element of C] means $1 in (dom f1 /\ dom f2)\((f1"{-infty} /\ f2"{+infty}) \/ (f1"{+infty} /\ f2"{-infty})); consider F be PartFunc of C,ExtREAL such that A1:for c being Element of C holds c in dom F iff P[c] and A2:for c being Element of C st c in dom F holds F.c = F(c) from LambdaPFD'; take F; thus dom F=(dom f1 /\ dom f2)\((f1"{-infty} /\ f2"{+infty}) \/ (f1"{+infty} /\ f2"{-infty})) proof for x being set st x in dom F holds x in (dom f1 /\ dom f2)\((f1"{-infty} /\ f2"{+infty}) \/ (f1"{+infty} /\ f2"{-infty})) proof let x be set; assume A3:x in dom F; dom F c= C by RELSET_1:12; then reconsider x as Element of C by A3; x in dom F by A3; hence thesis by A1; end; then A4: dom F c= (dom f1 /\ dom f2)\((f1"{-infty} /\ f2"{+infty}) \/ (f1"{+infty} /\ f2"{-infty})) by TARSKI:def 3; for x being set st x in (dom f1 /\ dom f2)\((f1"{-infty} /\ f2"{+infty}) \/ (f1"{+infty} /\ f2"{-infty})) holds x in dom F proof let x be set; assume A5:x in (dom f1 /\ dom f2)\((f1"{-infty} /\ f2"{+infty}) \/ (f1"{+infty} /\ f2"{-infty})); then x in dom f1 /\ dom f2 by XBOOLE_0:def 4; then A6: x in dom f1 by XBOOLE_0:def 3; dom f1 c= C by RELSET_1:12; then reconsider x as Element of C by A6; x in (dom f1 /\ dom f2)\((f1"{-infty} /\ f2"{+infty}) \/ (f1"{+infty} /\ f2"{-infty})) by A5; hence thesis by A1; end; then (dom f1 /\ dom f2)\((f1"{-infty} /\ f2"{+infty}) \/ (f1"{+infty} /\ f2"{-infty})) c= dom F by TARSKI:def 3; hence thesis by A4,XBOOLE_0:def 10; end; let c be Element of C; assume c in dom F; hence thesis by A2; end; uniqueness proof set X = (dom f1 /\ dom f2)\((f1"{-infty} /\ f2"{+infty}) \/ (f1"{+infty} /\ f2"{-infty})); thus for f,g being PartFunc of C,ExtREAL st (dom f=X & for c be Element of C st c in dom f holds f.c = F(c)) & (dom g=X & for c be Element of C st c in dom g holds g.c = F(c)) holds f = g from UnPartFuncD'; end; deffunc F(Element of C) = f1.$1 - f2.$1; func f1-f2 -> PartFunc of C,ExtREAL means dom it = (dom f1 /\ dom f2)\((f1"{+infty} /\ f2"{+infty}) \/ (f1"{-infty} /\ f2"{-infty})) & for c being Element of C st c in dom it holds it.c = f1.c - f2.c; existence proof defpred P[Element of C] means $1 in (dom f1 /\ dom f2)\((f1"{+infty} /\ f2"{+infty}) \/ (f1"{-infty} /\ f2"{-infty})); consider F be PartFunc of C,ExtREAL such that A7:for c being Element of C holds c in dom F iff P[c] and A8:for c being Element of C st c in dom F holds F.c = F(c) from LambdaPFD'; take F; thus dom F=(dom f1 /\ dom f2)\((f1"{+infty} /\ f2"{+infty}) \/ (f1"{-infty} /\ f2"{-infty})) proof for x being set st x in dom F holds x in (dom f1 /\ dom f2)\((f1"{+infty} /\ f2"{+infty}) \/ (f1"{-infty} /\ f2"{-infty})) proof let x be set; assume A9:x in dom F; dom F c= C by RELSET_1:12; then reconsider x as Element of C by A9; x in dom F by A9; hence thesis by A7; end; then A10: dom F c= (dom f1 /\ dom f2)\((f1"{+infty} /\ f2"{+infty}) \/ (f1"{-infty} /\ f2"{-infty})) by TARSKI:def 3; for x being set st x in (dom f1 /\ dom f2)\((f1"{+infty} /\ f2"{+infty}) \/ (f1"{-infty} /\ f2"{-infty})) holds x in dom F proof let x be set; assume A11:x in (dom f1 /\ dom f2)\((f1"{+infty} /\ f2"{+infty}) \/ (f1"{-infty} /\ f2"{-infty})); then x in dom f1 /\ dom f2 by XBOOLE_0:def 4; then A12: x in dom f1 by XBOOLE_0:def 3; dom f1 c= C by RELSET_1:12; then reconsider x as Element of C by A12; x in (dom f1 /\ dom f2)\((f1"{+infty} /\ f2"{+infty}) \/ (f1"{-infty} /\ f2"{-infty})) by A11; hence thesis by A7; end; then (dom f1 /\ dom f2)\((f1"{+infty} /\ f2"{+infty}) \/ (f1"{-infty} /\ f2"{-infty})) c= dom F by TARSKI:def 3; hence thesis by A10,XBOOLE_0:def 10; end; let c be Element of C; assume c in dom F; hence thesis by A8; end; uniqueness proof set X = (dom f1 /\ dom f2)\((f1"{+infty} /\ f2"{+infty}) \/ (f1"{-infty} /\ f2"{-infty})); thus for f,g being PartFunc of C,ExtREAL st (dom f=X & for c be Element of C st c in dom f holds f.c = F(c)) & (dom g=X & for c be Element of C st c in dom g holds g.c = F(c)) holds f = g from UnPartFuncD'; end; deffunc F(Element of C) = f1.$1 * f2.$1; func f1(#)f2 -> PartFunc of C,ExtREAL means :Def5: dom it = dom f1 /\ dom f2 & for c being Element of C st c in dom it holds it.c = f1.c * f2.c; existence proof defpred P[Element of C] means $1 in dom f1 /\ dom f2; consider F be PartFunc of C,ExtREAL such that A13:for c being Element of C holds c in dom F iff P[c] and A14:for c being Element of C st c in dom F holds F.c = F(c) from LambdaPFD'; take F; thus dom F=dom f1 /\ dom f2 proof for x being set st x in dom F holds x in dom f1 /\ dom f2 proof let x be set; assume A15:x in dom F; dom F c= C by RELSET_1:12; then reconsider x as Element of C by A15; x in dom F by A15; hence thesis by A13; end; then A16: dom F c= dom f1 /\ dom f2 by TARSKI:def 3; for x being set st x in dom f1 /\ dom f2 holds x in dom F proof let x be set; assume A17:x in dom f1 /\ dom f2; then A18: x in dom f1 by XBOOLE_0:def 3; dom f1 c= C by RELSET_1:12; then reconsider x as Element of C by A18; x in dom f1 /\ dom f2 by A17; hence thesis by A13; end; then dom f1 /\ dom f2 c= dom F by TARSKI:def 3; hence thesis by A16,XBOOLE_0:def 10; end; let c be Element of C; assume c in dom F; hence thesis by A14; end; uniqueness proof set X = dom f1 /\ dom f2; thus for F,G being PartFunc of C,ExtREAL st (dom F=X & for c be Element of C st c in dom F holds F.c = F(c)) & (dom G=X & for c be Element of C st c in dom G holds G.c = F(c)) holds F = G from UnPartFuncD'; end; end; definition let C be non empty set, f be PartFunc of C,ExtREAL, r be Real; deffunc F(Element of C) = (R_EAL r) * f.$1; func r(#)f -> PartFunc of C,ExtREAL means :Def6: dom it = dom f & for c being Element of C st c in dom it holds it.c = (R_EAL r) * f.c; existence proof defpred P[Element of C] means $1 in dom f; consider F be PartFunc of C,ExtREAL such that A1:for c being Element of C holds c in dom F iff P[c] and A2:for c being Element of C st c in dom F holds F.c = F(c) from LambdaPFD'; take F; thus dom F=dom f proof for x being set st x in dom F holds x in dom f proof let x be set; assume A3:x in dom F; dom F c= C by RELSET_1:12; then reconsider x as Element of C by A3; x in dom F by A3; hence thesis by A1; end; then A4: dom F c= dom f by TARSKI:def 3; for x being set st x in dom f holds x in dom F proof let x be set; assume A5:x in dom f; dom f c= C by RELSET_1:12; then reconsider x as Element of C by A5; x in dom f by A5; hence thesis by A1; end; then dom f c= dom F by TARSKI:def 3; hence thesis by A4,XBOOLE_0:def 10; end; let c be Element of C; assume c in dom F; hence thesis by A2; end; uniqueness proof set X = dom f; thus for F,G being PartFunc of C,ExtREAL st (dom F=X & for c be Element of C st c in dom F holds F.c = F(c)) & (dom G=X & for c be Element of C st c in dom G holds G.c = F(c)) holds F = G from UnPartFuncD'; end; end; theorem Th6: for C being non empty set, f being PartFunc of C,ExtREAL, r being Real st r <> 0 holds for c being Element of C st c in dom(r(#)f) holds f.c = (r(#)f).c / R_EAL r proof let C being non empty set; let f being PartFunc of C,ExtREAL; let r being Real; assume A1:r <> 0; then A2:R_EAL r <> 0. by MEASURE6:def 1,SUPINF_2:def 1; A3:R_EAL r = r by MEASURE6:def 1; then A4:R_EAL r <> +infty & R_EAL r <> -infty by SUPINF_1:31; let c being Element of C; assume c in dom(r(#)f); then A5: (r(#)f).c = (R_EAL r) * f.c by Def6; now per cases; suppose A6:f.c = +infty; now per cases by A2,SUPINF_1:22; suppose A7:0. <' R_EAL r; then (r(#)f).c = +infty by A5,A6,EXTREAL1:def 1; hence thesis by A4,A6,A7,EXTREAL1:def 2; suppose A8:R_EAL r <' 0.; then (r(#)f).c = -infty by A5,A6,EXTREAL1:def 1; hence thesis by A4,A6,A8,EXTREAL1:def 2; end; hence thesis; suppose A9:f.c = -infty; now per cases by A2,SUPINF_1:22; suppose A10:0. <' R_EAL r; then (r(#)f).c = -infty by A5,A9,EXTREAL1:def 1; hence thesis by A4,A9,A10,EXTREAL1:def 2; suppose A11:R_EAL r <' 0.; then (r(#)f).c = +infty by A5,A9,EXTREAL1:def 1; hence thesis by A4,A9,A11,EXTREAL1:def 2; end; hence thesis; suppose f.c <> +infty & f.c <> -infty; then reconsider a = f.c as Real by EXTREAL1:1; (r(#)f).c = r * a by A3,A5,EXTREAL1:13; then (r(#)f).c / R_EAL r = r*a/r by A1,A3,EXTREAL1:32,SUPINF_2:def 1 .= a/(r/r) by XCMPLX_1:78 .= a / 1 by A1,XCMPLX_1:60; hence thesis; end; hence thesis; end; definition let C be non empty set; let f be PartFunc of C,ExtREAL; deffunc F(Element of C) = -(f.