### The Mizar article:

### Darboux's Theorem

**by****Noboru Endou,****Katsumi Wasaki, and****Yasunari Shidama**

- Received December 7, 1999
Copyright (c) 1999 Association of Mizar Users

- MML identifier: INTEGRA3
- [ MML identifier index ]

environ vocabulary FINSEQ_1, ARYTM_1, INTEGRA1, MEASURE5, RELAT_1, ORDINAL2, BOOLE, FUNCT_1, CARD_1, FUNCT_3, SQUARE_1, ABSVALUE, RFUNCT_1, RLVECT_1, SEQ_2, LATTICES, JORDAN3, RCOMP_1, PARTFUN1, RFINSEQ, INTEGRA2, FDIFF_1, SEQ_1, ARYTM_3, PROB_1, FINSEQ_4, ARYTM; notation TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, REAL_1, NAT_1, RELAT_1, FUNCT_1, FUNCT_2, RELSET_1, SEQ_4, PARTFUN1, PSCOMP_1, FINSEQ_1, RFUNCT_1, RVSUM_1, INTEGRA1, SEQ_1, SEQ_2, JORDAN3, PRE_CIRC, SFMASTR3, RCOMP_1, FDIFF_1, ABSVALUE, GOBOARD1, CARD_1, SQUARE_1, FINSEQ_4, TOPREAL1, RFINSEQ, BINARITH, INTEGRA2; constructors REAL_1, REALSET1, PARTFUN1, RFUNCT_1, PRE_CIRC, SFMASTR3, FDIFF_1, SQUARE_1, FINSEQ_4, RFINSEQ, TOPREAL1, BINARITH, JORDAN3, INTEGRA2, ABSVALUE, PSCOMP_1, FINSOP_1; clusters XREAL_0, RELSET_1, FINSEQ_1, NAT_2, GOBOARD1, INTEGRA1, INTEGRA2, NAT_1, MEMBERED, NUMBERS, ORDINAL2; requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM; definitions TARSKI; theorems AXIOMS, REAL_1, SEQ_4, SUBSET_1, REAL_2, PARTFUN1, PSCOMP_1, INTEGRA1, RFUNCT_1, FUNCT_1, FINSEQ_1, RVSUM_1, SEQ_1, SEQ_2, FDIFF_1, ABSVALUE, PRE_CIRC, NAT_1, GOBOARD2, RFINSEQ, TARSKI, GOBOARD1, SQUARE_1, CARD_1, FINSEQ_5, TOPREAL1, FINSEQ_3, FINSEQ_4, JORDAN7, JORDAN3, SFMASTR3, RLVECT_1, BINARITH, INTEGRA2, RELAT_1, XREAL_0, SCMFSA_7, FUNCT_2, XBOOLE_0, XBOOLE_1, RCOMP_1, XCMPLX_1; schemes FINSEQ_2, NAT_1, BINARITH; begin :: Lemmas of Division reserve a,b,d,e,x,y for Real, i,j,k,n,m for Nat, x1 for set, p,q for FinSequence of REAL; Lm1: now let k; k <= k+1 by NAT_1:29; hence k-1 <= k by REAL_1:86; end; Lm2: 2-1=1; Lm3:j-'j+1 = 1 proof j-'j=j-j by SCMFSA_7:3 .= 0 by XCMPLX_1:14; hence thesis; end; Lm4:for n st 1 <= n & n <= 2 holds n = 1 or n = 2 proof let n; assume A1:1<=n & n<=2; per cases by A1,REAL_1:def 5; suppose n=1; hence thesis; suppose n>1; then n >= 1+1 by NAT_1:38; hence thesis by A1,AXIOMS:21; end; theorem Th1:for A be closed-interval Subset of REAL, D be Element of divs A st vol(A) <> 0 holds ex i st i in dom D & vol(divset(D,i)) > 0 proof let A be closed-interval Subset of REAL; let D be Element of divs A; assume A1:vol(A) <>0; assume A2:for i st i in dom D holds vol(divset(D,i))<=0; A3:i in dom D implies vol(divset(D,i))=0 proof assume i in dom D; then vol(divset(D,i)) <= 0 by A2; hence thesis by INTEGRA1:11; end; A4:i in dom D implies sup divset(D,i)=inf divset(D,i) proof assume i in dom D; then vol(divset(D,i))=0 by A3; then sup divset(D,i)-inf divset(D,i)=0 by INTEGRA1:def 6; hence thesis by XCMPLX_1:15; end; A5:len D in dom D by SCMFSA_7:12; len D = 1 proof assume A6:len D <> 1; then A7:inf divset(D,len D)=D.(len D-1) & sup divset(D,len D)=D.(len D) by A5,INTEGRA1:def 5; A8: len D-1 in dom D & len D-1 in NAT by A5,A6,INTEGRA1:9; len D < len D+1 by NAT_1:38; then len D-1 < len D by REAL_1:84; then inf divset(D,len D)<sup divset(D,len D) by A5,A7,A8,GOBOARD1:def 1; hence contradiction by A4,A5; end; then inf divset(D,len D)=inf A & sup divset(D,len D)=D.(len D) by A5,INTEGRA1:def 5; then inf divset(D,len D)=inf A & sup divset(D,len D)=sup A by INTEGRA1:def 2; then sup A = inf A+0 by A4,A5; then sup A- inf A = 0 by XCMPLX_1:26; hence contradiction by A1,INTEGRA1:def 6; end; theorem Th2: for A be closed-interval Subset of REAL, D be Element of divs A st x in A holds ex j st j in dom D & x in divset(D,j) proof let A be closed-interval Subset of REAL; let D be Element of divs A; assume x in A; then A1:inf A <= x & x <= sup A by INTEGRA2:1; rng D <> {}; then A2:1 in dom D by FINSEQ_3:34; per cases; suppose x in rng D; then consider j such that A3: j in dom D & D.j = x by PARTFUN1:26; x in divset(D,j) proof per cases; suppose j=1; then A4: inf divset(D,j)=inf A & sup divset(D,j)=D.j by A3,INTEGRA1:def 5; consider a,b such that A5: a <= b & a=inf divset(D,j) & b=sup divset(D,j) by INTEGRA1:4; thus thesis by A3,A4,A5,INTEGRA2:1; suppose j<>1; then A6: sup divset(D,j)=D.j by A3,INTEGRA1:def 5; consider a,b such that A7: a <= b & a=inf divset(D,j) & b=sup divset(D,j) by INTEGRA1:4; thus thesis by A3,A6,A7,INTEGRA2:1; end; hence thesis by A3; suppose A8:not x in rng D; defpred MIN[Nat] means x < sup divset(D,$1) & $1 in dom D; defpred MAX[Nat] means x >= inf divset(D,$1) & $1 in dom D; A9:len D in dom D by SCMFSA_7:12; sup divset(D,len D)=D.(len D) proof per cases; suppose len D=1; hence thesis by A9,INTEGRA1:def 5; suppose len D<>1; hence thesis by A9,INTEGRA1:def 5; end; then A10: sup divset(D,len D) = sup A by INTEGRA1:def 2; x < sup divset(D,len D) proof x <> sup A proof assume x = sup A; then x = D.(len D) by INTEGRA1:def 2; hence contradiction by A8,A9,FUNCT_1:def 5; end; hence thesis by A1,A10,REAL_1:def 5; end; then A11: ex k st MIN[k] by A9; consider k such that A12:MIN[k] & for n st MIN[n] holds k <= n from Min(A11); inf divset(D,1)=inf A by A2,INTEGRA1:def 5; then A13: ex k st MAX[k] by A1,A2; A14: for k holds MAX[k] implies k <= len D by FINSEQ_3:27; consider j such that A15:MAX[j] & for n st MAX[n] holds n <= j from Max(A14,A13); k=j proof assume A16:k<>j; now per cases by A16,AXIOMS:21; suppose A17:k < j; A18: 1 <= k & k <= len D by A12,FINSEQ_3:27; then A19: j-1 in dom D & j-1 in NAT by A15,A17,INTEGRA1:9; A20: sup divset(D,k)=D.k proof per cases; suppose k=1; hence thesis by A12,INTEGRA1:def 5; suppose k<>1; hence thesis by A12,INTEGRA1:def 5; end; D.(j-1) <= x by A15,A17,A18,INTEGRA1:def 5; then D.(j-1) < D.k by A12,A20,AXIOMS:22; then j-1 < k by A12,A19,GOBOARD2:18; then j < k+1 by REAL_1:84; hence contradiction by A17,NAT_1:38; suppose A21:k > j; x < sup divset(D,j) proof assume A22:x >= sup divset(D,j); A23: sup divset(D,j)=D.j proof per cases; suppose j=1; hence thesis by A15,INTEGRA1:def 5; suppose j<>1; hence thesis by A15,INTEGRA1:def 5; end; A24: j+1 in dom D & j+1 > 1 proof A25: 1 <= j & k <= len D by A12,A15,FINSEQ_3:27; j < j+1 & j+1 <= k by A21,NAT_1:38; then 1 <= j+1 & j+1 <= len D by A25,AXIOMS:22; hence j+1 in dom D by FINSEQ_3:27; thus thesis by A25,NAT_1:38; end; then inf divset(D,j+1) = D.(j+1-1) by INTEGRA1:def 5 .=D.j by XCMPLX_1:26; then j+1 <= j by A15,A22,A23,A24; hence contradiction by NAT_1:38; end; hence contradiction by A12,A15,A21; end; hence contradiction; end; then x in divset(D,k) & k in dom D by A12,A15,INTEGRA2:1; hence thesis; end; theorem Th3: for A be closed-interval Subset of REAL, D1,D2 be Element of divs A holds ex D be Element of divs A st D1 <= D & D2 <= D & rng D = rng D1 \/ rng D2 proof let A be closed-interval Subset of REAL; let D1,D2 be Element of divs A; consider D being FinSequence of REAL such that A1:rng D = rng(D1^D2) & len D = card rng(D1^D2) & D is increasing by GOBOARD2:21; reconsider D as increasing FinSequence of REAL by A1; D1^D2 <> {} by FINSEQ_1:48; then rng D <> {} by A1,FINSEQ_1:27; then reconsider D as non empty increasing FinSequence of REAL by FINSEQ_1:27 ; A2:rng(D1^D2) = rng D1 \/ rng D2 by FINSEQ_1:44; rng D1 c= A & rng D2 c= A by INTEGRA1:def 2; then A3:rng D c= A by A1,A2,XBOOLE_1:8; A4:rng D1 c= rng(D1^D2) by A2,XBOOLE_1:7; A5:rng D2 c= rng(D1^D2) by A2,XBOOLE_1:7; D.(len D) = sup A proof assume A6:D.(len D) <> sup A; A7: len D in dom D by SCMFSA_7:12; then D.(len D) in rng D by FUNCT_1:def 5; then D.(len D) <= sup A by A3,INTEGRA2:1; then A8: D.(len D) < sup A by A6,REAL_1:def 5; A9: D1.(len D1) = sup A by INTEGRA1:def 2; len D1 in dom D1 by SCMFSA_7:12; then D1.(len D1) in rng D1 by FUNCT_1:def 5; then consider k such that A10: k in dom D & D1.(len D1)=D.k by A1,A4,PARTFUN1:26; k > len D by A7,A8,A9,A10,GOBOARD2:18; hence contradiction by A10,FINSEQ_3:27; end; then D is DivisionPoint of A by A3,INTEGRA1:def 2; then reconsider D as Element of divs A by INTEGRA1:def 3; take D; A11:card rng D1 <= len D by A1,A4,CARD_1:80; D1 is one-to-one by JORDAN7:17; then len D1 <= len D by A11,FINSEQ_4:77; hence D1 <= D by A1,A4,INTEGRA1:def 20; A12:card rng D2 <= len D by A1,A5,CARD_1:80; D2 is one-to-one by JORDAN7:17; then len D2 <= len D by A12,FINSEQ_4:77; hence D2 <= D by A1,A5,INTEGRA1:def 20; thus thesis by A1,FINSEQ_1:44; end; theorem Th4:for A be closed-interval Subset of REAL, D,D1 be Element of divs A st delta(D1)<min rng upper_volume(chi(A,A),D) holds (for x,y,i st i in dom D1 & x in rng D /\ divset(D1,i) & y in rng D /\ divset(D1,i) holds x=y) proof let A be closed-interval Subset of REAL; let D,D1 be Element of divs A; assume A1:delta(D1)<min rng upper_volume(chi(A,A),D); let x,y,i; assume A2:i in dom D1; assume A3:x in rng D /\ divset(D1,i); assume A4:y in rng D /\ divset(D1,i); assume A5:x<>y; x in rng D by A3,XBOOLE_0:def 3; then consider n such that A6:n in dom D & x=D.n by PARTFUN1:26; y in rng D by A4,XBOOLE_0:def 3; then consider m such that A7:m in dom D & y=D.m by PARTFUN1:26; A8:abs( D.n-D.m ) >= min rng upper_volume(chi(A,A),D) proof per cases by A5,A6,A7,AXIOMS:21; suppose n<m; then A9: n+1<=m by NAT_1:38; n in Seg len D & m in Seg len D by A6,A7,FINSEQ_1:def 3; then A10: 1<=n & m<=len D by FINSEQ_1:3; then 1<=n+1 & n+1<=len D by A9,AXIOMS:22,NAT_1:37; then A11: n+1 in Seg len D by FINSEQ_1:3; then A12: n+1 in dom D by FINSEQ_1:def 3; -abs( D.n-D.m ) <= D.n-D.m by ABSVALUE:11; then A13: abs( D.n-D.m ) >= -(D.n-D.m) by REAL_2:110; D.m>=D.(n+1) by A7,A9,A12,GOBOARD2:18; then D.n-D.m <= D.n-D.(n+1) by REAL_2:106; then -(D.n-D.m) >= -(D.n-D.(n+1)) by REAL_1:50; then A14: -(D.n-D.m) >= D.(n+1)-D.n by XCMPLX_1:143; A15: D.(n+1)-D.n=(upper_volume(chi(A,A),D)).(n+1) proof n+1 <> 1 by A10,NAT_1:38; then A16: inf divset(D,n+1)=D.((n+1)-1) & sup divset(D,n+1)=D.(n+1) by A12,INTEGRA1:def 5; (n+1)-1=n by XCMPLX_1:26; then vol(divset(D,n+1))=D.(n+1)-D.n by A16,INTEGRA1:def 6; hence thesis by A11,INTEGRA1:22; end; n+1 in Seg len upper_volume(chi(A,A),D) by A11,INTEGRA1:def 7; then n+1 in dom upper_volume(chi(A,A),D) by FINSEQ_1:def 3; then (upper_volume(chi(A,A),D)).(n+1) in rng upper_volume(chi(A,A),D) by FUNCT_1:def 5; then D.(n+1)-D.n>=min rng upper_volume(chi(A,A),D) by A15,SFMASTR3:def 1 ; then -(D.n-D.m) >= min rng upper_volume(chi(A,A),D) by A14,AXIOMS:22; hence thesis by A13,AXIOMS:22; suppose n>m; then A17: m+1<=n by NAT_1:38; n in Seg len D & m in Seg len D by A6,A7,FINSEQ_1:def 3; then A18: 1<=m & n<=len D by FINSEQ_1:3; then A19: 1<=m+1 & m+1<=len D by A17,AXIOMS:22,NAT_1:37; then A20: m+1 in Seg len D by FINSEQ_1:3; A21: m+1 in dom D by A19,FINSEQ_3:27; A22: abs( D.n-D.m ) >= D.n-D.m by ABSVALUE:11; D.(m+1)<=D.n by A6,A17,A21,GOBOARD2:18; then A23: D.n-D.m >= D.(m+1)-D.m by REAL_1:49; A24: D.(m+1)-D.m=(upper_volume(chi(A,A),D)).(m+1) proof 1 < m+1 by A18,NAT_1:38; then A25: inf divset(D,m+1)=D.((m+1)-1) & sup divset(D,m+1)=D.(m+1) by A21,INTEGRA1:def 5; (m+1)-1=m by XCMPLX_1:26; then vol(divset(D,m+1))=D.(m+1)-D.m by A25,INTEGRA1:def 6; hence thesis by A20,INTEGRA1:22; end; m+1 in Seg len upper_volume(chi(A,A),D) by A20,INTEGRA1:def 7; then m+1 in dom upper_volume(chi(A,A),D) by FINSEQ_1:def 3; then (upper_volume(chi(A,A),D)).(m+1) in rng upper_volume(chi(A,A),D) by FUNCT_1:def 5; then D.(m+1)-D.m>=min rng upper_volume(chi(A,A),D) by A24,SFMASTR3:def 1 ; then D.n-D.m >= min rng upper_volume(chi(A,A),D) by A23,AXIOMS:22; hence thesis by A22,AXIOMS:22; end; abs (D.n-D.m) <= delta D1 proof per cases by A5,A6,A7,AXIOMS:21; suppose n<m; then D.n<D.m by A6,A7,GOBOARD1:def 1; then D.n-D.m<0 by REAL_2:105; then A26: abs( D.n-D.m )=-(D.n-D.m) by ABSVALUE:def 1 .= D.m-D.n by XCMPLX_1: 143; D.n in divset(D1,i) by A3,A6,XBOOLE_0:def 3; then A27: D.n >= inf divset(D1,i) by INTEGRA2:1; D.m in divset(D1,i) by A4,A7,XBOOLE_0:def 3; then D.m <= sup divset(D1,i) by INTEGRA2:1; then A28: D.m-inf divset(D1,i)<= sup divset(D1,i)-inf divset(D1,i) by REAL_1: 49; D.m-D.n <= D.m-inf divset(D1,i) by A27,REAL_2:106; then D.m-D.n <= sup divset(D1,i)-inf divset(D1,i) by A28,AXIOMS:22; then A29: D.m-D.n <= vol(divset(D1,i)) by INTEGRA1:def 6; A30: i in Seg len D1 by A2,FINSEQ_1:def 3; then A31: D.m-D.n <= upper_volume(chi(A,A),D1).i by A29,INTEGRA1:22; i in Seg len upper_volume(chi(A,A),D1) by A30,INTEGRA1:def 7; then i in dom upper_volume(chi(A,A),D1) by FINSEQ_1:def 3; then upper_volume(chi(A,A),D1).i in rng upper_volume(chi(A,A),D1) by FUNCT_1:def 5; then upper_volume(chi(A,A),D1).i <= max rng upper_volume(chi(A,A),D1) by PRE_CIRC:def 1; then upper_volume(chi(A,A),D1).i<=delta(D1) by INTEGRA1:def 19; hence thesis by A26,A31,AXIOMS:22; suppose n>m; then D.n>D.m by A6,A7,GOBOARD1:def 1; then D.n-D.m>0 by SQUARE_1:11; then A32: abs( D.n-D.m )=D.n-D.m by ABSVALUE:def 1; D.m in divset(D1,i) by A4,A7,XBOOLE_0:def 3; then A33: D.m >= inf divset(D1,i) by INTEGRA2:1; D.n in divset(D1,i) by A3,A6,XBOOLE_0:def 3; then D.n <= sup divset(D1,i) by INTEGRA2:1; then A34: D.n-inf divset(D1,i)<= sup divset(D1,i)-inf divset(D1,i) by REAL_1: 49; D.n-D.m <= D.n-inf divset(D1,i) by A33,REAL_2:106; then D.n-D.m <= sup divset(D1,i)-inf divset(D1,i) by A34,AXIOMS:22; then A35: D.n-D.m <= vol(divset(D1,i)) by INTEGRA1:def 6; A36: i in Seg len D1 by A2,FINSEQ_1:def 3; then A37: D.n-D.m <= upper_volume(chi(A,A),D1).i by A35,INTEGRA1:22; i in Seg len upper_volume(chi(A,A),D1) by A36,INTEGRA1:def 7; then i in dom upper_volume(chi(A,A),D1) by FINSEQ_1:def 3; then upper_volume(chi(A,A),D1).i in rng upper_volume(chi(A,A),D1) by FUNCT_1:def 5; then upper_volume(chi(A,A),D1).i <= max rng upper_volume(chi(A,A),D1) by PRE_CIRC:def 1; then upper_volume(chi(A,A),D1).i<=delta(D1) by INTEGRA1:def 19; hence thesis by A32,A37,AXIOMS:22; end; hence contradiction by A1,A8,AXIOMS:22; end; theorem Th5: for p,q st rng p = rng q & p is increasing & q is increasing holds p = q proof let p,q; assume A1:rng p = rng q; assume A2:p is increasing & q is increasing; then p is one-to-one & q is one-to-one by JORDAN7:17; then len p = len q by A1,RLVECT_1:106; hence thesis by A1,A2,GOBOARD2:22; end; theorem Th6: for A be closed-interval Subset of REAL, D,D1 be Element of divs A st D <= D1 & i in dom D & j in dom D & i <= j holds indx(D1,D,i) <= indx(D1,D,j) & indx(D1,D,i) in dom D1 proof let A be closed-interval Subset of REAL; let D,D1 be Element of divs A; assume A1:D <= D1 & i in dom D & j in dom D & i <= j; then A2:D.i = D1.indx(D1,D,i) & indx(D1,D,i) in dom D1 by INTEGRA1:def 21; A3:D.j = D1.indx(D1,D,j) & indx(D1,D,j) in dom D1 by A1,INTEGRA1:def 21; D.i <= D.j by A1,GOBOARD2:18; hence thesis by A2,A3,GOBOARD1:def 1; end; theorem Th7:for A be closed-interval Subset of REAL, D,D1 be Element of divs A st D <= D1 & i in dom D & j in dom D & i < j holds indx(D1,D,i) < indx(D1,D,j) proof let A be closed-interval Subset of REAL; let D,D1 be Element of divs A; assume A1:D <= D1 & i in dom D & j in dom D & i < j; then A2:D.i = D1.indx(D1,D,i) & indx(D1,D,i) in dom D1 by INTEGRA1:def 21; A3:D.j = D1.indx(D1,D,j) & indx(D1,D,j) in dom D1 by A1,INTEGRA1:def 21; D.i < D.j by A1,GOBOARD1:def 1; hence thesis by A2,A3,GOBOARD2:18; end; theorem Th8: for A be closed-interval Subset of REAL, D be Element of divs A holds delta(D) >= 0 proof let A be closed-interval Subset of REAL; let D be Element of divs A; consider y such that A1:y in rng D by SUBSET_1:10; consider n such that A2:n in dom D & y=D.n by A1,PARTFUN1:26; A3:n in Seg len D by A2,FINSEQ_1:def 3; then vol(divset(D,n))=upper_volume(chi(A,A),D).n by INTEGRA1:22; then A4:upper_volume(chi(A,A),D).n >= 0 by INTEGRA1:11; n in Seg len upper_volume(chi(A,A),D) by A3,INTEGRA1:def 7; then n in dom upper_volume(chi(A,A),D) by FINSEQ_1:def 3; then upper_volume(chi(A,A),D).n in rng upper_volume(chi(A,A),D) by FUNCT_1:def 5; then upper_volume(chi(A,A),D).n <= max rng upper_volume(chi(A,A),D) by PRE_CIRC:def 1; hence thesis by A4,INTEGRA1:def 19; end; Lm5:a+b-(d+e) = (a-d)+(b-e) proof a+b-(d+e)=a+b-d-e by XCMPLX_1:36 .=a-d+b-e by XCMPLX_1:29; hence thesis by XCMPLX_1:29; end; Lm6:for A be closed-interval Subset of REAL, g be Function of A,REAL st g is_bounded_on A holds sup rng g >= inf rng g proof let A be closed-interval Subset of REAL; let g be Function of A,REAL; assume A1: g is_bounded_on A; dom g <> {} by FUNCT_2:def 1; then A2: rng g <> {} by RELAT_1:65; g is_bounded_below_on A & g is_bounded_above_on A by A1,RFUNCT_1:def 11; then rng g is bounded_above & rng g is bounded_below by INTEGRA1:13,15; then rng g is bounded by SEQ_4:def 3; hence thesis by A2,SEQ_4:24; end; Lm7:for A,B be closed-interval Subset of REAL, f be Function of A,REAL st f is_bounded_on A & B c= A holds inf rng(f|B) >= inf rng f & inf rng f <= sup rng(f|B) & sup rng(f|B) <= sup rng f & inf rng(f|B) <= sup rng f proof let A,B be closed-interval Subset of REAL, f be Function of A,REAL; assume A1: f is_bounded_on A & B c= A; then f is_bounded_above_on A & f is_bounded_below_on A by RFUNCT_1:def 11; then A2:rng f is bounded_above & rng f is bounded_below by INTEGRA1:13,15; consider x such that A3:x in B by SUBSET_1:10; B c=dom f by A1,FUNCT_2:def 1; then A4:dom(f|B)=B by RELAT_1:91; then A5:(f|B).x in rng(f|B) by A3,FUNCT_1:def 5; A6:rng(f|B) <> {} by A4,RELAT_1:65; A7:rng(f|B) c= rng f by FUNCT_1:76; then A8:rng(f|B) is bounded_above & rng(f|B) is bounded_below by A2,RCOMP_1:3,4; thus A9:inf rng(f|B) >= inf rng f by A2,A6,A7,PSCOMP_1:12; A10:inf rng(f|B)<=(f|B).x by A5,A8,SEQ_4:def 5; sup rng(f|B)>=(f|B).x by A5,A8,SEQ_4:def 4; then A11: inf rng(f|B) <= sup