$1); func -f -> PartFunc of C,ExtREAL means dom it = dom f & for c being Element of C st c in dom it holds it.c = -(f.c); existence proof defpred P[Element of C] means $1 in dom f; consider F be PartFunc of C,ExtREAL such that A1:for c being Element of C holds c in dom F iff P[c] and A2:for c being Element of C st c in dom F holds F.c = F(c) from LambdaPFD'; take F; thus dom F=dom f proof for x being set st x in dom F holds x in dom f proof let x be set; assume A3:x in dom F; dom F c= C by RELSET_1:12; then reconsider x as Element of C by A3; x in dom F by A3; hence thesis by A1; end; then A4: dom F c= dom f by TARSKI:def 3; for x being set st x in dom f holds x in dom F proof let x be set; assume A5:x in dom f; dom f c= C by RELSET_1:12; then reconsider x as Element of C by A5; x in dom f by A5; hence thesis by A1; end; then dom f c= dom F by TARSKI:def 3; hence thesis by A4,XBOOLE_0:def 10; end; let c be Element of C; assume c in dom F; hence thesis by A2; end; uniqueness proof set X = dom f; thus for F,G being PartFunc of C,ExtREAL st (dom F=X & for c be Element of C st c in dom F holds F.c = F(c)) & (dom G=X & for c be Element of C st c in dom G holds G.c = F(c)) holds F = G from UnPartFuncD'; end; end; definition func 1. -> R_eal equals :Def8: 1; correctness by SUPINF_1:10; end; definition let C be non empty set; let f be PartFunc of C,ExtREAL; let r be Real; deffunc F(Element of C) = (R_EAL r)/(f.$1); func r/f -> PartFunc of C,ExtREAL means :Def9: dom it = dom f \ f"{0.} & for c being Element of C st c in dom it holds it.c = (R_EAL r)/(f.c); existence proof defpred P[Element of C] means $1 in dom f \ f"{0.}; consider F be PartFunc of C,ExtREAL such that A1:for c being Element of C holds c in dom F iff P[c] and A2:for c being Element of C st c in dom F holds F.c = F(c) from LambdaPFD'; take F; thus dom F=dom f \ f"{0.} proof for x being set st x in dom F holds x in dom f \ f"{0.} proof let x be set; assume A3:x in dom F; dom F c= C by RELSET_1:12; then reconsider x as Element of C by A3; x in dom F by A3; hence thesis by A1; end; then A4: dom F c= dom f \ f"{0.} by TARSKI:def 3; for x being set st x in dom f \ f"{0.} holds x in dom F proof let x be set; assume A5:x in dom f \ f"{0.}; dom f c= C & dom f \ f"{0.} c= dom f by RELSET_1:12,XBOOLE_1:36; then dom f \ f"{0.} c= C by XBOOLE_1:1; then reconsider x as Element of C by A5; x in dom f \ f"{0.} by A5; hence thesis by A1; end; then dom f \ f"{0.} c= dom F by TARSKI:def 3; hence thesis by A4,XBOOLE_0:def 10; end; let c be Element of C; assume c in dom F; hence thesis by A2; end; uniqueness proof set X = dom f \ f"{0.}; thus for F,G being PartFunc of C,ExtREAL st (dom F=X & for c be Element of C st c in dom F holds F.c = F(c)) & (dom G=X & for c be Element of C st c in dom G holds G.c = F(c)) holds F = G from UnPartFuncD'; end; end; theorem for C being non empty set, f being PartFunc of C,ExtREAL holds dom (1/f) = dom f \ f"{0.} & for c being Element of C st c in dom (1/f) holds (1/f).c = 1./(f.c) proof let C be non empty set; let f be PartFunc of C,ExtREAL; thus dom (1/f) = dom f \ f"{0.} by Def9; thus for c being Element of C st c in dom (1/f) holds (1/f).c = 1./(f.c) proof let c be Element of C; assume c in dom (1/f); then (1/f).c = (R_EAL 1)/(f.c) by Def9; hence thesis by Def8,MEASURE6:def 1; end; end; definition let C be non empty set; let f be PartFunc of C,ExtREAL; deffunc F(Element of C) = |. f.$1 .|; func |.f.| -> PartFunc of C,ExtREAL means dom it = dom f & for c being Element of C st c in dom it holds it.c = |. f.c .|; existence proof defpred P[Element of C] means $1 in dom f; consider F be PartFunc of C,ExtREAL such that A1:for c being Element of C holds c in dom F iff P[c] and A2:for c being Element of C st c in dom F holds F.c = F(c) from LambdaPFD'; take F; thus dom F=dom f proof for x being set st x in dom F holds x in dom f proof let x be set; assume A3:x in dom F; dom F c= C by RELSET_1:12; then reconsider x as Element of C by A3; x in dom F by A3; hence thesis by A1; end; then A4: dom F c= dom f by TARSKI:def 3; for x being set st x in dom f holds x in dom F proof let x be set; assume A5:x in dom f; dom f c= C by RELSET_1:12; then reconsider x as Element of C by A5; x in dom f by A5; hence thesis by A1; end; then dom f c= dom F by TARSKI:def 3; hence thesis by A4,XBOOLE_0:def 10; end; let c be Element of C; assume c in dom F; hence thesis by A2; end; uniqueness proof set X = dom f; thus for F,G being PartFunc of C,ExtREAL st (dom F=X & for c be Element of C st c in dom F holds F.c = F(c)) & (dom G=X & for c be Element of C st c in dom G holds G.c = F(c)) holds F = G from UnPartFuncD'; end; end; canceled; theorem Th9: for C being non empty set, f1,f2 being PartFunc of C,ExtREAL holds f1 + f2 = f2 + f1 proof let C be non empty set; let f1,f2 be PartFunc of C,ExtREAL; A1:dom (f1+f2) =(dom f1 /\ dom f2)\ ((f1"{-infty} /\ f2"{+infty}) \/ (f1"{+infty} /\ f2"{-infty})) by Def3 .= dom (f2+f1) by Def3; for x being Element of C st x in dom (f1+f2) holds (f1+f2).x = (f2+f1).x proof let x be Element of C; assume A2:x in dom (f1+f2); then (f1+f2).x = f1.x + f2.x by Def3; hence thesis by A1,A2,Def3; end; hence thesis by A1,PARTFUN1:34; end; theorem Th10: for C being non empty set, f1,f2 being PartFunc of C,ExtREAL holds f1 (#) f2 = f2 (#) f1 proof let C be non empty set; let f1,f2 be PartFunc of C,ExtREAL; A1:dom (f1(#)f2) = dom f1 /\ dom f2 by Def5 .= dom (f2(#)f1) by Def5; for x being Element of C st x in dom (f1(#)f2) holds (f1(#)f2).x = (f2(#) f1).x proof let x be Element of C; assume A2:x in dom(f1(#)f2); then (f1(#)f2).x = f1.x * f2.x by Def5; hence thesis by A1,A2,Def5; end; hence thesis by A1,PARTFUN1:34; end; definition let C be non empty set; let f1,f2 be PartFunc of C,ExtREAL; redefine func f1+f2; commutativity by Th9; redefine func f1(#)f2; commutativity by Th10; end; begin :: Level sets theorem Th11: for r being Real holds ex n being Nat st r <= n proof let r being Real; A1:r <= [/ r \] by INT_1:def 5; per cases; suppose [/ r \] >= 0; then reconsider n=[/ r \] as Nat by INT_1:16; take n; thus thesis by INT_1:def 5; suppose A2: [/ r \] < 0; take 0; thus thesis by A1,A2,AXIOMS:22; end; theorem Th12: for r being Real holds ex n being Nat st -n <= r proof let r being Real; A1:[\ r /] <= r by INT_1:def 4; [\ r /] is Element of INT by INT_1:def 2; then consider k being Nat such that A2:[\ r /] = k or [\ r /] = -k by INT_1:def 1; per cases; suppose [\ r /] < 0; hence thesis by A1,A2,NAT_1:18; suppose A3: [\ r /] >= 0; take 0; thus thesis by A3,INT_1:def 4; end; theorem Th13: for r,s being real number st r < s holds ex n being Nat st 1/(n+1) < s-r proof let r,s be real number; assume r < s; then s-r > 0 by SQUARE_1:11; then consider t being real number such that A1:0 < t & t < s-r by REAL_1:75; reconsider t as Real by XREAL_0:def 1; A2:1/t > 0 by A1,REAL_2:127; A3:[/ 1/t \] + 1 > 1/t by INT_1:57; set n = [/ 1/t \]; reconsider i0=0 as Integer; n + 1 > i0 by A2,A3,AXIOMS:22; then n >= i0 by INT_1:20; then reconsider n as Nat by INT_1:16; A4:n+1 > 0 by A2,A3,AXIOMS:22; (n+1)*t >= 1 by A1,A3,REAL_2:178; then 1/(n+1) <= t by A4,REAL_2:177; then 1/(n+1) < s-r by A1,AXIOMS:22; hence thesis; end; theorem Th14: for r,s being real number st for n being Nat holds r-1/(n+1) <= s holds r <= s proof let r,s be real number; assume A1:for n being Nat holds r-1/(n+1) <= s; assume r > s; then consider n being Nat such that A2:1/(n+1) < r - s by Th13; s + 1/(n+1) < r by A2,REAL_1:86; then s < r - 1/(n+1) by REAL_1:86; hence contradiction by A1; end; theorem Th15: for a being R_eal st (for r being Real holds R_EAL r <' a) holds a = +infty proof let a being R_eal; assume A1:for r being Real holds R_EAL r <' a; assume A2:not a = +infty; a <=' +infty by SUPINF_1:20; then a <' +infty by A2,SUPINF_1:22; then consider b being R_eal such that A3:a <' b & b <' +infty & b in REAL by MEASURE5:17; reconsider b as Real by A3; a <=' R_EAL b by A3,MEASURE6:def 1; hence contradiction by A1; end; theorem Th16: for a being R_eal st (for r being Real holds a <' R_EAL r) holds a = -infty proof let a being R_eal; assume A1:for r being Real holds a <' R_EAL r; assume A2:not a = -infty; -infty <=' a by SUPINF_1:21; then -infty <' a by A2,SUPINF_1:22; then consider b being R_eal such that A3:-infty <' b & b <' a & b in REAL by MEASURE5:17; reconsider b as Real by A3; R_EAL b <=' a by A3,MEASURE6:def 1; hence contradiction by A1; end; definition let X be set; let S be sigma_Field_Subset of X; let A be set; pred A is_measurable_on S means :Def11: A in S; end; theorem for X,A being set, S being sigma_Field_Subset of X holds A is_measurable_on S iff (for M being sigma_Measure of S holds A is_measurable M) proof let X,A be set; let S be sigma_Field_Subset of X; A1:A is_measurable_on S implies (for M being sigma_Measure of S holds A is_measurable M) proof assume A2:A is_measurable_on S; let M being sigma_Measure of S; A in S by A2,Def11; hence thesis by MEASURE1:def 12; end; (for M being sigma_Measure of S holds A is_measurable M) implies A is_measurable_on S proof assume A3:for M being sigma_Measure of S holds A is_measurable M; consider M being sigma_Measure of S; A is_measurable M by A3; then A in S by MEASURE1:def 12; hence thesis by Def11; end; hence thesis by A1; end; reserve X for non empty set; reserve x for Element of X; reserve f,g for PartFunc of X,ExtREAL; reserve S for sigma_Field_Subset of X; reserve F for Function of NAT,S; reserve A for set; reserve a for R_eal; reserve r,s for Real; reserve n,m for Nat; definition let X,f,a; func less_dom(f,a) -> Subset of X means :Def12: x in it iff x in dom f & ex y being R_eal st y=f.x & y <' a; existence proof defpred P[Element of X] means $1 in dom f & ex y being R_eal st y=f.