rng(f|B) by A10,AXIOMS:22; hence sup rng(f|B) >= inf rng f by A9,AXIOMS:22; thus sup rng(f|B) <= sup rng f by A2,A6,A7,PSCOMP_1:13; hence thesis by A11,AXIOMS:22; end; Lm8: for A be closed-interval Subset of REAL, D1 be Element of divs A st j in dom D1 holds vol(divset(D1,j)) <= delta(D1) proof let A be closed-interval Subset of REAL, D1 be Element of divs A; assume j in dom D1; then A1:j in Seg len D1 by FINSEQ_1:def 3; then j in Seg len upper_volume(chi(A,A),D1) by INTEGRA1:def 7; then j in dom upper_volume(chi(A,A),D1) by FINSEQ_1:def 3; then upper_volume(chi(A,A),D1).j in rng upper_volume(chi(A,A),D1) by FUNCT_1:def 5; then upper_volume(chi(A,A),D1).j <= max rng upper_volume(chi(A,A),D1) by PRE_CIRC:def 1; then vol(divset(D1,j)) <= max rng upper_volume(chi(A,A),D1) by A1,INTEGRA1:22 ; hence thesis by INTEGRA1:def 19; end; Lm9:for A be closed-interval Subset of REAL, j1 be Nat, D1,D2 be Element of divs A st j1 = len D1 - 1 & x in divset(D1,len D1) & len D1 >= 2 & D1 <= D2 & rng D2 = rng D1 \/ {x} holds rng (D2|indx(D2,D1,j1)) = rng (D1|j1) proof let A be closed-interval Subset of REAL, j1 be Nat, D1,D2 be Element of divs A; assume A1:j1 = len D1 - 1 & x in divset(D1,len D1) & len D1 >= 2; assume A2:D1<=D2 & rng D2 = rng D1 \/ {x}; A3: len D1 in dom D1 & len D2 in dom D2 by SCMFSA_7:12; A4: len D1 <> 1 by A1; then A5:len D1-1 in NAT & len D1-1 in dom D1 by A3,INTEGRA1:9; inf divset(D1,len D1) <= x & x <= sup divset(D1,len D1) by A1,INTEGRA2:1; then A6:D1.j1 <= x & x <= D1.(len D1) by A1,A3,A4,INTEGRA1:def 5; A7:indx(D2,D1,j1) in dom D2 by A1,A2,A5,INTEGRA1:def 21; then A8: 1 <= indx(D2,D1,j1) & indx(D2,D1,j1) <= len D2 by FINSEQ_3:27; A9:j1 in dom D1 by A1,A3,A4,INTEGRA1:9; then A10: 1 <= j1 & j1 <= len D1 by FINSEQ_3:27; for x1 st x1 in rng(D2|indx(D2,D1,j1)) holds x1 in rng(D1|j1) proof let x1; assume x1 in rng(D2|indx(D2,D1,j1)); then consider k such that A11: k in dom(D2|indx(D2,D1,j1)) & x1=(D2|indx(D2,D1,j1)).k by PARTFUN1:26; k in Seg len(D2|indx(D2,D1,j1)) by A11,FINSEQ_1:def 3; then A12: k in Seg indx(D2,D1,j1) by A8,TOPREAL1:3; then A13: (D2|indx(D2,D1,j1)).k = D2.k & k in dom D2 by A7,RFINSEQ:19; then A14: D2.k in rng D2 by FUNCT_1:def 5; 1 <= k & k <= indx(D2,D1,j1) by A12,FINSEQ_1:3; then D2.k <= D2.indx(D2,D1,j1) by A7,A13,GOBOARD2:18; then A15: D2.k <= D1.j1 by A1,A2,A5,INTEGRA1:def 21; A16: len(D1|j1) = j1 by A10,TOPREAL1:3; A17: D2.k in {x} implies D2.k = D1.j1 proof assume D2.k in {x}; then D1.j1 <= D2.k by A6,TARSKI:def 1; hence thesis by A15,AXIOMS:21; end; A18: D2.k in {x} implies D2.k in rng(D1|j1) proof assume A19:D2.k in {x}; j1 in dom(D1|j1) by A10,A16,FINSEQ_3:27; then A20: (D1|j1).j1 in rng(D1|j1) by FUNCT_1:def 5; j1 in Seg j1 by A10,FINSEQ_1:3; hence thesis by A9,A17,A19,A20,RFINSEQ:19; end; D2.k in rng D1 implies D2.k in rng(D1|j1) proof assume D2.k in rng D1; then consider m such that A21: m in dom D1 & D2.k = D1.m by PARTFUN1:26; m in Seg len D1 by A21,FINSEQ_1:def 3; then A22: 1 <= m & m <= j1 by A9,A15,A21,FINSEQ_1:3,GOBOARD1:def 1; then A23: m in dom (D1|j1) by A16,FINSEQ_3:27; m in Seg j1 by A22,FINSEQ_1:3; then D2.k = (D1|j1).m by A9,A21,RFINSEQ:19; hence thesis by A23,FUNCT_1:def 5; end; hence thesis by A2,A7,A11,A12,A14,A18,RFINSEQ:19,XBOOLE_0:def 2; end; then A24: rng(D2|indx(D2,D1,j1)) c= rng (D1|j1) by TARSKI:def 3; for x1 st x1 in rng(D1|j1) holds x1 in rng(D2|indx(D2,D1,j1)) proof let x1; assume x1 in rng(D1|j1); then consider k such that A25: k in dom(D1|j1) & x1=(D1|j1).k by PARTFUN1:26; k in Seg len(D1|j1) by A25,FINSEQ_1:def 3; then A26: k in Seg j1 by A10,TOPREAL1:3; then A27: (D1|j1).k = D1.k & k in dom D1 by A9,RFINSEQ:19; then D1.k in rng D1 by FUNCT_1:def 5; then x1 in rng D2 by A2,A25,A27,XBOOLE_0:def 2; then consider n such that A28: n in dom D2 & x1=D2.n by PARTFUN1:26; A29: indx(D2,D1,k) in dom D2 & D2.indx(D2,D1,k)=D2.n by A2,A25,A27,A28,INTEGRA1:def 21; k <= j1 by A26,FINSEQ_1:3; then D1.k <= D1.j1 by A1,A5,A27,GOBOARD2:18; then D2.indx(D2,D1,k) <= D1.j1 by A2,A27,INTEGRA1:def 21; then D2.indx(D2,D1,k)<=D2.indx(D2,D1,j1) by A1,A2,A5,INTEGRA1:def 21; then A30: n <= indx(D2,D1,j1) by A7,A28,A29,GOBOARD1:def 1; 1 <= n by A28,FINSEQ_3:27; then A31: n in Seg indx(D2,D1,j1) by A30,FINSEQ_1:3; then A32: D2.n = (D2|indx(D2,D1,j1)).n by A7,RFINSEQ:19; n in Seg len(D2|indx(D2,D1,j1)) by A8,A31,TOPREAL1:3; then n in dom(D2|indx(D2,D1,j1)) by FINSEQ_1:def 3; hence thesis by A28,A32,FUNCT_1:def 5; end; then rng(D1|j1) c= rng(D2|indx(D2,D1,j1)) by TARSKI:def 3; hence thesis by A24,XBOOLE_0:def 10; end; theorem Th9: for A be closed-interval Subset of REAL, g be Function of A,REAL, D1,D2 be Element of divs A st x in divset(D1,len D1) & len D1 >= 2 & D1<=D2 & rng D2 = rng D1 \/ {x} & g is_bounded_on A holds Sum lower_volume(g,D2)-Sum lower_volume(g,D1)<=(sup rng g-inf rng g)*delta(D1) proof let A be closed-interval Subset of REAL; let g be Function of A,REAL; let D1,D2 be Element of divs A; assume A1:x in divset(D1,len D1) & len D1 >= 2; assume A2:D1<=D2 & rng D2 = rng D1 \/ {x}; assume A3:g is_bounded_on A; deffunc PLg(Element of divs A,Nat) = (PartSums(lower_volume(g,$1))).$2; deffunc LVg(Element of divs A) = lower_volume(g,$1); set j = len D1; A4:len D1 <> 0 & len D2 <> 0 by FINSEQ_1:25; then A5:len D1 in Seg len D1 & len D2 in Seg len D2 by FINSEQ_1:5; A6:len D1 in dom D1 & len D2 in dom D2 by SCMFSA_7:12; A7:len D1 <> 1 by A1; A8:sup rng g >= inf rng g by A3,Lm6; A9:indx(D2,D1,j) in dom D2 & D2.indx(D2,D1,j)=D1.(len D1) by A2,A6,INTEGRA1:def 21; then A10:indx(D2,D1,j) in Seg len D2 by FINSEQ_1:def 3; A11: indx(D2,D1,j) >= len lower_volume(g,D2) proof assume indx(D2,D1,j) < len lower_volume(g,D2); then indx(D2,D1,j) < len D2 by INTEGRA1:def 8; then A12:D1.(len D1) < D2.(len D2) by A6,A9,GOBOARD1:def 1; A13:not D2.(len D2) in rng D1 proof assume A14:D2.(len D2) in rng D1; rng D1 c= A by INTEGRA1:def 2; then inf A <= D2.(len D2) & D2.(len D2) <= sup A by A14,INTEGRA2:1; hence contradiction by A12,INTEGRA1:def 2; end; D2.(len D2) in rng D2 by A6,FUNCT_1:def 5; then D2.(len D2) in rng D1 or D2.(len D2) in {x} by A2,XBOOLE_0:def 2; then D2.(len D2) = x by A13,TARSKI:def 1; then D2.(len D2) <= sup divset(D1,len D1) by A1,INTEGRA2:1; hence contradiction by A6,A7,A12,INTEGRA1:def 5; end; then A15:indx(D2,D1,j) in Seg len lower_volume(g,D2) & indx(D2,D1,j) >= len lower_volume(g,D2) by A10,INTEGRA1:def 8; A16:j in Seg len lower_volume(g,D1) & j >= len lower_volume(g,D1) by A5,INTEGRA1:def 8; A17:len D1-1 in NAT & len D1-1 in dom D1 by A6,A7,INTEGRA1:9; reconsider j1=len D1-1 as Element of NAT by A6,A7,INTEGRA1:9; A18:indx(D2,D1,j1) in dom D2 & 1 <= indx(D2,D1,j1) & indx(D2,D1,j1) <= len D2 proof thus indx(D2,D1,j1) in dom D2 by A2,A17,INTEGRA1:def 21; hence thesis by FINSEQ_3:27; end; then mid(D2,1,indx(D2,D1,j1)) is increasing by INTEGRA1:37; then A19:D2|indx(D2,D1,j1) is increasing by A18,JORDAN3:25; A20:j1 in dom D1 & 1 <= j1 & j1 <= len D1 proof thus j1 in dom D1 by A6,A7,INTEGRA1:9; hence thesis by FINSEQ_3:27; end; then mid(D1,1,j1) is increasing by INTEGRA1:37; then A21:D1|j1 is increasing by A20,JORDAN3:25; A22:rng (D2|indx(D2,D1,j1)) = rng (D1|j1) by A1,A2,Lm9; then A23:D2|indx(D2,D1,j1)=D1|j1 by A19,A21,Th5; A24:for k st 1 <= k & k <= j1 holds k=indx(D2,D1,k) proof let k; assume A25:1 <= k & k <= j1; assume A26:k<>indx(D2,D1,k); now per cases by A26,AXIOMS:21; suppose A27:k > indx(D2,D1,k); 1 <= k & k <= len D1 by A20,A25,AXIOMS:22; then A28: k in dom D1 by FINSEQ_3:27; then A29: indx(D2,D1,k) in dom D2 & D1.k=D2.indx(D2,D1,k) by A2,INTEGRA1:def 21; then indx(D2,D1,k) in Seg len D2 by FINSEQ_1:def 3; then A30: 1<=indx(D2,D1,k)&indx(D2,D1,k)<=indx(D2,D1,j1) by A2,A20,A25,A28,Th6,FINSEQ_1:3; then indx(D2,D1,k) in Seg indx(D2,D1,j1) by FINSEQ_1:3; then A31: (D2|indx(D2,D1,j1)).indx(D2,D1,k)=D2.indx(D2,D1,k) by A18,RFINSEQ: 19; A32: indx(D2,D1,k) < j1 by A25,A27,AXIOMS:22; then indx(D2,D1,k) <= len D1 by A20,AXIOMS:22; then indx(D2,D1,k) in dom D1 by A30,FINSEQ_3:27; then A33: D1.k > D1.indx(D2,D1,k) by A27,A28,GOBOARD1:def 1; indx(D2,D1,k) in Seg j1 by A30,A32,FINSEQ_1:3; hence contradiction by A20,A23,A29,A31,A33,RFINSEQ:19; suppose A34:k < indx(D2,D1,k); k in Seg j1 by A25,FINSEQ_1:3; then A35: D1.k = (D1|j1).k by A17,RFINSEQ:19; 1 <= k & k <= len D1 by A20,A25,AXIOMS:22; then A36: k in dom D1 by FINSEQ_3:27; then A37: indx(D2,D1,k) in dom D2 & D1.k=D2.indx(D2,D1,k) by A2,INTEGRA1:def 21; indx(D2,D1,k) <= indx(D2,D1,j1) & indx(D2,D1,k) in dom D2 & indx(D2,D1,j1) in dom D2 by A2,A20,A25,A36,Th6; then A38: k <= indx(D2,D1,j1) by A34,AXIOMS:22; then k <= len D2 by A18,AXIOMS:22; then k in dom D2 by A25,FINSEQ_3:27; then A39: D2.k < D2.indx(D2,D1,k) by A34,A37,GOBOARD1:def 1; k in Seg indx(D2,D1,j1) by A25,A38,FINSEQ_1:3; hence contradiction by A18,A23,A35,A37,A39,RFINSEQ:19; end; hence contradiction; end; A40:len (D2|indx(D2,D1,j1))=len (D1|j1) by A19,A21,A22,Th5; len (D2|indx(D2,D1,j1))=indx(D2,D1,j1) by A18,TOPREAL1:3; then A41:indx(D2,D1,j1) =j1 by A20,A40,TOPREAL1:3; j1 <= len D1 by A17,FINSEQ_3:27; then j1 <= len lower_volume(g,D1) by INTEGRA1:def 8; then A42:len(lower_volume(g,D1)|j1)=indx(D2,D1,j1) by A41,TOPREAL1:3; indx(D2,D1,j1) in dom D2 by A2,A17,INTEGRA1:def 21; then indx(D2,D1,j1) <= len D2 by FINSEQ_3:27; then indx(D2,D1,j1) <= len lower_volume(g,D2) by INTEGRA1:def 8; then A43:len(lower_volume(g,D1)|j1)=len(lower_volume(g,D2)|indx(D2,D1,j1)) by A42,TOPREAL1:3; for k st 1 <= k & k <= len(lower_volume(g,D1)|j1) holds (lower_volume(g,D1)|j1).k = (lower_volume(g,D2)|indx(D2,D1,j1)).k proof let k; assume A44:1 <= k & k <= len(lower_volume(g,D1)|j1); A45: len(lower_volume(g,D1)) = len D1 by INTEGRA1:def 8; then A46:1 <= k & k <= j1 by A20,A44,TOPREAL1:3; then A47: k in Seg j1 by FINSEQ_1:3; k <= len D1 by A20,A46,AXIOMS:22; then A48:k in Seg len D1 by A44,FINSEQ_1:3; A49:divset(D1,k)=divset(D2,indx(D2,D1,k)) proof A50: divset(D1,k)=[. inf divset(D1,k), sup divset(D1,k) .] by INTEGRA1:5; A51: k in dom D1 by A48,FINSEQ_1:def 3; then A52: indx(D2,D1,k) in dom D2 & D1.k=D2.indx(D2,D1,k) by A2,INTEGRA1:def 21; inf divset(D1,k)=inf divset(D2,indx(D2,D1,k)) & sup divset(D1,k)=sup divset(D2,indx(D2,D1,k)) proof per cases; suppose A53:k=1; then A54: inf divset(D1,k)=inf A & sup divset(D1,k)=D1.k by A51,INTEGRA1:def 5 ; indx(D2,D1,k)=1 by A20,A24,A53; hence thesis by A52,A54,INTEGRA1:def 5; suppose A55:k<>1; then A56: inf divset(D1,k)=D1.(k-1) & sup divset(D1,k)=D1.k by A51,INTEGRA1:def 5; A57: k-1 in dom D1 & D1.(k-1) in A & k-1 in NAT by A51,A55,INTEGRA1:9; reconsider k1=k-1 as Nat by A51,A55,INTEGRA1:9; k <= k+1 by NAT_1:29; then k1 <= k by REAL_1:86; then A58: k1 <= j1 by A46,AXIOMS:22; A59: 1 <= k1 by A57,FINSEQ_3:27; indx(D2,D1,k)<>1 by A24,A46,A55; then A60: inf divset(D2,indx(D2,D1,k))=D2.(indx(D2,D1,k)-1) & sup divset(D2,indx(D2,D1,k))=D2.indx(D2,D1,k) by A52,INTEGRA1:def 5; D2.(indx(D2,D1,k)-1)=D2.(k-1) by A24,A46 .= D2.indx(D2,D1,k1) by A24,A58,A59; hence thesis by A2,A51,A56,A57,A60,INTEGRA1:def 21; end; hence thesis by A50,INTEGRA1:5; end; j1 in Seg len(lower_volume(g,D1)) by A20,A45,FINSEQ_1:def 3; then j1 in dom(lower_volume(g,D1)) by FINSEQ_1:def 3; then A61: (lower_volume(g,D1)|j1).k = lower_volume(g,D1).k by A47,RFINSEQ:19 .=(inf(rng(g|divset(D2,indx(D2,D1,k)))))*vol(divset(D2,indx(D2,D1,k))) by A48,A49,INTEGRA1:def 8; indx(D2,D1,k) in Seg j1 by A24,A46,A47; then A62:indx(D2,D1,k) in Seg indx(D2,D1,j1) by A20,A24; then 1<=indx(D2,D1,k) & indx(D2,D1,k)<=indx(D2,D1,j1) by FINSEQ_1:3; then 1<=indx(D2,D1,k) & indx(D2,D1,k)<=len D2 by A18,AXIOMS:22; then A63:indx(D2,D1,k) in Seg len D2 by FINSEQ_1:3; indx(D2,D1,j1) in Seg len D2 by A18,FINSEQ_1:def 3; then indx(D2,D1,j1) in Seg len lower_volume(g,D2) by INTEGRA1:def 8; then A64:indx(D2,D1,j1) in dom lower_volume(g,D2) by FINSEQ_1:def 3; (lower_volume(g,D2)|indx(D2,D1,j1)).k =(lower_volume(g,D2)|indx(D2,D1,j1)).indx(D2,D1,k) by A24,A46 .=lower_volume(g,D2).indx(D2,D1,k) by A62,A64,RFINSEQ:19 .=(inf(rng(g|divset(D2,indx(D2,D1,k)))))*vol(divset(D2,indx(D2,D1,k))) by A63,INTEGRA1:def 8; hence thesis by A61; end; then A65:lower_volume(g,D2)|indx(D2,D1,j1)=lower_volume(g,D1)|j1 by A43,FINSEQ_1:18; indx(D2,D1,j1) in Seg len D2 by A18,FINSEQ_1:def 3; then A66:indx(D2,D1,j1) in Seg len lower_volume(g,D2) by INTEGRA1:def 8; j1 in Seg len D1 by A20,FINSEQ_1:def 3; then A67:j1 in Seg len lower_volume(g,D1) by INTEGRA1:def 8; A68:PLg(D2,indx(D2,D1,j1))=Sum(lower_volume(g,D2)|indx(D2,D1,j1)) by A66,INTEGRA1:def 22 .=PLg(D1,j1) by A65,A67,INTEGRA1:def 22; A69:indx(D2,D1,j1)+1 <= indx(D2,D1,len D1) proof len D1 < len D1+1 by NAT_1:38; then j1 < len D1 by REAL_1:84; then indx(D2,D1,j1) < indx(D2,D1,len D1) by A2,A6,A20,Th7; hence thesis by NAT_1:38; end; A70:Sum mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,len D1)) -Sum mid(lower_volume(g,D1),len D1,len D1) <= (sup rng g-inf rng g)*delta(D1) proof A71: 1 <= indx(D2,D1,j) & indx(D2,D1,j) <= len lower_volume(g,D2) proof indx(D2,D1,j) in dom D2 by A2,A6,INTEGRA1:def 21; then indx(D2,D1,j) in Seg len D2 by FINSEQ_1:def 3; then indx(D2,D1,j) in Seg len lower_volume(g,D2) by INTEGRA1:def 8; hence thesis by FINSEQ_1:3; end; then A72: 1 <= indx(D2,D1,j1)+1 & indx(D2,D1,j1)+1 <= len lower_volume(g,D2) by A18,A69,AXIOMS:22,NAT_1:38; A73:indx(D2,D1,j)-indx(D2,D1,j1) <= 2 proof assume A74:indx(D2,D1,j)-indx(D2,D1,j1) > 2; set ID1=indx(D2,D1,j1)+1; set ID2=ID1+1; A75:indx(D2,D1,j1) < ID1 & ID1 < ID2 & ID2 < indx(D2,D1,j) proof thus indx(D2,D1,j1) < ID1 by NAT_1:38; thus ID1 < ID2 by NAT_1:38; indx(D2,D1,j1)+(1+1)<indx(D2,D1,j) by A74,REAL_1:86; hence thesis by XCMPLX_1:1; end; A76:indx(D2,D1,j) in dom D2 by A2,A6,INTEGRA1:def 21; then A77:indx(D2,D1,j) <= len D2 by FINSEQ_3:27; A78:ID1 in dom D2 proof A79: 1 <= ID1 & ID1 <= indx(D2,D1,j) by A18,A75,AXIOMS:22; then ID1 <= len D2 by A77,AXIOMS:22; hence thesis by A79,FINSEQ_3:27; end; A80:ID2 in dom D2 proof A81: indx(D2,D1,j1) <= ID2 & ID2 <= len D2 by A75,A77,AXIOMS:22; then 1 <= ID2 by A18,AXIOMS:22; hence thesis by A81,FINSEQ_3:27; end; then A82:D2.indx(D2,D1,j1)<D2.ID1 & D2.ID1<D2.ID2 & D2.ID2<D2.indx(D2,D1,j) by A18,A75,A76,A78,GOBOARD1:def 1; A83: D2.ID1 in rng D2 & D2.ID2 in rng D2 by A78,A80,FUNCT_1:def 5; A84: D1.j1 = D2.indx(D2,D1,j1) & D1.j = D2.indx(D2,D1,j) by A2,A6,A20,INTEGRA1:def 21; A85:D2.ID1 in {x} & D2.ID2 in {x} proof not(D2.ID1 in rng D1) & not(D2.ID2 in rng D1) proof assume A86:D2.ID1 in rng D1 or D2.ID2 in rng D1; now per cases by A86; suppose D2.ID1 in rng D1; then consider n such that A87: n in dom D1 & D1.n=D2.ID1 by PARTFUN1:26; D2.indx(D2,D1,j1)<D2.ID1 & D2.ID1<D2.indx(D2,D1,j) by A82,AXIOMS:22; then j1<n & n<j by A6,A20,A84,A87,GOBOARD2:18; then j<n+1 & n<j by REAL_1:84; hence contradiction by NAT_1:38; suppose D2.ID2 in rng D1; then consider n such that A88: n in dom D1 & D1.n=D2.ID2 by PARTFUN1:26; D2.indx(D2,D1,j1)<D2.ID2 & D2.ID2<D2.indx(D2,D1,j) by A82,AXIOMS:22; then j1<n & n<j by A6,A20,A84,A88,GOBOARD2:18; then j<n+1 & n<j by REAL_1:84; hence contradiction by NAT_1:38; end; hence thesis; end; hence thesis by A2,A83,XBOOLE_0:def 2; end; then D2.ID1 = x by TARSKI:def 1; then D2.ID1=D2.ID2 by A85,TARSKI:def 1; hence contradiction by A75,A78,A80,GOBOARD2:19; end; 1 <= indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 & indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 <= 2 proof A89: indx(D2,D1,j)-'(indx(D2,D1,j1)+1) =indx(D2,D1,j)-(indx(D2,D1,j1)+1) by A69,SCMFSA_7:3; then indx(D2,D1,j)-'(indx(D2,D1,j1)+1) >= 0 by A69,SQUARE_1:12; then indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 >= 0+1 by AXIOMS:24; hence indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 >= 1; indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 =indx(D2,D1,j)-indx(D2,D1,j1)-1+1 by A89,XCMPLX_1:36 .=indx(D2,D1,j)-indx(D2,D1,j1)+1-1 by XCMPLX_1:29 .=indx(D2,D1,j)-indx(D2,D1,j1) by XCMPLX_1:26; hence thesis by A73; end; then A90: 1 <= len mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) & len mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j))<=2 by A69,A71,A72,JORDAN3:27; A91: 1 <= j & j <= len lower_volume(g,D1) by A5,FINSEQ_1:3,INTEGRA1:def 8; j-'j+1 = 1 by Lm3; then A92: len mid(lower_volume(g,D1),j,j)=1 by A91,JORDAN3:27; mid(lower_volume(g,D1),j,j).1 = lower_volume(g,D1).j by A91,JORDAN3:27; then A93:mid(lower_volume(g,D1),j,j)=<*lower_volume(g,D1).j*> by A92,FINSEQ_1: 57; indx(D2,D1,j1)+1 in Seg len lower_volume(g,D2) by A72,FINSEQ_1:3; then A94:indx(D2,D1,j1)+1 in Seg len D2 by INTEGRA1:def 8; A95:j in Seg len D1 by A4,FINSEQ_1:5; now per cases by A90,Lm4; suppose A96: len mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j))=1; A97: indx(D2,D1,j1)+1=indx(D2,D1,j) proof len mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) = indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 by A69,A71,A72,JORDAN3:27; then indx(D2,D1,j)-'(indx(D2,D1,j1)+1)=1-1 by A96,XCMPLX_1:26; then indx(D2,D1,j)-(indx(D2,D1,j1)+1)=0 by A69,SCMFSA_7:3; then indx(D2,D1,j)=0+(indx(D2,D1,j1)+1) by XCMPLX_1:27; hence thesis; end; A98: divset(D2,indx(D2,D1,j))=divset(D1,j) proof inf divset(D1,j)=D2.indx(D2,D1,j1) & sup divset(D1,j)=D2.indx(D2,D1,j) proof inf divset(D1,j)=D1.j1 & sup divset(D1,j)=D1.j by A6,A7,INTEGRA1:def 5; hence thesis by A2,A6,A20,INTEGRA1:def 21; end; then A99: divset(D1,j)=[. D2.indx(D2,D1,j1),D2.indx(D2,D1,j).] by INTEGRA1:5; A100: indx(D2,D1,j)-1=indx(D2,D1,j1) by A97,XCMPLX_1:26; A101: indx(D2,D1,j) in dom D2 by A2,A6,INTEGRA1:def 21; indx(D2,D1,j)<>1 by A18,A97,NAT_1:38; then inf divset(D2,indx(D2,D1,j))=D2.