$1 & y <' a; thus ex A being Subset of X st for x holds x in A iff P[x] from SubsetEx; end; uniqueness proof let D1,D2 be Subset of X such that A1:x in D1 iff x in dom f & ex y being R_eal st y=f.x & y <' a and A2:x in D2 iff x in dom f & ex y being R_eal st y=f.x & y <' a; now let x; thus x in D1 implies x in D2 proof assume x in D1; then x in dom f & ex y being R_eal st y=f.x & y <' a by A1; hence x in D2 by A2; end; assume x in D2; then x in dom f & ex y being R_eal st y=f.x & y <' a by A2; hence x in D1 by A1; end; hence thesis by SUBSET_1:8; end; correctness; func less_eq_dom(f,a) -> Subset of X means :Def13: x in it iff x in dom f & ex y being R_eal st y=f.x & y <=' a; existence proof defpred P[Element of X] means $1 in dom f & ex y being R_eal st y=f.$1 & y <=' a; thus ex A being Subset of X st for x holds x in A iff P[x] from SubsetEx; end; uniqueness proof let D1,D2 be Subset of X such that A3:x in D1 iff x in dom f & ex y being R_eal st y=f.x & y <=' a and A4:x in D2 iff x in dom f & ex y being R_eal st y=f.x & y <=' a; now let x; thus x in D1 implies x in D2 proof assume x in D1; then x in dom f & ex y being R_eal st y=f.x & y <=' a by A3; hence x in D2 by A4; end; assume x in D2; then x in dom f & ex y being R_eal st y=f.x & y <=' a by A4; hence x in D1 by A3; end; hence thesis by SUBSET_1:8; end; correctness; func great_dom(f,a) -> Subset of X means :Def14: x in it iff x in dom f & ex y being R_eal st y=f.x & a <' y; existence proof defpred P[Element of X] means $1 in dom f & ex y being R_eal st y=f.$1 & a <' y; thus ex A being Subset of X st for x holds x in A iff P[x] from SubsetEx; end; uniqueness proof let D1,D2 be Subset of X such that A5:x in D1 iff x in dom f & ex y being R_eal st y=f.x & a <' y and A6:x in D2 iff x in dom f & ex y being R_eal st y=f.x & a <' y; now let x; thus x in D1 implies x in D2 proof assume x in D1; then x in dom f & ex y being R_eal st y=f.x & a <' y by A5; hence x in D2 by A6; end; assume x in D2; then x in dom f & ex y being R_eal st y=f.x & a <' y by A6; hence x in D1 by A5; end; hence thesis by SUBSET_1:8; end; correctness; func great_eq_dom(f,a) -> Subset of X means :Def15: x in it iff x in dom f & ex y being R_eal st y=f.x & a <=' y; existence proof defpred P[Element of X] means $1 in dom f & ex y being R_eal st y=f.$1 & a <=' y; thus ex A being Subset of X st for x holds x in A iff P[x] from SubsetEx; end; uniqueness proof let D1,D2 be Subset of X such that A7:x in D1 iff x in dom f & ex y being R_eal st y=f.x & a <=' y and A8:x in D2 iff x in dom f & ex y being R_eal st y=f.x & a <=' y; now let x; thus x in D1 implies x in D2 proof assume x in D1; then x in dom f & ex y being R_eal st y=f.x & a <=' y by A7; hence x in D2 by A8; end; assume x in D2; then x in dom f & ex y being R_eal st y=f.x & a <=' y by A8; hence x in D1 by A7; end; hence thesis by SUBSET_1:8; end; correctness; func eq_dom(f,a) -> Subset of X means :Def16: x in it iff x in dom f & ex y being R_eal st y=f.x & a=y; existence proof defpred P[Element of X] means $1 in dom f & ex y being R_eal st y=f.$1 & a = y; thus ex A being Subset of X st for x holds x in A iff P[x] from SubsetEx; end; uniqueness proof let D1,D2 be Subset of X such that A9:x in D1 iff x in dom f & ex y being R_eal st y=f.x & a=y and A10:x in D2 iff x in dom f & ex y being R_eal st y=f.x & a=y; now let x; thus x in D1 implies x in D2 proof assume x in D1; then x in dom f & ex y being R_eal st y=f.x & a=y by A9; hence x in D2 by A10; end; assume x in D2; then x in dom f & ex y being R_eal st y=f.x & a=y by A10; hence x in D1 by A9; end; hence thesis by SUBSET_1:8; end; correctness; end; theorem Th18: for X, S, f, A, a st A c= dom f holds A /\ great_eq_dom(f,a) = A\(A /\ less_dom(f,a)) proof let X, S, f, A, a; assume that A1:A c= dom f; dom f c= X by RELSET_1:12; then A2:A c= X by A1,XBOOLE_1:1; A3:A /\ great_eq_dom(f,a) c= A\(A /\ less_dom(f,a)) proof for x being set st x in A /\ great_eq_dom(f,a) holds x in A\(A /\ less_dom(f,a)) proof let x be set; assume x in A /\ great_eq_dom(f,a); then A4: x in A & x in great_eq_dom(f,a) by XBOOLE_0:def 3; then consider y1 being R_eal such that A5: y1=f.x & a <=' y1 by Def15; not x in less_dom(f,a) proof assume x in less_dom(f,a); then consider y2 being R_eal such that A6: y2=f.x & y2 <' a by Def12; thus contradiction by A5,A6; end; then not(x in (A /\ less_dom(f,a))) by XBOOLE_0:def 3; hence thesis by A4,XBOOLE_0:def 4; end; hence thesis by TARSKI:def 3; end; A\(A /\ less_dom(f,a)) c= A /\ great_eq_dom(f,a) proof for x being set st x in A\(A /\ less_dom(f,a)) holds x in A /\ great_eq_dom(f,a) proof let x be set; assume x in A\(A /\ less_dom(f,a)); then A7: x in A & not(x in A /\ less_dom(f,a)) by XBOOLE_0:def 4; then A8: not (x in less_dom(f,a)) by XBOOLE_0:def 3; reconsider x as Element of X by A2,A7; reconsider y=f.x as R_eal; not(y <' a) by A1,A7,A8,Def12; then x in great_eq_dom(f,a) by A1,A7,Def15; hence thesis by A7,XBOOLE_0:def 3; end; hence thesis by TARSKI:def 3; end; hence thesis by A3,XBOOLE_0:def 10; end; theorem Th19: for X, S, f, A, a st A c= dom f holds A /\ great_dom(f,a) = A\(A /\ less_eq_dom(f,a)) proof let X,S,f,A,a; assume A1:A c= dom f; dom f c= X by RELSET_1:12; then A2:A c= X by A1,XBOOLE_1:1; A3:A /\ great_dom(f,a) c= A\(A /\ less_eq_dom(f,a)) proof for x being set st x in A /\ great_dom(f,a) holds x in A\(A /\ less_eq_dom(f,a)) proof let x be set; assume x in A /\ great_dom(f,a); then A4: x in A & x in great_dom(f,a) by XBOOLE_0:def 3; then consider y1 being R_eal such that A5: y1=f.x & a <' y1 by Def14; not x in less_eq_dom(f,a) proof assume x in less_eq_dom(f,a); then consider y2 being R_eal such that A6: y2=f.x & y2 <=' a by Def13; thus contradiction by A5,A6; end; then not(x in (A /\ less_eq_dom(f,a))) by XBOOLE_0:def 3; hence thesis by A4,XBOOLE_0:def 4; end; hence thesis by TARSKI:def 3; end; A\(A /\ less_eq_dom(f,a)) c= A /\ great_dom(f,a) proof for x being set st x in A\(A /\ less_eq_dom(f,a)) holds x in A /\ great_dom(f,a) proof let x be set; assume x in A\(A /\ less_eq_dom(f,a)); then A7: x in A & not(x in A /\ less_eq_dom(f,a)) by XBOOLE_0:def 4; then A8: not x in less_eq_dom(f,a) by XBOOLE_0:def 3; reconsider x as Element of X by A2,A7; reconsider y=f.x as R_eal; not y <=' a by A1,A7,A8,Def13; then x in great_dom(f,a) by A1,A7,Def14; hence thesis by A7,XBOOLE_0:def 3; end; hence thesis by TARSKI:def 3; end; hence thesis by A3,XBOOLE_0:def 10; end; theorem Th20: for X, S, f, A, a st A c= dom f holds A /\ less_eq_dom(f,a) = A\(A /\ great_dom(f,a)) proof let X, S, f, A, a; assume A1:A c= dom f; dom f c= X by RELSET_1:12; then A2:A c= X by A1,XBOOLE_1:1; A3:A /\ less_eq_dom(f,a) c= A\(A /\ great_dom(f,a)) proof let x be set; assume x in A /\ less_eq_dom(f,a); then A4: x in A & x in less_eq_dom(f,a) by XBOOLE_0:def 3; then consider y1 being R_eal such that A5: y1=f.x & y1 <=' a by Def13; not x in great_dom(f,a) proof assume x in great_dom(f,a); then consider y2 being R_eal such that A6: y2=f.x & a <' y2 by Def14; thus contradiction by A5,A6; end; then not(x in (A /\ great_dom(f,a))) by XBOOLE_0:def 3; hence thesis by A4,XBOOLE_0:def 4; end; A\(A /\ great_dom(f,a)) c= A /\ less_eq_dom(f,a) proof for x being set st x in A\(A /\ great_dom(f,a)) holds x in A /\ less_eq_dom(f,a) proof let x be set; assume x in A\(A /\ great_dom(f,a)); then A7: x in A & not(x in A /\ great_dom(f,a)) by XBOOLE_0:def 4; then A8: not (x in great_dom(f,a)) by XBOOLE_0:def 3; reconsider x as Element of X by A2,A7; reconsider y=f.x as R_eal; not(a <' y) by A1,A7,A8,Def14; then x in less_eq_dom(f,a) by A1,A7,Def13; hence thesis by A7,XBOOLE_0:def 3; end; hence thesis by TARSKI:def 3; end; hence thesis by A3,XBOOLE_0:def 10; end; theorem Th21: for X, S, f, A, a st A c= dom f holds A /\ less_dom(f,a) = A\(A /\ great_eq_dom(f,a)) proof let X,S,f,A,a; assume A1:A c= dom f; dom f c= X by RELSET_1:12; then A2:A c= X by A1,XBOOLE_1:1; A3:A /\ less_dom(f,a) c= A\(A /\ great_eq_dom(f,a)) proof for x being set st x in A /\ less_dom(f,a) holds x in A\(A /\ great_eq_dom(f,a)) proof let x be set; assume x in A /\ less_dom(f,a); then A4: x in A & x in less_dom(f,a) by XBOOLE_0:def 3; then consider y1 being R_eal such that A5: y1=f.x & y1 <' a by Def12; not x in great_eq_dom(f,a) proof assume x in great_eq_dom(f,a); then consider y2 being R_eal such that A6: y2=f.x & a <=' y2 by Def15; thus contradiction by A5,A6; end; then not(x in (A /\ great_eq_dom(f,a))) by XBOOLE_0:def 3; hence thesis by A4,XBOOLE_0:def 4; end; hence thesis by TARSKI:def 3; end; A\(A /\ great_eq_dom(f,a)) c= A /\ less_dom(f,a) proof for x being set st x in A\(A /\ great_eq_dom(f,a)) holds x in A /\ less_dom(f,a) proof let x be set; assume x in A\(A /\ great_eq_dom(f,a)); then A7: x in A & not(x in A /\ great_eq_dom(f,a)) by XBOOLE_0:def 4; then A8: not (x in great_eq_dom(f,a)) by XBOOLE_0:def 3; reconsider x as Element of X by A2,A7; reconsider y=f.x as R_eal; not(a <=' y) by A1,A7,A8,Def15; then x in less_dom(f,a) by A1,A7,Def12; hence thesis by A7,XBOOLE_0:def 3; end; hence thesis by TARSKI:def 3; end; hence thesis by A3,XBOOLE_0:def 10; end; theorem for X, S, f, A, a holds A /\ eq_dom(f,a) = A /\ great_eq_dom(f,a) /\ less_eq_dom(f,a) proof let X,S,f,A,a; A1:A /\ eq_dom(f,a) c= A /\ great_eq_dom(f,a) /\ less_eq_dom(f,a) proof for x being set st x in A /\ eq_dom(f,a) holds x in A /\ great_eq_dom(f,a) /\ less_eq_dom(f,a) proof let x be set; assume x in A /\ eq_dom(f,a); then A2: x in A & x in eq_dom(f,a) by XBOOLE_0:def 3; then consider y1 being R_eal such that A3: y1=f.x & a = y1 by Def16; reconsider x as Element of X by A2; A4: x in dom f by A2,Def16; then x in great_eq_dom(f,a) by A3,Def15; then A5: x in A /\ great_eq_dom(f,a) by A2,XBOOLE_0:def 3; x in less_eq_dom(f,a) by A3,A4,Def13; hence thesis by A5,XBOOLE_0:def 3; end; hence thesis by TARSKI:def 3; end; A /\ great_eq_dom(f,a) /\ less_eq_dom(f,a) c= A /\ eq_dom(f,a) proof for x being set st x in A /\ great_eq_dom(f,a) /\ less_eq_dom(f,a) holds x in A /\ eq_dom(f,a) proof let x being set; assume x in A /\ great_eq_dom(f,a) /\ less_eq_dom(f,a); then A6: x in A /\ great_eq_dom(f,a) & x in less_eq_dom(f,a) by XBOOLE_0:def 3 ; then A7: x in A & x in great_eq_dom(f,a) by XBOOLE_0:def 3; then consider y1 being R_eal such that A8: y1=f.x & a <=' y1 by Def15; consider y2 being R_eal such that A9: y2=f.x & y2 <=' a by A6,Def13; reconsider x as Element of X by A6; A10: x in dom f by A6,Def13; a = y1 by A8,A9,SUPINF_1:22; then x in eq_dom(f,a) by A8,A10,Def16; hence thesis by A7,XBOOLE_0:def 3; end; hence thesis by TARSKI:def 3; end; hence thesis by A1,XBOOLE_0:def 10; end; theorem for X, S, F, f, A, r st for n holds F.n = A /\ great_dom(f,R_EAL(r-1/(n+1))) holds A /\ great_eq_dom(f,R_EAL r) = meet rng F proof let X,S,F,f,A,r; assume A1:for n holds F.n = A /\ great_dom(f,R_EAL(r-1/(n+1))); A2:A /\ great_eq_dom(f,R_EAL r) c= meet rng F proof for x being set st x in A /\ great_eq_dom(f,R_EAL r) holds x in meet rng F proof let x be set; assume x in A /\ great_eq_dom(f,R_EAL r); then A3: x in A & x in great_eq_dom(f,R_EAL r) by XBOOLE_0:def 3; for Y being set holds Y in rng F implies x in Y proof let Y be set; Y in rng F implies x in Y proof assume Y in rng F; then consider m being Nat such that A4: m in dom F & Y = F.m by PARTFUN1:26; A5: Y = A /\ great_dom(f,R_EAL(r-1/(m+1))) by A1,A4; A6: x in dom f by A3,Def15; reconsider x as Element of X by A3; consider y being R_eal such that A7: y=f.x & R_EAL r <=' y by A3,Def15; R_EAL(r-1/(m+1))<' R_EAL r proof m+1 > 0 by NAT_1:19; then (m+1)" > 0 by REAL_1:72; then 1/(m+1) > 0 by XCMPLX_1:217; then r < r+1/(m+1) by REAL_1:69; then A8: r-1/(m+1) < r by REAL_1:84; r-1/(m+1)=R_EAL(r-1/(m+1)) & r=R_EAL r by MEASURE6:def 1; hence thesis by A8,HAHNBAN:12; end; then R_EAL(r-1/(m+1)) <' y by A7,SUPINF_1:29; then x in great_dom(f,R_EAL(r-1/(m+1))) by A6,A7,Def14; hence thesis by A3,A5,XBOOLE_0:def 3; end; hence thesis; end; hence thesis by SETFAM_1:def 1; end; hence thesis by TARSKI:def 3; end; meet rng F c= A /\ great_eq_dom(f,R_EAL r) proof for x being set st x in meet rng F holds x in A /\ great_eq_dom(f,R_EAL r ) proof let x be set; assume A9:x in meet rng F; A10: for m holds x in A & x in dom f & ex y being R_eal st y=f.x & R_EAL(r-1/(m+1)) <' y proof let m; m in NAT; then m in dom F by FUNCT_2:def 1; then F.m in rng F by FUNCT_1:def 5; then x in F.m by A9,SETFAM_1:def 1; then x in A /\ great_dom(f,R_EAL(r-1/(m+1))) by A1; then x in A & x in great_dom(f,R_EAL(r-1/(m+1))) by XBOOLE_0:def 3; hence thesis by Def14; end; reconsider y=f.x as R_eal; 1 in NAT; then 1 in dom F by FUNCT_2:def 1; then F.1 in rng F by FUNCT_1:def 5; then x in F.1 by A9,SETFAM_1:def 1; then x in A /\ great_dom(f,R_EAL(r-1/(1+1))) by A1; then x in great_dom(f,R_EAL(r-1/(1+1))) by XBOOLE_0:def 3; then reconsider x as Element of X; R_EAL r <=' y proof A11: for m holds R_EAL(r-1/(m+1)) <' y proof let m; consider y1 being R_eal such that A12: y1=f.x & R_EAL(r-1/(m+1)) <' y1 by A10; thus thesis by A12; end; now per cases; suppose y=+infty; hence thesis by SUPINF_1:20; suppose not y=+infty; then A13: not +infty <=' y by SUPINF_1:24; R_EAL(r-1/(1+1))<'y by A11; then reconsider y1=y as Real by A13,MEASURE6:15; A14: R_EAL y1 = y by MEASURE6:def 1; for m holds r-1/(m+1) <= y1 proof let m; R_EAL(r-1/(m+1)) <' R_EAL y1 by A11,A14; hence thesis by MEASURE6:14; end; then r <= y1 by Th14; hence thesis by A14,MEASURE6:13; end; hence thesis; end; then x in great_eq_dom(f,R_EAL r) by A10,Def15; hence thesis by A10,XBOOLE_0:def 3; end; hence thesis by TARSKI:def 3; end; hence thesis by A2,XBOOLE_0:def 10; end; theorem Th24: for X, S, F, f, A for r being real number st for n holds F.n = A /\ less_dom(f,R_EAL(r+1/(n+1))) holds A /\ less_eq_dom(f,R_EAL r) = meet rng F proof let X,S,F,f,A; let r be real number; assume A1:for n holds F.n = A /\ less_dom(f,R_EAL(r+1/(n+1))); A2:A /\ less_eq_dom(f,R_EAL r) c= meet rng F proof for x being set st x in A /\ less_eq_dom(f,R_EAL r) holds x in meet rng F proof let x be set; assume x in A /\ less_eq_dom(f,R_EAL r); then A3: x in A & x in less_eq_dom(f,R_EAL r) by XBOOLE_0:def 3; for Y being set holds Y in rng F implies x in Y proof let Y be set; Y in rng F implies x in Y proof assume Y in rng F; then consider m being Nat such that A4: m in dom F & Y = F.m by PARTFUN1:26; A5: Y = A /\ less_dom(f,R_EAL(r+1/(m+1))) by A1,A4; A6: x in dom f by A3,Def13; reconsider x as Element of X by A3; consider y being R_eal such that A7: y=f.x & y <=' R_EAL r by A3,Def13; R_EAL r <' R_EAL(r+1/(m+1)) proof m+1 > 0 by NAT_1:19; then (m+1)" > 0 by REAL_1:72; then 1/(m+1) > 0 by XCMPLX_1:217; then A8: r < r+1/(m+1) by REAL_1:69; r+1/(m+1)=R_EAL(r+1/(m+1)) & r=R_EAL r by MEASURE6:def 1; hence thesis by A8,HAHNBAN:12; end; then y <' R_EAL(r+1/(m+1)) by A7,SUPINF_1:29; then x in less_dom(f,R_EAL(r+1/(m+1))) by A6,A7,Def12; hence thesis by A3,A5,XBOOLE_0:def 3; end; hence thesis; end; hence thesis by SETFAM_1:def 1; end; hence thesis by TARSKI:def 3; end; meet rng F c= A /\ less_eq_dom(f,R_EAL r) proof for x being set st x in meet rng F holds x in A /\ less_eq_dom(f,R_EAL r) proof let x be set; assume A9:x in meet rng F; A10: for m holds x in A & x in dom f & ex y being R_eal st y=f.x & y <' R_EAL(r+1/(m+1)) proof let m; m in NAT; then m in dom F by FUNCT_2:def 1; then F.m in rng F by FUNCT_1:def 5; then x in F.m by A9,SETFAM_1:def 1; then x in A /\ less_dom(f,R_EAL(r+1/(m+1))) by A1; then x in A & x in less_dom(f,R_EAL(r+1/(m+1))) by XBOOLE_0:def 3; hence thesis by Def12; end; reconsider y=f.x as R_eal; 1 in NAT; then 1 in dom F by FUNCT_2:def 1; then F.1 in rng F by FUNCT_1:def 5; then x in F.1 by A9,SETFAM_1:def 1; then x in A /\ less_dom(f,R_EAL(r+1/(1+1))) by A1; then x in less_dom(f,R_EAL(r+1/(1+1))) by XBOOLE_0:def 3; then reconsider x as Element of X; y <=' R_EAL r proof A11: for m holds y <' R_EAL(r+1/(m+1)) proof let m; consider y1 being R_eal such that A12: y1=f.x & y1 <' R_EAL(r+1/(m+1)) by A10; thus thesis by A12; end; now per cases; suppose y=-infty; hence thesis by SUPINF_1:21; suppose