(indx(D2,D1,j)-1) & sup divset(D2,indx(D2,D1,j))=D2.indx(D2,D1,j) by A101,INTEGRA1:def 5; hence thesis by A99,A100,INTEGRA1:5; end; mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)).1 =lower_volume(g,D2).(indx(D2,D1,j1)+1) by A71,A72,JORDAN3:27; then mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =<*lower_volume(g,D2).(indx(D2,D1,j1)+1)*> by A96,FINSEQ_1:57; then A102: Sum mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =lower_volume(g,D2).(indx(D2,D1,j1)+1) by RVSUM_1:103 .=(inf(rng(g|divset(D2,(indx(D2,D1,j1)+1))))) *vol(divset(D2,(indx(D2,D1,j1)+1))) by A94,INTEGRA1:def 8 .=lower_volume(g,D1).j by A95,A97,A98,INTEGRA1:def 8 .=Sum mid(lower_volume(g,D1),j,j) by A93,RVSUM_1:103; A103: delta(D1) >= 0 by Th8; sup rng g - inf rng g >= 0 by A8,SQUARE_1:12; then (sup rng g - inf rng g)*delta(D1) >= 0*delta(D1) by A103,AXIOMS:25; hence Sum mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) -Sum mid(lower_volume(g,D1),j,j) <= (sup rng g-inf rng g)*delta(D1) by A102,XCMPLX_1:14; suppose A104: len mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j))=2; A105: mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)).1 =lower_volume(g,D2).(indx(D2,D1,j1)+1) by A71,A72,JORDAN3:27; mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)).2 =lower_volume(g,D2).(indx(D2,D1,j1)+2) proof A106: 2+(indx(D2,D1,j1)+1)>=0+1 by A72,REAL_1:55; mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)).2 =LVg(D2).(2+(indx(D2,D1,j1)+1)-'1) by A69,A71,A72,A104,JORDAN3:27 .=LVg(D2).(2+(indx(D2,D1,j1)+1)-1) by A106,SCMFSA_7:3 .=LVg(D2).((indx(D2,D1,j1)+1)+(2-1)) by XCMPLX_1:29 .=LVg(D2).(indx(D2,D1,j1)+(1+1)) by XCMPLX_1:1; hence thesis; end; then mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =<*lower_volume(g,D2).(indx(D2,D1,j1)+1), lower_volume(g,D2).(indx(D2,D1,j1)+2)*> by A104,A105,FINSEQ_1:61; then A107: Sum mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =lower_volume(g,D2).(indx(D2,D1,j1)+1) +lower_volume(g,D2).(indx(D2,D1,j1)+2) by RVSUM_1:107; A108: vol(divset(D1,j)) =vol(divset(D2,indx(D2,D1,j1)+1))+vol(divset(D2,indx(D2,D1,j1)+2)) proof A109: inf divset(D1,j)=D2.indx(D2,D1,j1)&sup divset(D1,j)=D2.indx(D2,D1,j) proof inf divset(D1,j)=D1.j1 & sup divset(D1,j)=D1.j by A6,A7,INTEGRA1:def 5 ; hence thesis by A2,A6,A20,INTEGRA1:def 21; end; A110: indx(D2,D1,j)=indx(D2,D1,j1)+2 proof indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1=2 by A69,A71,A72,A104,JORDAN3:27; then indx(D2,D1,j)-(indx(D2,D1,j1)+1)+1=2 by A69,SCMFSA_7:3; then indx(D2,D1,j)-indx(D2,D1,j1)-1+1=2 by XCMPLX_1:36; then indx(D2,D1,j)-indx(D2,D1,j1)-(1-1)=2 by XCMPLX_1:37; hence thesis by XCMPLX_1:27; end; A111: inf divset(D2,(indx(D2,D1,j1)+2))=D2.(indx(D2,D1,j1)+1) & sup divset(D2,(indx(D2,D1,j1)+2))=D2.(indx(D2,D1,j1)+2) & inf divset(D2,(indx(D2,D1,j1)+1))=D2.indx(D2,D1,j1) & sup divset(D2,(indx(D2,D1,j1)+1))=D2.(indx(D2,D1,j1)+1) proof A112: indx(D2,D1,j1)+2 in dom D2 by A2,A6,A110,INTEGRA1:def 21; indx(D2,D1,j1)+1 in Seg len lower_volume(g,D2) by A72,FINSEQ_1:3; then indx(D2,D1,j1)+1 in Seg len D2 by INTEGRA1:def 8; then A113: indx(D2,D1,j1)+1 in dom D2 by FINSEQ_1:def 3; A114: indx(D2,D1,j1)+1 <> 1 by A18,NAT_1:38; indx(D2,D1,j1)+1+1 > 1 by A72,NAT_1:38; then A115: indx(D2,D1,j1)+(1+1) > 1 by XCMPLX_1:1; A116: indx(D2,D1,j1)+2-1=indx(D2,D1,j1)+1 by Lm2,XCMPLX_1:29; A117: indx(D2,D1,j1)+1-1=indx(D2,D1,j1)+(1-1) by XCMPLX_1:29 .=indx(D2,D1,j1)+0; thus inf divset(D2,(indx(D2,D1,j1)+2))=D2.(indx(D2,D1,j1)+1) & sup divset(D2,(indx(D2,D1,j1)+2))=D2.(indx(D2,D1,j1)+2) by A112,A115,A116,INTEGRA1:def 5; thus thesis by A113,A114,A117,INTEGRA1:def 5; end; vol(divset(D1,j))=sup divset(D1,j)-inf divset(D1,j) by INTEGRA1:def 6 .=D2.indx(D2,D1,j)-(D2.(indx(D2,D1,j1)+1) -D2.(indx(D2,D1,j1)+1))-D2.indx(D2,D1,j1) by A109,XCMPLX_1:17 .=D2.(indx(D2,D1,j1)+2)-D2.(indx(D2,D1,j1)+1) +D2.(indx(D2,D1,j1)+1)-D2.indx(D2,D1,j1) by A110,XCMPLX_1:37; then vol(divset(D1,j))=vol(divset(D2,indx(D2,D1,j1)+2)) +D2.(indx(D2,D1,j1)+1)-D2.indx(D2,D1,j1) by A111,INTEGRA1:def 6 .=vol(divset(D2,indx(D2,D1,j1)+2)) +(sup divset(D2,indx(D2,D1,j1)+1)-inf divset(D2,indx(D2,D1,j1)+1)) by A111,XCMPLX_1:29; hence thesis by INTEGRA1:def 6; end; then A118: lower_volume(g,D1).j=(inf(rng(g|divset(D1,j))))* (vol(divset(D2,indx(D2,D1,j1)+1))+vol(divset(D2,indx(D2,D1,j1)+2))) by A95,INTEGRA1:def 8; A119: vol(divset(D2,indx(D2,D1,j1)+1))>=0 by INTEGRA1:11; A120: vol(divset(D2,indx(D2,D1,j1)+2))>=0 by INTEGRA1:11; A121: Sum mid(LVg(D2),indx(D2,D1,j1)+1,indx(D2,D1,j))-Sum mid(LVg(D1),j,j) <=(sup rng g - inf rng g)*(vol(divset(D2,indx(D2,D1,j1)+2)) +vol(divset(D2,indx(D2,D1,j1)+1))) proof A122: indx(D2,D1,j) in dom D2 by A2,A6,INTEGRA1:def 21; A123: indx(D2,D1,j)=indx(D2,D1,j1)+2 proof indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1=2 by A69,A71,A72,A104,JORDAN3:27; then indx(D2,D1,j)-(indx(D2,D1,j1)+1)+1=2 by A69,SCMFSA_7:3; then indx(D2,D1,j)-indx(D2,D1,j1)-1+1=2 by XCMPLX_1:36; then indx(D2,D1,j)-indx(D2,D1,j1)-(1-1)=2 by XCMPLX_1:37; hence thesis by XCMPLX_1:27; end; then A124: indx(D2,D1,j1)+2 in Seg len D2 by A122,FINSEQ_1:def 3; set ID1=indx(D2,D1,j1)+1, ID2=indx(D2,D1,j1)+2; A125: Sum mid(LVg(D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =(inf rng(g|divset(D2,indx(D2,D1,j1)+2))) *vol(divset(D2,indx(D2,D1,j1)+2)) +LVg(D2).(indx(D2,D1,j1)+1) by A107,A124,INTEGRA1:def 8 .=(inf rng(g|divset(D2,indx(D2,D1,j1)+2))) *vol(divset(D2,indx(D2,D1,j1)+2)) +(inf rng(g|divset(D2,indx(D2,D1,j1)+1))) *vol(divset(D2,indx(D2,D1,j1)+1)) by A94,INTEGRA1:def 8; divset(D2,ID2)c=A by A122,A123,INTEGRA1:10; then inf rng(g|divset(D2,ID2)) <= sup rng g by A3,Lm7; then A126: (inf rng(g|divset(D2,ID2)))*vol(divset(D2,ID2)) <=(sup rng g)*vol(divset(D2,ID2)) by A120,AXIOMS:25; ID1 in dom D2 by A94,FINSEQ_1:def 3; then divset(D2,ID1) c= A by INTEGRA1:10; then inf rng(g|divset(D2,ID1)) <= sup rng g by A3,Lm7; then A127: (inf rng(g|divset(D2,ID1)))*vol(divset(D2,ID1)) <=(sup rng g)*vol(divset(D2,ID1)) by A119,AXIOMS:25; divset(D1,j)c=A by A6,INTEGRA1:10; then inf rng(g|divset(D1,j)) >= inf rng g by A3,Lm7; then A128: (inf rng(g|divset(D1,j)))*vol(divset(D2,ID2)) >=(inf rng g)*vol(divset(D2,ID2)) & (inf rng(g|divset(D1,j)))*vol(divset(D2,ID1)) >=(inf rng g)*vol(divset(D2,ID1)) by A119,A120,AXIOMS:25; Sum mid(LVg(D2),ID1,indx(D2,D1,j)) -(inf rng(g|divset(D2,ID1)))*vol(divset(D2,ID1)) <=(sup rng g)*vol(divset(D2,ID2)) by A125,A126,XCMPLX_1:26; then Sum mid(LVg(D2),ID1,indx(D2,D1,j)) <=(sup rng g)*vol(divset(D2,ID2)) +(inf rng(g|divset(D2,ID1)))*vol(divset(D2,ID1)) by REAL_1:86; then Sum mid(LVg(D2),ID1,indx(D2,D1,j))-(sup rng g)*vol(divset(D2,ID2)) <=(inf rng(g|divset(D2,ID1)))*vol(divset(D2,ID1)) by REAL_1:86; then Sum mid(LVg(D2),ID1,indx(D2,D1,j))-(sup rng g)*vol(divset(D2,ID2)) <=(sup rng g)*vol(divset(D2,ID1)) by A127,AXIOMS:22; then A129: Sum mid(LVg(D2),ID1,indx(D2,D1,j))<=(sup rng g)*vol(divset(D2,ID2)) + (sup rng g)*vol(divset(D2,ID1)) by REAL_1:86; set IR = (inf rng g)*vol(divset(D2,ID2)); Sum mid(LVg(D1),j,j)=(inf rng(g|divset(D1,j))) *(vol(divset(D2,indx(D2,D1,j1)+1)) +vol(divset(D2,indx(D2,D1,j1)+2))) by A93,A118,RVSUM_1:103 .=(inf rng(g|divset(D1,j)))*vol(divset(D2,indx(D2,D1,j1)+2)) +(inf rng(g|divset(D1,j)))*vol(divset(D2,indx(D2,D1,j1)+1)) by XCMPLX_1:8; then Sum mid(LVg(D1),j,j)-(inf rng(g|divset(D1,j)))*vol(divset(D2,ID1)) >=IR by A128,XCMPLX_1:26; then Sum mid(LVg(D1),j,j)>=(inf rng(g|divset(D1,j)))*vol(divset(D2,ID1)) +IR by REAL_1:84; then Sum mid(LVg(D1),j,j)-(inf rng g)*vol(divset(D2,ID2)) >=(inf rng(g|divset(D1,j)))*vol(divset(D2,ID1)) by REAL_1:84; then Sum mid(LVg(D1),j,j)-(inf rng g)*vol(divset(D2,ID2)) >=(inf rng g)*vol(divset(D2,ID1)) by A128,AXIOMS:22; then Sum mid(LVg(D1),j,j) >=IR+(inf rng g)*vol(divset(D2,ID1)) by REAL_1:84; then Sum mid(LVg(D2),ID1,indx(D2,D1,j))-Sum mid(LVg(D1),j,j) <=(sup rng g)*vol(divset(D2,ID2))+(sup rng g)*vol(divset(D2,ID1)) -(IR+(inf rng g)*vol(divset(D2,ID1))) by A129,REAL_1:92; then Sum mid(LVg(D2),ID1,indx(D2,D1,j))-Sum mid(LVg(D1),j,j) <=((sup rng g)*vol(divset(D2,ID2))-(inf rng g)*vol(divset(D2,ID2))) +((sup rng g)*vol(divset(D2,ID1))-(inf rng g)*vol(divset(D2,ID1))) by Lm5; then Sum mid(LVg(D2),ID1,indx(D2,D1,j))-Sum mid(LVg(D1),j,j) <=(sup rng g - inf rng g)*vol(divset(D2,ID2)) +((sup rng g)*vol(divset(D2,ID1))-(inf rng g)*vol(divset(D2,ID1))) by XCMPLX_1:40; then Sum mid(LVg(D2),ID1,indx(D2,D1,j))-Sum mid(LVg(D1),j,j) <=(sup rng g - inf rng g)*vol(divset(D2,ID2)) +(sup rng g - inf rng g)*vol(divset(D2,ID1)) by XCMPLX_1:40; hence thesis by XCMPLX_1:8; end; A130: vol(divset(D1,j)) <= delta(D1) by A6,Lm8; sup rng g - inf rng g >= 0 by A8,SQUARE_1:12; then (sup rng g - inf rng g)*(vol(divset(D1,j))) <=(sup rng g - inf rng g)*delta(D1) by A130,AXIOMS:25; hence thesis by A108,A121,AXIOMS:22; end; hence thesis; end; A131:PLg(D2,indx(D2,D1,j1)) +Sum mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =PLg(D2,indx(D2,D1,j)) proof indx(D2,D1,j) in dom D2 by A2,A6,INTEGRA1:def 21; then A132: indx(D2,D1,j) in Seg len D2 by FINSEQ_1:def 3; then A133:1 <= indx(D2,D1,j) & indx(D2,D1,j) <= len D2 by FINSEQ_1:3; then A134: indx(D2,D1,j) <= len LVg(D2) by INTEGRA1:def 8; A135: indx(D2,D1,j) in Seg len LVg(D2) by A132,INTEGRA1:def 8; A136: indx(D2,D1,j1) < indx(D2,D1,j) by A69,NAT_1:38; indx(D2,D1,j1) in Seg len D2 by A18,FINSEQ_1:def 3; then indx(D2,D1,j1) in Seg len LVg(D2) by INTEGRA1:def 8; then PLg(D2,indx(D2,D1,j1)) =Sum(LVg(D2)|indx(D2,D1,j1)) by INTEGRA1:def 22; then PLg(D2,indx(D2,D1,j1)) +Sum mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =Sum(LVg(D2)|indx(D2,D1,j1) ^mid(LVg(D2),(indx(D2,D1,j1)+1),indx(D2,D1,j))) by RVSUM_1:105 .