not y=-infty; then A13: not y <=' -infty by SUPINF_1:23; y <' R_EAL(r+1/(1+1)) by A11; then reconsider y1=y as Real by A13,MEASURE6:15; A14: R_EAL y1 = y by MEASURE6:def 1; for m holds y1-1/(m+1) <= r proof let m; R_EAL y1 <' R_EAL(r+1/(m+1)) by A11,A14; then y1 < r+1/(m+1) by MEASURE6:14; hence thesis by REAL_1:86; end; then y1 <= r by Th14; hence thesis by A14,MEASURE6:13; end; hence thesis; end; then x in less_eq_dom(f,R_EAL r) by A10,Def13; hence thesis by A10,XBOOLE_0:def 3; end; hence thesis by TARSKI:def 3; end; hence thesis by A2,XBOOLE_0:def 10; end; theorem Th25: for X, S, F, f, A for r being real number st for n holds F.n = A /\ less_eq_dom(f,R_EAL(r-1/(n+1))) holds A /\ less_dom(f,R_EAL r) = union rng F proof let X,S,F,f,A; let r be real number; A1: r in REAL by XREAL_0:def 1; assume A2:for n holds F.n = A /\ less_eq_dom(f,R_EAL(r-1/(n+1))); A3:A /\ less_dom(f,R_EAL r) c= union rng F proof for x being set st x in A /\ less_dom(f,R_EAL r) holds x in union rng F proof let x be set; assume x in A /\ less_dom(f,R_EAL r); then A4: x in A & x in less_dom(f,R_EAL r) by XBOOLE_0:def 3; ex Y being set st x in Y & Y in rng F proof reconsider x as Element of X by A4; A5: x in dom f by A4,Def12; consider y being R_eal such that A6: y=f.x & y <' R_EAL r by A4,Def12; ex m st y <=' R_EAL(r-1/(m+1)) proof per cases; suppose A7: y = -infty; take 1; thus thesis by A7,SUPINF_1:21; suppose not y=-infty; then not y <=' -infty by SUPINF_1:23; then reconsider y1=y as Real by A6,MEASURE6:15; A8: R_EAL y1 = y by MEASURE6:def 1; then y1 < r by A6,MEASURE6:14; then consider m such that A9: 1/(m+1) < r-y1 by Th13; y1+1/(m+1) < r by A9,REAL_1:86; then y1 < r-1/(m+1) by REAL_1:86; then y <=' R_EAL(r-1/(m+1)) by A8,MEASURE6:13; hence thesis; end; then consider m such that A10: y <=' R_EAL(r-1/(m+1)); x in less_eq_dom(f,R_EAL(r-1/(m+1))) by A5,A6,A10,Def13; then A11: x in A /\ less_eq_dom(f,R_EAL(r-1/(m+1))) by A4,XBOOLE_0:def 3; m in NAT; then A12: m in dom F by FUNCT_2:def 1; take F.m; thus thesis by A2,A11,A12,FUNCT_1:def 5; end; hence thesis by TARSKI:def 4; end; hence thesis by TARSKI:def 3; end; union rng F c= A /\ less_dom(f,R_EAL r) proof for x being set st x in union rng F holds x in A /\ less_dom(f,R_EAL r) proof let x be set; assume x in union rng F; then consider Y being set such that A13: x in Y & Y in rng F by TARSKI:def 4; consider m such that A14: m in dom F & F.m = Y by A13,PARTFUN1:26; x in A /\ less_eq_dom(f,R_EAL(r-1/(m+1))) by A2,A13,A14; then A15: x in A & x in less_eq_dom(f,R_EAL(r-1/(m+1))) by XBOOLE_0:def 3; then A16: x in dom f by Def13; consider y being R_eal such that A17: y = f.x & y <=' R_EAL(r-1/(m+1)) by A15,Def13; A18: r-1/(m+1) in REAL by XREAL_0:def 1; reconsider x as Element of X by A15; y <' R_EAL r proof now per cases; suppose A19:y=-infty; R_EAL r = r by MEASURE6:def 1; hence thesis by A1,A19,SUPINF_1:31; suppose not y=-infty; then A20: not y <=' -infty by SUPINF_1:23; R_EAL (r-1/(m+1)) = r-1/(m+1) by MEASURE6:def 1; then reconsider y1 = y as Real by A17,A18,A20,MEASURE6:18; A21: R_EAL y1 = y by MEASURE6:def 1; then A22: y1 <= r-1/(m+1) by A17,MEASURE6:13; m+1 > 0 by NAT_1:19; then 1/(m+1) > 0 by REAL_2:127; then r < r+1/(m+1) by REAL_1:69; then r-1/(m+1) < r by REAL_1:84; then y1 < r by A22,AXIOMS:22; hence thesis by A21,MEASURE6:14; end; hence thesis; end; then x in less_dom(f,R_EAL r) by A16,A17,Def12; hence thesis by A15,XBOOLE_0:def 3; end; hence thesis by TARSKI:def 3; end; hence thesis by A3,XBOOLE_0:def 10; end; theorem Th26: for X, S, F, f, A, r st for n holds F.n = A /\ great_eq_dom(f,R_EAL(r+1/(n+1))) holds A /\ great_dom(f,R_EAL r) = union rng F proof let X,S,F,f,A,r; assume A1:for n holds F.n = A /\ great_eq_dom(f,R_EAL(r+1/(n+1))); A2:A /\ great_dom(f,R_EAL r) c= union rng F proof for x being set st x in A /\ great_dom(f,R_EAL r) holds x in union rng F proof let x be set; assume x in A /\ great_dom(f,R_EAL r); then A3: x in A & x in great_dom(f,R_EAL r) by XBOOLE_0:def 3; ex Y being set st x in Y & Y in rng F proof reconsider x as Element of X by A3; A4: x in dom f by A3,Def14; consider y being R_eal such that A5: y=f.x & R_EAL r <' y by A3,Def14; ex m st R_EAL(r+1/(m+1)) <=' y proof per cases; suppose A6: y = +infty; take 1; thus thesis by A6,SUPINF_1:20; suppose not y=+infty; then not +infty <=' y by SUPINF_1:24; then reconsider y1=y as Real by A5,MEASURE6:15; A7: R_EAL y1 = y by MEASURE6:def 1; then r < y1 by A5,MEASURE6:14; then consider m such that A8: 1/(m+1) < y1-r by Th13; A9: r+1/(m+1) < y1 by A8,REAL_1:86; take m; thus thesis by A7,A9,MEASURE6:13; end; then consider m such that A10: R_EAL(r+1/(m+1)) <=' y; x in great_eq_dom(f,R_EAL(r+1/(m+1))) by A4,A5,A10,Def15; then A11: x in A /\ great_eq_dom(f,R_EAL(r+1/(m+1))) by A3,XBOOLE_0:def 3; m in NAT; then A12: m in dom F by FUNCT_2:def 1; take F.m; thus thesis by A1,A11,A12,FUNCT_1:def 5; end; hence thesis by TARSKI:def 4; end; hence thesis by TARSKI:def 3; end; union rng F c= A /\ great_dom(f,R_EAL r) proof for x being set st x in union rng F holds x in A /\ great_dom(f,R_EAL r) proof let x be set; assume x in union rng F; then consider Y being set such that A13: x in Y & Y in rng F by TARSKI:def 4; consider m such that A14: m in dom F & F.m = Y by A13,PARTFUN1:26; x in A /\ great_eq_dom(f,R_EAL(r+1/(m+1))) by A1,A13,A14; then A15: x in A & x in great_eq_dom(f,R_EAL(r+1/(m+1))) by XBOOLE_0:def 3; then A16: x in dom f by Def15; consider y being R_eal such that A17: y = f.x & R_EAL(r+1/(m+1)) <=' y by A15,Def15; reconsider x as Element of X by A15; R_EAL r <' y proof now per cases; suppose A18:y=+infty; R_EAL r = r by MEASURE6:def 1; hence thesis by A18,SUPINF_1:31; suppose not y=+infty; then A19: not +infty <=' y by SUPINF_1:24; R_EAL (r+1/(m+1)) = r+1/(m+1) by MEASURE6:def 1; then reconsider y1 = y as Real by A17,A19,MEASURE6:17; A20: R_EAL y1 = y by MEASURE6:def 1; then A21: r+1/(m+1) <= y1 by A17,MEASURE6:13; m+1 > 0 by NAT_1:19; then 1/(m+1) > 0 by REAL_2:127; then r < r+1/(m+1) by REAL_1:69; then r < y1 by A21,AXIOMS:22; hence thesis by A20,MEASURE6:14; end; hence thesis; end; then x in great_dom(f,R_EAL r) by A16,A17,Def14; hence thesis by A15,XBOOLE_0:def 3; end; hence thesis by TARSKI:def 3; end; hence thesis by A2,XBOOLE_0:def 10; end; theorem Th27: for X, S, F, f, A st for n holds F.n = A /\ great_dom(f,R_EAL n) holds A /\ eq_dom(f,+infty) = meet rng F proof let X,S,F,f,A; assume A1:for n holds F.n = A /\ great_dom(f,R_EAL n); A2:A /\ eq_dom(f,+infty) c= meet rng F proof for x being set st x in A /\ eq_dom(f,+infty) holds x in meet rng F proof let x being set; assume x in A /\ eq_dom(f,+infty); then A3: x in A & x in eq_dom(f,+infty) by XBOOLE_0:def 3; for Y being set holds Y in rng F implies x in Y proof let Y be set; Y in rng F implies x in Y proof assume Y in rng F; then consider m being Nat such that A4: m in dom F & Y = F.m by PARTFUN1:26; A5: Y = A /\ great_dom(f,R_EAL m) by A1,A4; A6: x in dom f by A3,Def16; reconsider x as Element of X by A3; consider y being R_eal such that A7: y=f.x & y = +infty by A3,Def16; R_EAL m = m by MEASURE6:def 1; then not +infty <=' R_EAL m by SUPINF_1:18; then x in great_dom(f,R_EAL m) by A6,A7,Def14; hence thesis by A3,A5,XBOOLE_0:def 3; end; hence thesis; end; hence thesis by SETFAM_1:def 1; end; hence thesis by TARSKI:def 3; end; meet rng F c= A /\ eq_dom(f,+infty) proof for x being set st x in meet rng F holds x in A /\ eq_dom(f,+infty) proof let x be set; assume A8:x in meet rng F; A9: for m holds x in A & x in dom f & ex y being R_eal st y=f.x & y = +infty proof let m; m in NAT; then m in dom F by FUNCT_2:def 1; then F.m in rng F by FUNCT_1:def 5; then x in F.m by A8,SETFAM_1:def 1; then x in A /\ great_dom(f,R_EAL m) by A1; then A10: x in A & x in great_dom(f,R_EAL m) by XBOOLE_0:def 3; then consider y being R_eal such that A11: y = f.x & R_EAL m <' y by Def14; for r holds R_EAL r <' y proof let r; consider n such that A12: r <= n by Th11; n in NAT; then n in dom F by FUNCT_2:def 1; then F.n in rng F by FUNCT_1:def 5; then x in F.n by A8,SETFAM_1:def 1; then x in A /\ great_dom(f,R_EAL n) by A1; then x in great_dom(f,R_EAL n) by XBOOLE_0:def 3; then consider y1 being R_eal such that A13: y1=f.x & R_EAL n <' y1 by Def14; R_EAL r <=' R_EAL n by A12,MEASURE6:13; hence thesis by A11,A13,SUPINF_1:29; end; then y = +infty by Th15; hence thesis by A10,A11,Def14; end; 1 in NAT; then 1 in dom F by FUNCT_2:def 1; then F.1 in rng F by FUNCT_1:def 5; then x in F.1 by A8,SETFAM_1:def 1; then x in A /\ great_dom(f,R_EAL 1) by A1; then x in great_dom(f,R_EAL 1) by XBOOLE_0:def 3; then reconsider x as Element of X; x in eq_dom(f,+infty) by A9,Def16; hence thesis by A9,XBOOLE_0:def 3; end; hence thesis by TARSKI:def 3; end; hence thesis by A2,XBOOLE_0:def 10; end; theorem Th28: for X, S, F, f, A st for n holds F.n = A /\ less_dom(f,R_EAL n) holds A /\ less_dom(f,+infty) = union rng F proof let X,S,F,f,A; assume A1:for n holds F.n = A /\ less_dom(f,R_EAL n); A2:A /\ less_dom(f,+infty) c= union rng F proof for x being set st x in A /\ less_dom(f,+infty) holds x in union rng F proof let x be set; assume x in A /\ less_dom(f,+infty); then A3: x in A & x in less_dom(f,+infty) by XBOOLE_0:def 3; then A4: x in dom f by Def12; consider y being R_eal such that A5: y = f.x & y <' +infty by A3,Def12; ex n being Nat st y <' R_EAL n proof per cases; suppose A6:y = -infty; 0 in REAL; then A7: R_EAL 0 in REAL by MEASURE6:def 1; take 0; thus thesis by A6,A7,SUPINF_1:31; suppose not y = -infty; then not y <=' -infty by SUPINF_1:23; then reconsider y1=y as Real by A5,MEASURE6:15; consider n1 being Nat such that A8: y1 <= n1 by Th11; A9: n1 < n1+1 by NAT_1:38; reconsider m=n1+1 as Nat; y1 < m by A8,A9,AXIOMS:22; then A10: R_EAL y1 <' R_EAL m by MEASURE6:14; take m; thus thesis by A10,MEASURE6:def 1; end; then consider n being Nat such that A11: y <' R_EAL n; reconsider x as Element of X by A3; x in less_dom(f,R_EAL n) by A4,A5,A11,Def12; then x in A /\ less_dom(f,R_EAL n) by A3,XBOOLE_0:def 3; then A12: x in F.n by A1; n in NAT; then n in dom F by FUNCT_2:def 1; then F.n in rng F by FUNCT_1:def 5; hence thesis by A12,TARSKI:def 4; end; hence thesis by TARSKI:def 3; end; union rng F c= A /\ less_dom(f,+infty) proof for x being set st x in union rng F holds x in A /\ less_dom(f,+infty) proof let x being set; assume x in union rng F; then consider Y being set such that A13: x in Y & Y in rng F by TARSKI:def 4; consider m such that A14: m in dom F & F.m = Y by A13,PARTFUN1:26; x in A /\ less_dom(f,R_EAL m) by A1,A13,A14; then A15: x in A & x in less_dom(f,R_EAL m) by XBOOLE_0:def 3; then A16: x in dom f by Def12; consider y being R_eal such that A17: y = f.x & y <' R_EAL m by A15,Def12; reconsider x as Element of X by A15; R_EAL m = m by MEASURE6:def 1; then R_EAL m <' +infty by SUPINF_1:31; then y <' +infty by A17,SUPINF_1:29; then x in less_dom(f,+infty) by A16,A17,Def12; hence thesis by A15,XBOOLE_0:def 3; end; hence thesis by TARSKI:def 3; end; hence thesis by A2,XBOOLE_0:def 10; end; theorem Th29: for X, S, F, f, A st for n holds F.n = A /\ less_dom(f,R_EAL (-n)) holds A /\ eq_dom(f,-infty) = meet rng F proof let X,S,F,f,A; assume A1:for n holds F.n = A /\ less_dom(f,R_EAL (-n)); A2:A /\ eq_dom(f,-infty) c= meet rng F proof for x being set st x in A /\ eq_dom(f,-infty) holds x in meet rng F proof let x being set; assume x in A /\ eq_dom(f,-infty); then A3: x in A & x in eq_dom(f,-infty) by XBOOLE_0:def 3; for Y being set holds Y in rng F implies x in Y proof let Y be set; Y in rng F implies x in Y proof assume Y in rng F; then consider m being Nat such that A4: m in dom F & Y = F.m by PARTFUN1:26; A5: Y = A /\ less_dom(f,R_EAL (-m)) by A1,A4; A6: x in dom f by A3,Def16; reconsider x as Element of X by A3; consider y being R_eal such that A7: y=f.x & y = -infty by A3,Def16; R_EAL (-m) = -m by MEASURE6:def 1; then not R_EAL (-m) <=' -infty by SUPINF_1:17; then x in less_dom(f,R_EAL (-m)) by A6,A7,Def12; hence thesis by A3,A5,XBOOLE_0:def 3; end; hence thesis; end; hence thesis by SETFAM_1:def 1; end; hence thesis by TARSKI:def 3; end; meet rng F c= A /\ eq_dom(f,-infty) proof for x being set st x in meet rng F holds x in A /\ eq_dom(f,-infty) proof let x be set; assume A8:x in meet rng F; A9: for m holds x in A & x in dom f & ex y being R_eal st y=f.x & y = -infty proof let m; m in NAT; then m in dom F by FUNCT_2:def 1; then F.m in rng F by FUNCT_1:def 5; then x in F.m by A8,SETFAM_1:def 1; then x in A /\ less_dom(f,R_EAL (-m)) by A1; then A10: x in A & x in less_dom(f,R_EAL (-m)) by XBOOLE_0:def 3; then consider y being R_eal such that A11: y = f.x & y <' R_EAL (-m) by Def12; for r holds y <' R_EAL r proof let r; consider n such that A12: -n <= r by Th12; n in NAT; then n in dom F by FUNCT_2:def 1; then F.n in rng F by FUNCT_1:def 5; then x in F.n by A8,SETFAM_1:def 1; then x in A /\ less_dom(f,R_EAL (-n)) by A1; then x in less_dom(f,R_EAL (-n)) by XBOOLE_0:def 3; then consider y1 being R_eal such that A13: y1=f.x & y1 <' R_EAL (-n) by Def12; R_EAL (-n) <=' R_EAL r by A12,MEASURE6:13; hence thesis by A11,A13,SUPINF_1:29; end; then y = -infty by Th16; hence thesis by A10,A11,Def12; end; 1 in NAT; then 1 in dom F by FUNCT_2:def 1; then F.1 in rng F by FUNCT_1:def 5; then x in F.1 by A8,SETFAM_1:def 1; then x in A /\ less_dom(f,R_EAL (-1)) by A1; then x in less_dom(f,R_EAL (-1)) by XBOOLE_0:def 3; then reconsider x as Element of X; x in eq_dom(f,-infty) by A9,Def16; hence thesis by A9,XBOOLE_0:def 3; end; hence thesis by TARSKI:def 3; end; hence thesis by A2,XBOOLE_0:def 10; end; theorem Th30: for X, S, F, f, A st for n holds F.n = A /\ great_dom(f,R_EAL (-n)) holds A /\ great_dom(f,-infty) = union rng F proof let X,S,F,f,A; assume A1:for n holds F.n = A /\ great_dom(f,R_EAL (-n)); A2:A /\ great_dom(f,-infty) c= union rng F proof for x being set st x in A /\ great_dom(f,-infty) holds x in union rng F proof let x be set; assume x in A /\ great_dom(f,-infty); then A3: x in A & x in great_dom(f,-infty) by XBOOLE_0:def 3; then A4: x in dom f by Def14; consider y being R_eal such that A5: y = f.x & -infty <' y by A3,Def14; ex n being Nat st R_EAL (-n) <' y proof per cases; suppose A6:y = +infty; 0 in REAL; then A7: R_EAL 0 in REAL by MEASURE6:def 1; take 0; thus thesis by A6,A7,SUPINF_1:31; suppose not y = +infty; then not +infty <=' y by SUPINF_1:24; then reconsider y1=y as Real by A5,MEASURE6:15; consider n1 being Nat such that A8: -n1 <= y1 by Th12; n1 < n1+1 by NAT_1:38; then A9: -(n1+1) < -n1 by REAL_1:50; reconsider m=n1+1 as Nat; -m < y1 by A8,A9,AXIOMS:22; then A10: R_EAL (-m) <' R_EAL y1 by MEASURE6:14; take m; thus thesis by A10,MEASURE6:def 1; end; then consider n being Nat such that A11: R_EAL (-n) <' y; reconsider x as Element of X by A3; x in great_dom(f,R_EAL (-n)) by A4,A5,A11,Def14; then x in A /\ great_dom(f,R_EAL (-n)) by A3,XBOOLE_0:def 3; then A12: x in F.n by A1; n in NAT; then n in dom F by FUNCT_2:def 1; then F.n in rng F by FUNCT_1:def 5; hence thesis by A12,TARSKI:def 4; end; hence thesis by TARSKI:def 3; end; union rng F c= A /\ great_dom(f,-infty) proof for x being set st x in union rng F holds x in A /\ great_dom(f,-infty) proof let x being set; assume x in union rng F; then consider Y being set such that A13: x in Y & Y in rng F by TARSKI:def 4; consider m such that A14: m in dom F & F.m = Y by A13,PARTFUN1:26; x in A /\ great_dom(f,R_EAL (-m)) by A1,A13,A14; then A15: x in A & x in great_dom(f,R_EAL (-m)) by XBOOLE_0:def 3; then A16: x in dom f by Def14; consider y being R_eal such that A17: y = f.x & R_EAL (-m) <' y by A15,Def14; reconsider x as Element of X by A15; R_EAL (-m) = -m by MEASURE6:def 1; then -infty <' R_EAL (-m) by SUPINF_1:31; then -infty <' y by A17,SUPINF_1:29; then x in great_dom(f,-infty) by A16,A17,Def14; hence thesis by A15,XBOOLE_0:def 3; end; hence thesis by TARSKI:def 3; end; hence thesis by A2,XBOOLE_0:def 10; end; begin :: Measurable functions definition let X be non empty set; let S be sigma_Field_Subset of X; let f be PartFunc of X,ExtREAL; let A be Element of S; pred f is_measurable_on A means :Def17: for r being real number holds A /\ less_dom(f,R_EAL r) is_measurable_on S; end; reserve A,B for Element of S; theorem for X,S,f,A st A c= dom f holds f is_measurable_on A iff (for r being real number holds A /\ great_eq_dom(f,R_EAL r) is_measurable_on S) proof let X,S,f,A; assume A1:A c= dom f; A2:f is_measurable_on A implies for r being real number holds A /\ great_eq_dom(f,R_EAL r) is_measurable_on S proof assume A3:f is_measurable_on A; for r being real number holds A /\ great_eq_dom(f,R_EAL r) is_measurable_on S proof let r be real number; A /\ less_dom(f,R_EAL r) is_measurable_on S by A3,Def17; then A4: A /\ less_dom(f,R_EAL r) in S by Def11; A /\ great_eq_dom(f,R_EAL r) = A\(A /\ less_dom(f,R_EAL r)) by A1,Th18; then A /\ great_eq_dom(f,R_EAL r) in S by A4,MEASURE1:47; hence thesis by Def11; end; hence thesis; end; (for r being real number holds A /\ great_eq_dom(f,R_EAL r) is_measurable_on S) implies f is_measurable_on A proof assume A5: for r being real