=Sum(mid(LVg(D2),1,indx(D2,D1,j1)) ^mid(LVg(D2),indx(D2,D1,j1)+1,indx(D2,D1,j))) by A18,JORDAN3:25 .=Sum(mid(LVg(D2),1,indx(D2,D1,j))) by A18,A134,A136,INTEGRA2:4 .=Sum(LVg(D2)|indx(D2,D1,j)) by A133,JORDAN3:25; hence thesis by A135,INTEGRA1:def 22; end; PLg(D1,j1)+Sum mid(lower_volume(g,D1),j,j)=PLg(D1,j) proof A137: 1 <= j & j <= len LVg(D1) by A5,FINSEQ_1:3,INTEGRA1:def 8; A138: j in Seg len LVg(D1) by A5,INTEGRA1:def 8; A139: j1+1=j by XCMPLX_1:27; j < j+1 by NAT_1:38; then A140: j1 < j by REAL_1:84; j1 in Seg len D1 by A20,FINSEQ_1:def 3; then j1 in Seg len LVg(D1) by INTEGRA1:def 8; then PLg(D1,j1)=Sum(LVg(D1)|j1) by INTEGRA1:def 22; then PLg(D1,j1)+Sum mid(LVg(D1),j,j) =Sum((LVg(D1)|j1)^mid(LVg(D1),j,j)) by RVSUM_1:105 .=Sum(mid(LVg(D1),1,j1)^mid(LVg(D1),j1+1,j)) by A20,A139,JORDAN3:25 .=Sum(mid(LVg(D1),1,j)) by A20,A137,A140,INTEGRA2:4 .=Sum(LVg(D1)|j) by A137,JORDAN3:25; hence thesis by A138,INTEGRA1:def 22; end; then A141:PLg(D2,indx(D2,D1,j))-PLg(D1,j) =(PLg(D2,indx(D2,D1,j1))-PLg(D1,j1)) +(Sum mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,len D1)) -Sum mid(lower_volume(g,D1),j,j)) by A131,Lm5 .=0+(Sum mid(lower_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,len D1)) -Sum mid(lower_volume(g,D1),j,j)) by A68,XCMPLX_1:14; A142:PLg(D2,indx(D2,D1,j))=Sum(lower_volume(g,D2)|indx(D2,D1,j)) by A15,INTEGRA1:def 22 .=Sum lower_volume(g,D2) by A11,TOPREAL1:2; PLg(D1,j)=Sum(lower_volume(g,D1)|j) by A16,INTEGRA1:def 22 .=Sum lower_volume(g,D1) by A16,TOPREAL1:2; hence thesis by A70,A141,A142; end; theorem Th10: for A be closed-interval Subset of REAL, g be Function of A,REAL, D1,D2 be Element of divs A st x in divset(D1,len D1) & len D1 >= 2 & D1<=D2 & rng D2 = rng D1 \/ {x} & g is_bounded_on A holds Sum upper_volume(g,D1)-Sum upper_volume(g,D2)<=(sup rng g-inf rng g)*delta(D1) proof let A be closed-interval Subset of REAL; let g be Function of A,REAL; let D1,D2 be Element of divs A; assume A1:x in divset(D1,len D1) & len D1 >= 2; assume A2:D1<=D2 & rng D2 = rng D1 \/ {x}; assume A3:g is_bounded_on A; deffunc PUg(Element of divs A,Nat) = (PartSums(upper_volume(g,$1))).$2; deffunc UVg(Element of divs A) = upper_volume(g,$1); set j = len D1; len D1 <> 0 & len D2 <> 0 by FINSEQ_1:25; then A4:len D1 in Seg len D1 & len D2 in Seg len D2 by FINSEQ_1:5; then A5:len D1 in dom D1 & len D2 in dom D2 by FINSEQ_1:def 3; A6:len D1 <> 1 by A1; A7:sup rng g >= inf rng g by A3,Lm6; A8:indx(D2,D1,j) in dom D2 & D2.indx(D2,D1,j)=D1.(len D1) by A2,A5,INTEGRA1:def 21; then A9:indx(D2,D1,j) in Seg len D2 by FINSEQ_1:def 3; A10: indx(D2,D1,j) >= len upper_volume(g,D2) proof assume indx(D2,D1,j) < len upper_volume(g,D2); then indx(D2,D1,j) < len D2 by INTEGRA1:def 7; then A11:D1.(len D1) < D2.(len D2) by A5,A8,GOBOARD1:def 1; A12:not(D2.(len D2) in rng D1) proof assume A13:D2.(len D2) in rng D1; rng D1 c= A by INTEGRA1:def 2; then inf A <= D2.(len D2) & D2.(len D2) <= sup A by A13,INTEGRA2:1; hence contradiction by A11,INTEGRA1:def 2; end; D2.(len D2) in rng D2 by A5,FUNCT_1:def 5; then D2.(len D2) in rng D1 or D2.(len D2) in {x} by A2,XBOOLE_0:def 2; then D2.(len D2) = x by A12,TARSKI:def 1; then D2.(len D2) <= sup divset(D1,len D1) by A1,INTEGRA2:1; hence contradiction by A5,A6,A11,INTEGRA1:def 5; end; then A14:indx(D2,D1,j) in Seg len upper_volume(g,D2) & indx(D2,D1,j) >= len upper_volume(g,D2) by A9,INTEGRA1:def 7; A15:j in Seg len upper_volume(g,D1) & j >= len upper_volume(g,D1) by A4,INTEGRA1:def 7; A16:len D1-1 in NAT & len D1-1 in dom D1 by A5,A6,INTEGRA1:9; reconsider j1=len D1-1 as Element of NAT by A5,A6,INTEGRA1:9; A17:indx(D2,D1,j1) in dom D2 & 1 <= indx(D2,D1,j1) & indx(D2,D1,j1) <= len D2 proof thus indx(D2,D1,j1) in dom D2 by A2,A16,INTEGRA1:def 21; hence thesis by FINSEQ_3:27; end; then mid(D2,1,indx(D2,D1,j1)) is increasing by INTEGRA1:37; then A18:D2|indx(D2,D1,j1) is increasing by A17,JORDAN3:25; A19:j1 in dom D1 & 1 <= j1 & j1 <= len D1 by A16,FINSEQ_3:27; then mid(D1,1,j1) is increasing by INTEGRA1:37; then A20: D1|j1 is increasing by A19,JORDAN3:25; A21: rng (D2|indx(D2,D1,j1)) = rng (D1|j1) by A1,A2,Lm9; then A22:D2|indx(D2,D1,j1)=D1|j1 by A18,A20,Th5; A23:for k st 1 <= k & k <= j1 holds k=indx(D2,D1,k) proof let k; assume A24:1 <= k & k <= j1; assume A25:k<>indx(D2,D1,k); now per cases by A25,AXIOMS:21; suppose A26:k > indx(D2,D1,k); 1 <= k & k <= len D1 by A19,A24,AXIOMS:22; then A27: k in dom D1 by FINSEQ_3:27; then A28: indx(D2,D1,k) in dom D2 & D1.k=D2.indx(D2,D1,k) by A2,INTEGRA1:def 21; then indx(D2,D1,k) in Seg len D2 by FINSEQ_1:def 3; then A29: 1<=indx(D2,D1,k)&indx(D2,D1,k)<=indx(D2,D1,j1) by A2,A16,A24,A27,Th6,FINSEQ_1:3; then indx(D2,D1,k) in Seg indx(D2,D1,j1) by FINSEQ_1:3; then A30: (D2|indx(D2,D1,j1)).indx(D2,D1,k)=D2.indx(D2,D1,k) by A17,RFINSEQ: 19; A31: indx(D2,D1,k) < j1 by A24,A26,AXIOMS:22; then indx(D2,D1,k) <= len D1 by A19,AXIOMS:22; then indx(D2,D1,k) in Seg len D1 by A29,FINSEQ_1:3; then indx(D2,D1,k) in dom D1 by FINSEQ_1:def 3; then A32: D1.k > D1.indx(D2,D1,k) by A26,A27,GOBOARD1:def 1; indx(D2,D1,k) in Seg j1 by A29,A31,FINSEQ_1:3; hence contradiction by A16,A22,A28,A30,A32,RFINSEQ:19; suppose A33:k < indx(D2,D1,k); k in Seg j1 by A24,FINSEQ_1:3; then A34: D1.k = (D1|j1).k by A16,RFINSEQ:19; 1 <= k & k <= len D1 by A19,A24,AXIOMS:22; then A35: k in dom D1 by FINSEQ_3:27; then A36: indx(D2,D1,k) in dom D2 & D1.k=D2.indx(D2,D1,k) by A2,INTEGRA1:def 21; indx(D2,D1,k) <= indx(D2,D1,j1) & indx(D2,D1,k) in dom D2 & indx(D2,D1,j1) in dom D2 by A2,A16,A24,A35,Th6; then A37: k <= indx(D2,D1,j1) by A33,AXIOMS:22; then k <= len D2 by A17,AXIOMS:22; then k in dom D2 by A24,FINSEQ_3:27; then A38: D2.k < D2.indx(D2,D1,k) by A33,A36,GOBOARD1:def 1; k in Seg indx(D2,D1,j1) by A24,A37,FINSEQ_1:3; hence contradiction by A17,A22,A34,A36,A38,RFINSEQ:19; end; hence contradiction; end; A39: len (D2|indx(D2,D1,j1))=len (D1|j1) by A18,A20,A21,Th5; len (D2|indx(D2,D1,j1))=indx(D2,D1,j1) by A17,TOPREAL1:3; then A40: indx(D2,D1,j1) =j1 by A19,A39,TOPREAL1:3; j1 in Seg len D1 by A16,FINSEQ_1:def 3; then j1 <= len D1 by FINSEQ_1:3; then j1 <= len upper_volume(g,D1) by INTEGRA1:def 7; then A41:len(upper_volume(g,D1)|j1)=indx(D2,D1,j1) by A40,TOPREAL1:3; indx(D2,D1,j1) in dom D2 by A2,A16,INTEGRA1:def 21; then indx(D2,D1,j1) <= len D2 by FINSEQ_3:27; then indx(D2,D1,j1) <= len upper_volume(g,D2) by INTEGRA1:def 7; then A42:len(upper_volume(g,D1)|j1)=len(upper_volume(g,D2)|indx(D2,D1,j1)) by A41,TOPREAL1:3; for k st 1 <= k & k <= len(upper_volume(g,D1)|j1) holds (upper_volume(g,D1)|j1).k = (upper_volume(g,D2)|indx(D2,D1,j1)).k proof let k; assume A43:1 <= k & k <= len(upper_volume(g,D1)|j1); A44: len(upper_volume(g,D1)) = len D1 by INTEGRA1:def 7; then A45:1 <= k & k <= j1 by A19,A43,TOPREAL1:3; then A46: k in Seg j1 by FINSEQ_1:3; 1 <= k & k <= len D1 by A19,A45,AXIOMS:22; then A47:k in Seg len D1 by FINSEQ_1:3; A48:divset(D1,k)=divset(D2,indx(D2,D1,k)) proof A49: divset(D1,k)=[. inf divset(D1,k), sup divset(D1,k) .] by INTEGRA1:5; A50: k in dom D1 by A47,FINSEQ_1:def 3; then A51: indx(D2,D1,k) in dom D2 & D1.k=D2.indx(D2,D1,k) by A2,INTEGRA1:def 21; inf divset(D1,k)=inf divset(D2,indx(D2,D1,k)) & sup divset(D1,k)=sup divset(D2,indx(D2,D1,k)) proof per cases; suppose A52:k=1; then A53: inf divset(D1,k)=inf A&sup divset(D1,k)=D1.k by A50,INTEGRA1:def 5; indx(D2,D1,k)=1 by A19,A23,A52; hence thesis by A51,A53,INTEGRA1:def 5; suppose A54:k<>1; then A55: inf divset(D1,k)=D1.(k-1) & sup divset(D1,k)=D1.k by A50,INTEGRA1:def 5; A56: k-1 in dom D1 & D1.(k-1) in A & k-1 in NAT by A50,A54,INTEGRA1:9; reconsider k1=k-1 as Nat by A50,A54,INTEGRA1:9; k <= k+1 by NAT_1:29; then k1 <= k by REAL_1:86; then A57: k1 <= j1 by A45,AXIOMS:22; 1 <= k1 by A56,FINSEQ_3:27; then A58: k1=indx(D2,D1,k1) by A23,A57; indx(D2,D1,k)<>1 by A23,A45,A54; then A59: inf divset(D2,indx(D2,D1,k))=D2.