number holds A /\ great_eq_dom(f,R_EAL r) is_measurable_on S; for r being real number holds A /\ less_dom(f,R_EAL r) is_measurable_on S proof let r be real number; A /\ great_eq_dom(f,R_EAL r) is_measurable_on S by A5; then A6: A /\ great_eq_dom(f,R_EAL r) in S by Def11; A /\ less_dom(f,R_EAL r) = A\(A /\ great_eq_dom(f,R_EAL r)) by A1,Th21; then A /\ less_dom(f,R_EAL r) in S by A6,MEASURE1:47; hence thesis by Def11; end; hence thesis by Def17; end; hence thesis by A2; end; theorem Th32: for X,S,f,A holds f is_measurable_on A iff (for r being real number holds A /\ less_eq_dom(f,R_EAL r) is_measurable_on S) proof let X,S,f,A; A1:f is_measurable_on A implies (for r being real number holds A /\ less_eq_dom(f,R_EAL r) is_measurable_on S) proof assume A2:f is_measurable_on A; for r being real number holds A /\ less_eq_dom(f,R_EAL r) is_measurable_on S proof let r be real number; A3: for n holds A /\ less_dom(f,R_EAL(r+1/(n+1))) in S proof let n; A /\ less_dom(f,R_EAL(r+1/(n+1))) is_measurable_on S by A2,Def17; hence thesis by Def11; end; defpred P[Nat,Element of S] means A /\ less_dom(f,R_EAL(r+1/($1+1))) = $2; A4: for n ex y being Element of S st P[n,y] proof let n; reconsider y=A /\ less_dom(f,R_EAL(r+1/(n+1))) as Element of S by A3; take y; thus thesis; end; consider F being Function of NAT,S such that A5: for n holds P[n,F.n] from FuncExD(A4); A /\ less_eq_dom(f,R_EAL r) = meet rng F by A5,Th24; hence thesis by Def11; end; hence thesis; end; (for r being real number holds A /\ less_eq_dom(f,R_EAL r) is_measurable_on S) implies f is_measurable_on A proof assume A6:for r being real number holds A /\ less_eq_dom(f,R_EAL r) is_measurable_on S; for r being real number holds A /\ less_dom(f,R_EAL r) is_measurable_on S proof let r be real number; A7: for n being real number holds A /\ less_eq_dom(f,R_EAL(r-1/(n+1))) in S proof let n be real number; A /\ less_eq_dom(f,R_EAL(r-1/(n+1))) is_measurable_on S by A6; hence thesis by Def11; end; defpred P[Nat,Element of S] means A /\ less_eq_dom(f,R_EAL(r-1/($1+1))) = $2; A8: for n ex y being Element of S st P[n,y] proof let n; reconsider y=A /\ less_eq_dom(f,R_EAL(r-1/(n+1))) as Element of S by A7; take y; thus thesis; end; consider F being Function of NAT,S such that A9: for n holds P[n,F.n] from FuncExD(A8); A /\ less_dom(f,R_EAL r) = union rng F by A9,Th25; hence thesis by Def11; end; hence thesis by Def17; end; hence thesis by A1; end; theorem Th33: for X,S,f,A st A c= dom f holds f is_measurable_on A iff (for r being real number holds A /\ great_dom(f,R_EAL r) is_measurable_on S) proof let X,S,f,A; assume that A1:A c= dom f; A2:f is_measurable_on A implies (for r being real number holds A /\ great_dom(f,R_EAL r) is_measurable_on S) proof assume A3:f is_measurable_on A; for r being real number holds A /\ great_dom(f,R_EAL r) is_measurable_on S proof let r be real number; A /\ less_eq_dom(f,R_EAL r) is_measurable_on S by A3,Th32; then A4: A /\ less_eq_dom(f,R_EAL r) in S by Def11; A /\ great_dom(f,R_EAL r) = A\(A /\ less_eq_dom(f,R_EAL r)) by A1,Th19; then A /\ great_dom(f,R_EAL r) in S by A4,MEASURE1:47; hence thesis by Def11; end; hence thesis; end; (for r being real number holds A /\ great_dom(f,R_EAL r) is_measurable_on S ) implies f is_measurable_on A proof assume A5: for r being real number holds A /\ great_dom(f,R_EAL r) is_measurable_on S; for r being real number holds A /\ less_eq_dom(f,R_EAL r) is_measurable_on S proof let r be real number; A /\ great_dom(f,R_EAL r) is_measurable_on S by A5; then A6: A /\ great_dom(f,R_EAL r) in S by Def11; A /\ less_eq_dom(f,R_EAL r) = A\(A /\ great_dom(f,R_EAL r)) by A1,Th20; then A /\ less_eq_dom(f,R_EAL r) in S by A6,MEASURE1:47; hence thesis by Def11; end; hence thesis by Th32; end; hence thesis by A2; end; theorem for X,S,f,A,B st B c= A & f is_measurable_on A holds f is_measurable_on B proof let X,S,f,A,B; assume that A1:B c= A and A2:f is_measurable_on A; for r being real number holds (B /\ less_dom(f,R_EAL r)) is_measurable_on S proof let r be real number; A /\ less_dom(f,R_EAL r) is_measurable_on S by A2,Def17; then A /\ less_dom(f,R_EAL r) in S by Def11; then A3: B /\ (A /\ less_dom(f,R_EAL r)) in S by MEASURE1:46; B /\ (A /\ less_dom(f,R_EAL r)) = B /\ A /\ less_dom(f,R_EAL r) by XBOOLE_1:16 .= B /\ less_dom(f,R_EAL r) by A1,XBOOLE_1:28; hence thesis by A3,Def11; end; hence thesis by Def17; end; theorem for X,S,f,A,B st f is_measurable_on A & f is_measurable_on B holds f is_measurable_on (A \/ B) proof let X,S,f,A,B; assume that A1:f is_measurable_on A and A2:f is_measurable_on B; for r being real number holds ((A \/ B) /\ less_dom(f,R_EAL r)) is_measurable_on S proof let r be real number; A /\ less_dom(f,R_EAL r) is_measurable_on S by A1,Def17; then A3: A /\ less_dom(f,R_EAL r) in S by Def11; B /\ less_dom(f,R_EAL r) is_measurable_on S by A2,Def17; then B /\ less_dom(f,R_EAL r) in S by Def11; then (A /\ less_dom(f,R_EAL r)) \/ (B /\ less_dom(f,R_EAL r)) in S by A3,MEASURE1:46; then ((A \/ B) /\ less_dom(f,R_EAL r)) in S by XBOOLE_1:23; hence thesis by Def11; end; hence thesis by Def17; end; theorem for X,S,f,A,r,s st f is_measurable_on A & A c= dom f holds (A /\ great_dom(f,R_EAL r) /\ less_dom(f,R_EAL s)) is_measurable_on S proof let X,S,f,A,r,s; assume that A1:f is_measurable_on A and A2:A c= dom f; A /\ less_dom(f,R_EAL s) is_measurable_on S by A1,Def17; then A3:A /\ less_dom(f,R_EAL s) in S by Def11; A4:for r1 being Real holds A /\ great_eq_dom(f,R_EAL r1) is_measurable_on S proof let r1 be Real; A /\ less_dom(f,R_EAL r1) is_measurable_on S by A1,Def17; then A5: A /\ less_dom(f,R_EAL r1) in S by Def11; A /\ great_eq_dom(f,R_EAL r1) = A\(A /\ less_dom(f,R_EAL r1)) by A2,Th18 ; then A /\ great_eq_dom(f,R_EAL r1) in S by A5,MEASURE1:47; hence thesis by Def11; end; for r1 being Real holds A /\ great_dom(f,R_EAL r1) is_measurable_on S proof let r1 being Real; A6: for n holds A /\ great_eq_dom(f,R_EAL(r1+1/(n+1))) in S proof let n; A /\ great_eq_dom(f,R_EAL(r1+1/(n+1))) is_measurable_on S by A4; hence thesis by Def11; end; defpred P[Nat,Element of S] means A /\ great_eq_dom(f,R_EAL(r1+1/($1+1))) = $2; A7: for n ex y being Element of S st P[n,y] proof let n; reconsider y=A /\ great_eq_dom(f,R_EAL(r1+1/(n+1))) as Element of S by A6; take y; thus thesis; end; consider F being Function of NAT,S such that A8: for n holds P[n,F.n] from FuncExD(A7); A /\ great_dom(f,R_EAL r1) = union rng F by A8,Th26; hence thesis by Def11; end; then A /\ great_dom(f,R_EAL r) is_measurable_on S; then A /\ great_dom(f,R_EAL r) in S by Def11; then A9:(A /\ great_dom(f,R_EAL r))/\(A /\ less_dom(f,R_EAL s)) in S by A3,MEASURE1:46; (A /\ great_dom(f,R_EAL r))/\(A /\ less_dom(f,R_EAL s)) = (A /\ great_dom(f,R_EAL r) /\ A) /\ less_dom(f,R_EAL s) by XBOOLE_1:16 .= (great_dom(f,R_EAL r) /\ (A /\ A)) /\ less_dom(f,R_EAL s) by XBOOLE_1:16; hence thesis by A9,Def11; end; theorem for X,S,f,A st f is_measurable_on A & A c= dom f holds A /\ eq_dom(f,+infty) is_measurable_on S proof let X,S,f,A; assume that A1:f is_measurable_on A and A2:A c= dom f; A3:for n holds A /\ great_dom(f,R_EAL n) in S proof let n; A /\ great_dom(f,R_EAL n) is_measurable_on S by A1,A2,Th33; hence thesis by Def11; end; defpred P[Nat,Element of S]means A /\ great_dom(f,R_EAL $1) = $2; A4:for n ex y being Element of S st P[n,y] proof let n; reconsider y=A /\ great_dom(f,R_EAL n) as Element of S by A3; take y; thus thesis; end; consider F being Function of NAT,S such that A5:for n holds P[n,F.n] from FuncExD(A4); A /\ eq_dom(f,+infty) = meet rng F by A5,Th27; hence thesis by Def11; end; theorem for X,S,f,A st f is_measurable_on A holds A /\ eq_dom(f,-infty) is_measurable_on S proof let X,S,f,A; assume A1:f is_measurable_on A; A2:for n holds A /\ less_dom(f,R_EAL (-n)) in S proof let n; A /\ less_dom(f,R_EAL (-n)) is_measurable_on S by A1,Def17; hence thesis by Def11; end; defpred P[Nat,Element of S] means A /\ less_dom(f,R_EAL (-$1)) = $2; A3:for n ex y being Element of S st P[n,y] proof let n; reconsider y=A /\ less_dom(f,R_EAL (-n)) as Element of S by A2; take y; thus thesis; end; consider F being Function of NAT,S such that A4:for n holds P[n,F.n] from FuncExD(A3); A /\ eq_dom(f,-infty) = meet rng F by A4,Th29; hence thesis by Def11; end; theorem for X,S,f,A st f is_measurable_on A & A c= dom f holds A /\ great_dom(f,-infty) /\ less_dom(f,+infty) is_measurable_on S proof let X,S,f,A; assume that A1:f is_measurable_on A and A2:A c= dom f; A3:A /\ great_dom(f,-infty) in S proof A4: for n holds A /\ great_dom(f,R_EAL (-n)) in S proof let n; A /\ great_dom(f,R_EAL (-n)) is_measurable_on S by A1,A2,Th33; hence thesis by Def11; end; defpred P[Nat,Element of S] means A /\ great_dom(f,R_EAL (-$1)) = $2; A5: for n ex y being Element of S st P[n,y] proof let n; reconsider y=A /\ great_dom(f,R_EAL (-n)) as Element of S by A4; take y; thus thesis; end; consider F being Function of NAT,S such that A6: for n holds P[n,F.n] from FuncExD(A5); A /\ great_dom(f,-infty) = union rng F by A6,Th30; hence thesis; end; A /\ less_dom(f,+infty) in S proof A7: for n holds A /\ less_dom(f,R_EAL n) in S proof let n; A /\ less_dom(f,R_EAL n) is_measurable_on S by A1,Def17; hence thesis by Def11; end; defpred P[Nat,Element of S] means A /\ less_dom(f,R_EAL $1) = $2; A8: for n ex y being Element of S st P[n,y] proof let n; reconsider y=A /\ less_dom(f,R_EAL n) as Element of S by A7; take y; thus thesis; end; consider F being Function of NAT,S such that A9:for n holds P[n,F.n] from FuncExD(A8); A /\ less_dom(f,+infty) = union rng F by A9,Th28; hence thesis; end; then A10:(A /\ great_dom(f,-infty)) /\ (A /\ less_dom(f,+infty)) in S by A3,MEASURE1:46; (A /\ great_dom(f,-infty)) /\ (A /\ less_dom(f,+infty)) = (A /\ great_dom(f,-infty) /\ A) /\ less_dom(f,+infty) by XBOOLE_1:16 .