(indx(D2,D1,k)-1) & sup divset(D2,indx(D2,D1,k))=D2.indx(D2,D1,k) by A51,INTEGRA1:def 5; D2.(indx(D2,D1,k)-1)= D2.indx(D2,D1,k1) by A23,A45,A58; hence thesis by A2,A50,A55,A56,A59,INTEGRA1:def 21; end; hence thesis by A49,INTEGRA1:5; end; j1 in Seg len(upper_volume(g,D1)) by A16,A44,FINSEQ_1:def 3; then j1 in dom(upper_volume(g,D1)) by FINSEQ_1:def 3; then A60: (upper_volume(g,D1)|j1).k = upper_volume(g,D1).k by A46,RFINSEQ:19 .=(sup(rng(g|divset(D2,indx(D2,D1,k)))))*vol(divset(D2,indx(D2,D1,k))) by A47,A48,INTEGRA1:def 7; indx(D2,D1,k) in Seg j1 by A23,A45,A46; then A61:indx(D2,D1,k) in Seg indx(D2,D1,j1) by A19,A23; then 1<=indx(D2,D1,k) & indx(D2,D1,k)<=indx(D2,D1,j1) by FINSEQ_1:3; then 1<=indx(D2,D1,k) & indx(D2,D1,k)<=len D2 by A17,AXIOMS:22; then A62:indx(D2,D1,k) in Seg len D2 by FINSEQ_1:3; indx(D2,D1,j1) in Seg len D2 by A17,FINSEQ_1:def 3; then indx(D2,D1,j1) in Seg len upper_volume(g,D2) by INTEGRA1:def 7; then A63:indx(D2,D1,j1) in dom upper_volume(g,D2) by FINSEQ_1:def 3; (upper_volume(g,D2)|indx(D2,D1,j1)).k =(upper_volume(g,D2)|indx(D2,D1,j1)).indx(D2,D1,k) by A23,A45 .=upper_volume(g,D2).indx(D2,D1,k) by A61,A63,RFINSEQ:19 .=(sup(rng(g|divset(D2,indx(D2,D1,k)))))*vol(divset(D2,indx(D2,D1,k))) by A62,INTEGRA1:def 7; hence thesis by A60; end; then A64:upper_volume(g,D2)|indx(D2,D1,j1)=upper_volume(g,D1)|j1 by A42,FINSEQ_1:18; indx(D2,D1,j1) in Seg len D2 by A17,FINSEQ_1:def 3; then A65:indx(D2,D1,j1) in Seg len upper_volume(g,D2) by INTEGRA1:def 7; j1 in Seg len D1 by A16,FINSEQ_1:def 3; then A66:j1 in Seg len upper_volume(g,D1) by INTEGRA1:def 7; A67:PUg(D2,indx(D2,D1,j1))=Sum(upper_volume(g,D2)|indx(D2,D1,j1)) by A65,INTEGRA1:def 22 .=PUg(D1,j1) by A64,A66,INTEGRA1:def 22; A68:indx(D2,D1,j1)+1 <= indx(D2,D1,len D1) proof len D1 < len D1+1 by NAT_1:38; then j1 < len D1 by REAL_1:84; then indx(D2,D1,j1) < indx(D2,D1,len D1) by A2,A5,A16,Th7; hence thesis by NAT_1:38; end; A69:Sum mid(upper_volume(g,D1),len D1,len D1) -Sum mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,len D1)) <= (sup rng g-inf rng g)*delta(D1) proof A70: 1 <= indx(D2,D1,j) & indx(D2,D1,j) <= len upper_volume(g,D2) proof indx(D2,D1,j) in dom D2 by A2,A5,INTEGRA1:def 21; then indx(D2,D1,j) in Seg len D2 by FINSEQ_1:def 3; then indx(D2,D1,j) in Seg len upper_volume(g,D2) by INTEGRA1:def 7; hence thesis by FINSEQ_1:3; end; then A71: 1 <= indx(D2,D1,j1)+1 & indx(D2,D1,j1)+1 <= len upper_volume(g,D2) by A17,A68,AXIOMS:22,NAT_1:38; A72:indx(D2,D1,j)-indx(D2,D1,j1) <= 2 proof assume A73:indx(D2,D1,j)-indx(D2,D1,j1) > 2; reconsider ID1=indx(D2,D1,j1)+1 as Nat; reconsider ID2=ID1+1 as Nat; A74:indx(D2,D1,j1) < ID1 & ID1 < ID2 & ID2 < indx(D2,D1,j) proof thus indx(D2,D1,j1) < ID1 by NAT_1:38; thus ID1 < ID2 by NAT_1:38; indx(D2,D1,j1)+(1+1)<indx(D2,D1,j) by A73,REAL_1:86; hence thesis by XCMPLX_1:1; end; A75:indx(D2,D1,j) in dom D2 by A2,A5,INTEGRA1:def 21; then A76:indx(D2,D1,j) <= len D2 by FINSEQ_3:27; A77:ID1 in dom D2 proof A78: 1 <= ID1 & ID1 <= indx(D2,D1,j) by A17,A74,AXIOMS:22; then ID1 <= len D2 by A76,AXIOMS:22; hence thesis by A78,FINSEQ_3:27; end; A79:ID2 in dom D2 proof A80: indx(D2,D1,j1) <= ID2 & ID2 <= len D2 by A74,A76,AXIOMS:22; then 1 <= ID2 by A17,AXIOMS:22; hence thesis by A80,FINSEQ_3:27; end; then A81:D2.indx(D2,D1,j1)<D2.ID1 & D2.ID1<D2.ID2 & D2.ID2<D2.indx(D2,D1,j) by A17,A74,A75,A77,GOBOARD1:def 1; A82:D2.ID1 in {x} & D2.ID2 in {x} proof A83: D2.ID1 in rng D2 & D2.ID2 in rng D2 by A77,A79,FUNCT_1:def 5; A84: D1.j1 = D2.indx(D2,D1,j1) & D1.j = D2.indx(D2,D1,j) by A2,A5,A16,INTEGRA1:def 21; not(D2.ID1 in rng D1) & not(D2.ID2 in rng D1) proof assume A85:D2.ID1 in rng D1 or D2.ID2 in rng D1; now per cases by A85; suppose D2.ID1 in rng D1; then consider n such that A86: n in dom D1 & D1.n=D2.ID1 by PARTFUN1:26; D2.indx(D2,D1,j1)<D2.ID1 & D2.ID1<D2.indx(D2,D1,j) by A81,AXIOMS:22; then j1<n & n<j by A5,A16,A84,A86,GOBOARD2:18; then j<n+1 & n<j by REAL_1:84; hence contradiction by NAT_1:38; suppose D2.ID2 in rng D1; then consider n such that A87: n in dom D1 & D1.n=D2.ID2 by PARTFUN1:26; D2.indx(D2,D1,j1)<D2.ID2 & D2.ID2<D2.indx(D2,D1,j) by A81,AXIOMS:22; then j1<n & n<j by A5,A16,A84,A87,GOBOARD2:18; then j<n+1 & n<j by REAL_1:84; hence contradiction by NAT_1:38; end; hence thesis; end; hence thesis by A2,A83,XBOOLE_0:def 2; end; then D2.ID1 = x by TARSKI:def 1; then D2.ID1=D2.ID2 by A82,TARSKI:def 1; hence contradiction by A74,A77,A79,GOBOARD2:19; end; 1 <= indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 & indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 <= 2 proof A88: indx(D2,D1,j)-'(indx(D2,D1,j1)+1) =indx(D2,D1,j)-(indx(D2,D1,j1)+1) by A68,SCMFSA_7:3; then indx(D2,D1,j)-'(indx(D2,D1,j1)+1) >= 0 by A68,SQUARE_1:12; then indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 >= 0+1 by AXIOMS:24; hence indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 >= 1; indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 =indx(D2,D1,j)-indx(D2,D1,j1)-1+1 by A88,XCMPLX_1:36 .=indx(D2,D1,j)-indx(D2,D1,j1)+1-1 by XCMPLX_1:29 .=indx(D2,D1,j)-indx(D2,D1,j1) by XCMPLX_1:26; hence thesis by A72; end; then A89: 1 <= len mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) & len mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j))<=2 by A68,A70,A71,JORDAN3:27; A90: 1 <= j & j <= len upper_volume(g,D1) by A4,FINSEQ_1:3,INTEGRA1:def 7; j-'j+1 = 1 by Lm3; then A91: len mid(upper_volume(g,D1),j,j)=1 by A90,JORDAN3:27; mid(upper_volume(g,D1),j,j).1 = upper_volume(g,D1).j by A90,JORDAN3:27; then mid(upper_volume(g,D1),j,j)=<*upper_volume(g,D1).j*> by A91,FINSEQ_1: 57; then A92:Sum mid(upper_volume(g,D1),j,j)=upper_volume(g,D1).j by RVSUM_1:103; A93:indx(D2,D1,j1)+1 in Seg len D2 proof indx(D2,D1,j1)+1 in Seg len upper_volume(g,D2) by A71,FINSEQ_1:3; hence thesis by INTEGRA1:def 7; end; now per cases by A89,Lm4; suppose A94: len mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j))=1; A95: indx(D2,D1,j1)+1=indx(D2,D1,j) proof len mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) = indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 by A68,A70,A71,JORDAN3:27; then indx(D2,D1,j)-'(indx(D2,D1,j1)+1)=1-1 by A94,XCMPLX_1:26; then indx(D2,D1,j)-(indx(D2,D1,j1)+1)=0 by A68,SCMFSA_7:3; then indx(D2,D1,j)=0+(indx(D2,D1,j1)+1) by XCMPLX_1:27; hence thesis; end; A96: divset(D2,indx(D2,D1,j))=divset(D1,j) proof inf divset(D1,j)=D1.j1 & sup divset(D1,j)=D1.j by A5,A6,INTEGRA1:def 5; then inf divset(D1,j)=D2.indx(D2,D1,j1) & sup divset(D1,j)=D2.indx(D2,D1,j) by A2,A5,A16,INTEGRA1:def 21; then A97: divset(D1,j)=[. D2.indx(D2,D1,j1),D2.indx(D2,D1,j).] by INTEGRA1:5; A98: indx(D2,D1,j)-1=indx(D2,D1,j1) by A95,XCMPLX_1:26; A99: indx(D2,D1,j) in dom D2 by A2,A5,INTEGRA1:def 21; indx(D2,D1,j)<>1 by A17,A95,NAT_1:38; then inf divset(D2,indx(D2,D1,j))=D2.(indx(D2,D1,j)-1) & sup divset(D2,indx(D2,D1,j))=D2.indx(D2,D1,j) by A99,INTEGRA1:def 5; hence thesis by A97,A98,INTEGRA1:5; end; mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)).1 =upper_volume(g,D2).(indx(D2,D1,j1)+1) by A70,A71,JORDAN3:27; then mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =<*upper_volume(g,D2).(indx(D2,D1,j1)+1)*> by A94,FINSEQ_1:57; then A100: Sum mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =upper_volume(g,D2).(indx(D2,D1,j1)+1) by RVSUM_1:103 .=(sup(rng(g|divset(D2,(indx(D2,D1,j1)+1))))) *vol(divset(D2,(indx(D2,D1,j1)+1))) by A93,INTEGRA1:def 7 .=Sum mid(upper_volume(g,D1),j,j) by A4,A92,A95,A96,INTEGRA1:def 7; A101: delta(D1) >= 0 by Th8; sup rng g - inf rng g >= 0 by A7,SQUARE_1:12; then (sup rng g - inf rng g)*delta(D1) >= 0*delta(D1) by A101,AXIOMS:25; hence Sum mid(upper_volume(g,D1),j,j)- Sum mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) <= (sup rng g-inf rng g)*delta(D1) by A100,XCMPLX_1:14; suppose A102: len mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j))=2; A103: mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)).1 =upper_volume(g,D2).(indx(D2,D1,j1)+1) by A70,A71,JORDAN3:27; mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)).2 =upper_volume(g,D2).(indx(D2,D1,j1)+2) proof A104: 2+(indx(D2,D1,j1)+1)>=0+1 by A71,REAL_1:55; mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)).2 =UVg(D2).(2+(indx(D2,D1,j1)+1)-'1) by A68,A70,A71,A102,JORDAN3:27 .=UVg(D2).(2+(indx(D2,D1,j1)+1)-1) by A104,SCMFSA_7:3 .=UVg(D2).((indx(D2,D1,j1)+1)+(2-1)) by XCMPLX_1:29 .=UVg(D2).(indx(D2,D1,j1)+(1+1)) by XCMPLX_1:1; hence thesis; end; then mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =<*upper_volume(g,D2).(indx(D2,D1,j1)+1), upper_volume(g,D2).(indx(D2,D1,j1)+2)*> by A102,A103,FINSEQ_1:61; then A105: Sum mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =upper_volume(g,D2).