= (great_dom(f,-infty) /\ (A /\ A)) /\ less_dom(f,+infty) by XBOOLE_1:16 ; hence thesis by A10,Def11; end; theorem for X,S,f,g,A,r st f is_measurable_on A & g is_measurable_on A & A c= dom g holds A /\ less_dom(f,R_EAL r) /\ great_dom(g,R_EAL r) is_measurable_on S proof let X,S,f,g,A,r; assume A1:f is_measurable_on A & g is_measurable_on A & A c= dom g; then A /\ less_dom(f,R_EAL r) is_measurable_on S by Def17; then A2:A /\ less_dom(f,R_EAL r) in S by Def11; A /\ great_dom(g,R_EAL r) is_measurable_on S by A1,Th33; then A /\ great_dom(g,R_EAL r) in S by Def11; then A3:(A /\ less_dom(f,R_EAL r))/\(A /\ great_dom(g,R_EAL r)) in S by A2,MEASURE1:46; (A /\ less_dom(f,R_EAL r)) /\ (A /\ great_dom(g,R_EAL r)) =((A /\ less_dom(f,R_EAL r)) /\ A) /\ great_dom(g,R_EAL r) by XBOOLE_1:16 .=((A /\ A) /\ less_dom(f,R_EAL r)) /\ great_dom(g,R_EAL r) by XBOOLE_1:16 .=A /\ less_dom(f,R_EAL r) /\ great_dom(g,R_EAL r); hence thesis by A3,Def11; end; theorem for X,S,f,A,r st f is_measurable_on A & A c= dom f holds r(#)f is_measurable_on A proof let X,S,f,A,r; assume A1:f is_measurable_on A & A c= dom f; for r1 being real number holds A /\ less_dom(r(#)f,R_EAL r1) is_measurable_on S proof let r1 being real number; now per cases; suppose A2:r<>0; then A3:R_EAL r1=r1 & R_EAL r=r & R_EAL r<>0. by MEASURE6:def 1,SUPINF_2:def 1; A4: r1 in REAL by XREAL_0:def 1; reconsider r0=r1/r as Real by XREAL_0:def 1; A5:r1=r*r0 by A2,XCMPLX_1:88; A6:R_EAL r0 = r0 by MEASURE6:def 1; A7:R_EAL r1 <>+infty & R_EAL r1 <>-infty & R_EAL r <>+infty & R_EAL r <> -infty by A3,A4,SUPINF_1:31; now per cases by A2; suppose r > 0; then A8: 0. <' R_EAL r by A3,EXTREAL1:46,SUPINF_2:def 1; less_dom(f,R_EAL r0) = less_dom(r(#)f,R_EAL r1) proof for x being Element of X st x in less_dom(f,R_EAL r0) holds x in less_dom(r(#)f,R_EAL r1) proof let x be Element of X; assume A9:x in less_dom(f,R_EAL r0); then x in dom f & ex y being R_eal st y = f.x & y <' R_EAL r0 by Def12; then A10: x in dom(r(#)f) by Def6; consider y being R_eal such that A11: y = f.x & y <' R_EAL r0 by A9,Def12; A12: r1 in REAL by XREAL_0:def 1; then y <' R_EAL r1 / R_EAL r by A2,A11,EXTREAL1:39; then A13: y * R_EAL r <' R_EAL r1 / R_EAL r * R_EAL r by A7,A8,EXTREAL1:44; A14: r1/r in REAL by XREAL_0:def 1; R_EAL r1 / R_EAL r = r1 / r by A3,A12,EXTREAL1:32; then R_EAL r1 / R_EAL r * R_EAL r = (r1/r)*r by A3,A14,EXTREAL1:13 .= r1/(r/r) by XCMPLX_1:82 .= r1/1 by A2,XCMPLX_1:60 .= r1; then A15: R_EAL r * y <' R_EAL r1 by A13,MEASURE6:def 1; (r(#)f).x = R_EAL r * y by A10,A11,Def6; hence thesis by A10,A15,Def12; end; then A16: less_dom(f,R_EAL r0) c= less_dom(r(#)f,R_EAL r1) by SUBSET_1:7; for x being Element of X st x in less_dom(r(#)f,R_EAL r1) holds x in less_dom(f,R_EAL r0) proof let x being Element of X; assume A17:x in less_dom(r(#)f,R_EAL r1); then A18: x in dom(r(#)f) & ex y being R_eal st y=(r(#)f).x & y <' R_EAL r1 by Def12 ; then A19: x in dom f by Def6; consider y being R_eal such that A20: y = (r(#)f).x & y <' R_EAL r1 by A17,Def12; y <' R_EAL r * R_EAL r0 by A5,A20,EXTREAL1:38; then A21: y / R_EAL r <' R_EAL r * R_EAL r0 / R_EAL r by A7,A8,EXTREAL1:51; R_EAL r * R_EAL r0 = r * r0 by A3,A6,EXTREAL1:13; then R_EAL r * R_EAL r0 / R_EAL r = (r*r0)/r by A3,EXTREAL1:32 .= r0/(r/r) by XCMPLX_1:78 .= r0/1 by A2,XCMPLX_1:60 .= r0; then A22: y / R_EAL r <' R_EAL r0 by A21,MEASURE6:def 1; f.x = y / R_EAL r by A2,A18,A20,Th6; hence thesis by A19,A22,Def12; end; then less_dom(r(#)f,R_EAL r1) c= less_dom(f,R_EAL r0) by SUBSET_1:7; hence thesis by A16,XBOOLE_0:def 10; end; hence thesis by A1,Def17; suppose r < 0; then A23: R_EAL r <' 0. by A3,EXTREAL1:46,SUPINF_2:def 1; great_dom(f,R_EAL r0) = less_dom(r(#)f,R_EAL r1) proof for x being Element of X st x in great_dom(f,R_EAL r0) holds x in less_dom(r(#)f,R_EAL r1) proof let x be Element of X; assume A24:x in great_dom(f,R_EAL r0); then x in dom f & ex y being R_eal st y = f.x & R_EAL r0 <' y by Def14; then A25: x in dom(r(#)f) by Def6; consider y being R_eal such that A26: y = f.x & R_EAL r0 <' y by A24,Def14; R_EAL r1 / R_EAL r <' y by A2,A4,A26,EXTREAL1:39; then A27: y * R_EAL r <' R_EAL r1 / R_EAL r * R_EAL r by A7,A23,EXTREAL1:45; A28: r1/r in REAL by XREAL_0:def 1; R_EAL r1 / R_EAL r = r1 / r by A3,A4,EXTREAL1:32; then R_EAL r1 / R_EAL r * R_EAL r = (r1/r)*r by A3,A28,EXTREAL1:13 .= r1/(r/r) by XCMPLX_1:82 .= r1/1 by A2,XCMPLX_1:60 .= r1; then A29: R_EAL r * y <' R_EAL r1 by A27,MEASURE6:def 1; (r(#)f).x = R_EAL r * y by A25,A26,Def6; hence thesis by A25,A29,Def12; end; then A30: great_dom(f,R_EAL r0) c= less_dom(r(#)f,R_EAL r1) by SUBSET_1:7; for x being Element of X st x in less_dom(r(#)f,R_EAL r1) holds x in great_dom(f,R_EAL r0) proof let x being Element of X; assume A31:x in less_dom(r(#)f,R_EAL r1); then A32: x in dom(r(#)f) & ex y being R_eal st y=(r(#) f).x & y <' R_EAL r1 by Def12 ; then A33: x in dom f by Def6; consider y being R_eal such that A34: y = (r(#)f).x & y <' R_EAL r1 by A31,Def12; y <' R_EAL r * R_EAL r0 by A5,A34,EXTREAL1:38; then A35: R_EAL r * R_EAL r0 / R_EAL r <' y / R_EAL r by A7,A23,EXTREAL1:52; R_EAL r * R_EAL r0 = r * r0 by A3,A6,EXTREAL1:13; then R_EAL r * R_EAL r0 / R_EAL r = (r*r0)/r by A3,EXTREAL1:32 .= r0/(r/r) by XCMPLX_1:78 .= r0/1 by A2,XCMPLX_1:60 .= r0; then A36: R_EAL r0 <' y / R_EAL r by A35,MEASURE6:def 1; f.x = y / R_EAL r by A2,A32,A34,Th6; hence thesis by A33,A36,Def14; end; then less_dom(r(#)f,R_EAL r1) c= great_dom(f,R_EAL r0) by SUBSET_1:7; hence thesis by A30,XBOOLE_0:def 10; end; hence thesis by A1,Th33; end; hence thesis; suppose A37:r=0; now per cases; suppose A38:r1>0; A /\ less_dom(r(#)f,R_EAL r1) = A proof A39: A /\ less_dom(r(#)f,R_EAL r1) c= A by XBOOLE_1:17; A c= A /\ less_dom(r(#)f,R_EAL r1) proof for x1 being set holds x1 in A implies x1 in A /\ less_dom(r(#)f,R_EAL r1) proof let x1 being set; assume A40:x1 in A; then reconsider x1 as Element of X; x1 in dom f by A1,A40; then A41: x1 in dom (r(#)f) by Def6; reconsider y = (r(#)f).x1 as R_eal; A42: R_EAL r = 0. by A37,MEASURE6:def 1,SUPINF_2:def 1; A43: y = (R_EAL r) * f.x1 by A41,Def6 .= 0. by A42,EXTREAL1:def 1; 0. <' R_EAL r1 by A37,A38,A42,MEASURE6:14; then x1 in less_dom(r(#)f,R_EAL r1) by A41,A43,Def12; hence thesis by A40,XBOOLE_0:def 3; end; hence thesis by TARSKI:def 3; end; hence thesis by A39,XBOOLE_0:def 10; end; hence thesis by Def11; suppose A44:r1<=0; less_dom(r(#)f,R_EAL r1) = {} proof assume less_dom(r(#)f,R_EAL r1) <> {}; then consider x1 being Element of X such that A45: x1 in less_dom(r(#)f,R_EAL r1) by SUBSET_1:10; A46: x1 in dom (r(#)f) & ex y being R_eal st y=(r(#)f).x1 & y <' R_EAL r1 by A45,Def12; consider y1 being R_eal such that A47: y1 = (r(#)f).x1 & y1 <' R_EAL r1 by A45,Def12; A48: y1 in rng(r(#)f) by A46,A47,FUNCT_1:def 5; for y being R_eal st y in rng(r(#)f) holds not y <' R_EAL r1 proof let y being R_eal; assume y in rng(r(#)f); then consider x such that A49: x in dom(r(#)f) & y = (r(#)f).x by PARTFUN1:26; A50: R_EAL r = 0. by A37,MEASURE6:def 1,SUPINF_2:def 1; y = (R_EAL r) * f.x by A49,Def6 .= 0. by A50,EXTREAL1:def 1; hence thesis by A37,A44,A50,MEASURE6:13; end; hence contradiction by A47,A48; end; then A /\ less_dom(r(#)f,R_EAL r1) in S by MEASURE1:45; hence thesis by Def11; end; hence thesis; end; hence thesis; end; hence thesis by Def17; end;