(indx(D2,D1,j1)+1) +upper_volume(g,D2).(indx(D2,D1,j1)+2) by RVSUM_1:107; A106: vol(divset(D1,j)) =vol(divset(D2,indx(D2,D1,j1)+1))+vol(divset(D2,indx(D2,D1,j1)+2)) proof A107: inf divset(D1,j)=D2.indx(D2,D1,j1)&sup divset(D1,j)=D2.indx(D2,D1,j) proof inf divset(D1,j)=D1.j1 & sup divset(D1,j)=D1.j by A5,A6,INTEGRA1:def 5 ; hence thesis by A2,A5,A16,INTEGRA1:def 21; end; A108: indx(D2,D1,j)=indx(D2,D1,j1)+2 proof indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1=2 by A68,A70,A71,A102,JORDAN3:27; then indx(D2,D1,j)-(indx(D2,D1,j1)+1)+1=2 by A68,SCMFSA_7:3; then indx(D2,D1,j)-indx(D2,D1,j1)-1+1=2 by XCMPLX_1:36; then indx(D2,D1,j)-indx(D2,D1,j1)-(1-1)=2 by XCMPLX_1:37; hence thesis by XCMPLX_1:27; end; A109: inf divset(D2,(indx(D2,D1,j1)+2))=D2.(indx(D2,D1,j1)+1) & sup divset(D2,(indx(D2,D1,j1)+2))=D2.(indx(D2,D1,j1)+2) & inf divset(D2,(indx(D2,D1,j1)+1))=D2.indx(D2,D1,j1) & sup divset(D2,(indx(D2,D1,j1)+1))=D2.(indx(D2,D1,j1)+1) proof A110: indx(D2,D1,j1)+2 in dom D2 by A2,A5,A108,INTEGRA1:def 21; indx(D2,D1,j1)+1 in Seg len upper_volume(g,D2) by A71,FINSEQ_1:3; then indx(D2,D1,j1)+1 in Seg len D2 by INTEGRA1:def 7; then A111: indx(D2,D1,j1)+1 in dom D2 by FINSEQ_1:def 3; A112: indx(D2,D1,j1)+1 <> 1 by A17,NAT_1:38; indx(D2,D1,j1)+1+1 > 1 by A71,NAT_1:38; then A113: indx(D2,D1,j1)+(1+1) > 1 by XCMPLX_1:1; A114: indx(D2,D1,j1)+2-1=indx(D2,D1,j1)+1 by Lm2,XCMPLX_1:29; A115: indx(D2,D1,j1)+1-1=indx(D2,D1,j1)+(1-1) by XCMPLX_1:29 .=indx(D2,D1,j1)+0; thus inf divset(D2,(indx(D2,D1,j1)+2))=D2.(indx(D2,D1,j1)+1) & sup divset(D2,(indx(D2,D1,j1)+2))=D2.(indx(D2,D1,j1)+2) by A110,A113,A114,INTEGRA1:def 5; thus thesis by A111,A112,A115,INTEGRA1:def 5; end; vol(divset(D1,j)) =sup divset(D1,j)-inf divset(D1,j) by INTEGRA1:def 6 .=D2.indx(D2,D1,j)-(D2.(indx(D2,D1,j1)+1) -D2.(indx(D2,D1,j1)+1))-D2.indx(D2,D1,j1) by A107,XCMPLX_1:17 .=D2.(indx(D2,D1,j1)+2)-D2.(indx(D2,D1,j1)+1) +D2.(indx(D2,D1,j1)+1)-D2.indx(D2,D1,j1) by A108,XCMPLX_1:37; then vol(divset(D1,j)) =vol(divset(D2,indx(D2,D1,j1)+2)) +D2.(indx(D2,D1,j1)+1)-D2.indx(D2,D1,j1) by A109,INTEGRA1:def 6 .=vol(divset(D2,indx(D2,D1,j1)+2)) +(sup divset(D2,indx(D2,D1,j1)+1)-inf divset(D2,indx(D2,D1,j1)+1)) by A109,XCMPLX_1:29; hence thesis by INTEGRA1:def 6; end; then A116: upper_volume(g,D1).j=(sup(rng(g|divset(D1,j))))* (vol(divset(D2,indx(D2,D1,j1)+1))+vol(divset(D2,indx(D2,D1,j1)+2))) by A4,INTEGRA1:def 7; A117: vol(divset(D2,indx(D2,D1,j1)+1))>=0 by INTEGRA1:11; A118: vol(divset(D2,indx(D2,D1,j1)+2))>=0 by INTEGRA1:11; A119: Sum mid(UVg(D1),j,j)-Sum mid(UVg(D2),indx(D2,D1,j1)+1,indx(D2,D1,j)) <=(sup rng g - inf rng g)*(vol(divset(D2,indx(D2,D1,j1)+2)) +vol(divset(D2,indx(D2,D1,j1)+1))) proof A120: indx(D2,D1,j) in dom D2 by A2,A5,INTEGRA1:def 21; then A121: indx(D2,D1,j) in Seg len D2 by FINSEQ_1:def 3; A122: indx(D2,D1,j)=indx(D2,D1,j1)+2 proof indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1=2 by A68,A70,A71,A102,JORDAN3:27; then indx(D2,D1,j)-(indx(D2,D1,j1)+1)+1=2 by A68,SCMFSA_7:3; then indx(D2,D1,j)-indx(D2,D1,j1)-1+1=2 by XCMPLX_1:36; then indx(D2,D1,j)-indx(D2,D1,j1)-(1-1)=2 by XCMPLX_1:37; hence thesis by XCMPLX_1:27; end; set ID1=indx(D2,D1,j1)+1, ID2=indx(D2,D1,j1)+2; A123: Sum mid(UVg(D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =(sup rng(g|divset(D2,indx(D2,D1,j1)+2))) *vol(divset(D2,indx(D2,D1,j1)+2)) +UVg(D2).(indx(D2,D1,j1)+1) by A105,A121,A122,INTEGRA1:def 7 .=(sup rng(g|divset(D2,indx(D2,D1,j1)+2))) *vol(divset(D2,indx(D2,D1,j1)+2)) +(sup rng(g|divset(D2,indx(D2,D1,j1)+1))) *vol(divset(D2,indx(D2,D1,j1)+1)) by A93,INTEGRA1:def 7; divset(D2,ID2)c=A by A120,A122,INTEGRA1:10; then sup rng(g|divset(D2,ID2)) >= inf rng g by A3,Lm7; then A124: (sup rng(g|divset(D2,ID2)))*vol(divset(D2,ID2)) >=(inf rng g)*vol(divset(D2,ID2)) by A118,AXIOMS:25; ID1 in dom D2 by A93,FINSEQ_1:def 3; then divset(D2,ID1)c=A by INTEGRA1:10; then sup rng(g|divset(D2,ID1)) >= inf rng g by A3,Lm7; then A125: (sup rng(g|divset(D2,ID1)))*vol(divset(D2,ID1)) >=(inf rng g)*vol(divset(D2,ID1)) by A117,AXIOMS:25; divset(D1,j)c=A by A5,INTEGRA1:10; then sup rng(g|divset(D1,j)) <= sup rng g by A3,Lm7; then A126: (sup rng(g|divset(D1,j)))*vol(divset(D2,ID2)) <=(sup rng g)*vol(divset(D2,ID2)) & (sup rng(g|divset(D1,j)))*vol(divset(D2,ID1)) <=(sup rng g)*vol(divset(D2,ID1)) by A117,A118,AXIOMS:25; Sum mid(UVg(D2),ID1,indx(D2,D1,j)) -(sup rng(g|divset(D2,ID1)))*vol(divset(D2,ID1)) >=(inf rng g)*vol(divset(D2,ID2)) by A123,A124,XCMPLX_1:26; then Sum mid(UVg(D2),ID1,indx(D2,D1,j)) >=(inf rng g)*vol(divset(D2,ID2)) +(sup rng(g|divset(D2,ID1)))*vol(divset(D2,ID1)) by REAL_1:84; then Sum mid(UVg(D2),ID1,indx(D2,D1,j))-(inf rng g)*vol(divset(D2,ID2)) >=(sup rng(g|divset(D2,ID1)))*vol(divset(D2,ID1)) by REAL_1:84; then Sum mid(UVg(D2),ID1,indx(D2,D1,j))-(inf rng g)*vol(divset(D2,ID2)) >=(inf rng g)*vol(divset(D2,ID1)) by A125,AXIOMS:22; then A127: Sum mid(UVg(D2),ID1,indx(D2,D1,j))>=(inf rng g)*vol(divset(D2,ID2)) + (inf rng g)*vol(divset(D2,ID1)) by REAL_1:84; Sum mid(UVg(D1),j,j) =(sup rng(g|divset(D1,j)))*vol(divset(D2,indx(D2,D1,j1)+2)) +(sup rng(g|divset(D1,j)))*vol(divset(D2,indx(D2,D1,j1)+1)) by A92,A116,XCMPLX_1:8; then Sum mid(UVg(D1),j,j)-(sup rng(g|divset(D1,j)))*vol(divset(D2,ID1)) =(sup rng(g|divset(D1,j)))*vol(divset(D2,ID2)) by XCMPLX_1:26; then Sum mid(UVg(D1),j,j)<=(sup rng(g|divset(D1,j)))*vol(divset(D2,ID1)) +(sup rng g)*vol(divset(D2,ID2)) by A126,REAL_1:86; then Sum mid(UVg(D1),j,j)-(sup rng g)*vol(divset(D2,ID2)) <=(sup rng(g|divset(D1,j)))*vol(divset(D2,ID1)) by REAL_1:86; then Sum mid(UVg(D1),j,j)-(sup rng g)*vol(divset(D2,ID2)) <=(sup rng g)*vol(divset(D2,ID1)) by A126,AXIOMS:22; then Sum mid(UVg(D1),j,j) <=(sup rng g)*vol(divset(D2,ID2))+(sup rng g)*vol(divset(D2,ID1)) by REAL_1:86; then Sum mid(UVg(D1),j,j)-Sum mid(UVg(D2),ID1,indx(D2,D1,j)) <=(sup rng g)*vol(divset(D2,ID2))+(sup rng g)*vol(divset(D2,ID1)) -((inf rng g)*vol(divset(D2,ID2))+(inf rng g)*vol(divset(D2,ID1))) by A127,REAL_1:92; then Sum mid(UVg(D1),j,j)-Sum mid(UVg(D2),ID1,indx(D2,D1,j)) <=((sup rng g)*vol(divset(D2,ID2))-(inf rng g)*vol(divset(D2,ID2))) +((sup rng g)*vol(divset(D2,ID1))-(inf rng g)*vol(divset(D2,ID1))) by Lm5; then Sum mid(UVg(D1),j,j)-Sum mid(UVg(D2),ID1,indx(D2,D1,j)) <=(sup rng g - inf rng g)*vol(divset(D2,ID2)) +((sup rng g)*vol(divset(D2,ID1))-(inf rng g)*vol(divset(D2,ID1))) by XCMPLX_1:40; then Sum mid(UVg(D1),j,j)-Sum mid(UVg(D2),ID1,indx(D2,D1,j)) <=(sup rng g - inf rng g)*vol(divset(D2,ID2)) +(sup rng g - inf rng g)*vol(divset(D2,ID1)) by XCMPLX_1:40; hence thesis by XCMPLX_1:8; end; A128: vol(divset(D1,j)) <= delta(D1) by A5,Lm8; sup rng g - inf rng g >= 0 by A7,SQUARE_1:12; then (sup rng g - inf rng g)*(vol(divset(D1,j))) <=(sup rng g - inf rng g)*delta(D1) by A128,AXIOMS:25; hence thesis by A106,A119,AXIOMS:22; end; hence thesis; end; A129:PUg(D2,indx(D2,D1,j1)) +Sum mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =PUg(D2,indx(D2,D1,j)) proof indx(D2,D1,j) in dom D2 by A2,A5,INTEGRA1:def 21; then A130: indx(D2,D1,j) in Seg len D2 by FINSEQ_1:def 3; then A131: 1 <= indx(D2,D1,j) & indx(D2,D1,j) <= len D2 by FINSEQ_1:3; then A132: 1 <= indx(D2,D1,j) & indx(D2,D1,j) <= len UVg(D2) by INTEGRA1:def 7; A133: indx(D2,D1,j) in Seg len UVg(D2) by A130,INTEGRA1:def 7; A134: indx(D2,D1,j1) < indx(D2,D1,j) by A68,NAT_1:38; indx(D2,D1,j1) in Seg len D2 by A17,FINSEQ_1:def 3; then indx(D2,D1,j1) in Seg len UVg(D2) by INTEGRA1:def 7; then PUg(D2,indx(D2,D1,j1)) =Sum(UVg(D2)|indx(D2,D1,j1)) by INTEGRA1:def 22; then PUg(D2,indx(D2,D1,j1)) +Sum mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =Sum(UVg(D2)|indx(D2,D1,j1) ^mid(UVg(D2),(indx(D2,D1,j1)+1),indx(D2,D1,j))) by RVSUM_1:105 .=Sum(mid(UVg(D2),1,indx(D2,D1,j1)) ^mid(UVg(D2),indx(D2,D1,j1)+1,indx(D2,D1,j))) by A17,JORDAN3:25 .=Sum(mid(UVg(D2),1,indx(D2,D1,j))) by A17,A132,A134,INTEGRA2:4 .=Sum(UVg(D2)|indx(D2,D1,j)) by A131,JORDAN3:25; hence thesis by A133,INTEGRA1:def 22; end; PUg(D1,j1)+Sum mid(upper_volume(g,D1),j,j)=PUg(D1,j) proof A135: 1 <= j & j <= len UVg(D1) by A4,FINSEQ_1:3,INTEGRA1:def 7; A136: j in Seg len UVg(D1) by A4,INTEGRA1:def 7; A137: j1+1=j by XCMPLX_1:27; j < j+1 by NAT_1:38; then A138: j1 < j by REAL_1:84; j1 in Seg len D1 by A16,FINSEQ_1:def 3; then j1 in Seg len UVg(D1) by INTEGRA1:def 7; then PUg(D1,j1)=Sum(UVg(D1)|j1) by INTEGRA1:def 22; then PUg(D1,j1)+Sum mid(UVg(D1),j,j) =Sum((UVg(D1)|j1)^mid(UVg(D1),j,j)) by RVSUM_1:105 .=Sum(mid(UVg(D1),1,j1)^mid(UVg(D1),j1+1,j)) by A19,A137,JORDAN3:25 .=Sum(mid(UVg(D1),1,j)) by A19,A135,A138,INTEGRA2:4 .=Sum(UVg(D1)|j) by A135,JORDAN3:25; hence thesis by A136,INTEGRA1:def 22; end; then A139:PUg(D1,j)-PUg(D2,indx(D2,D1,j)) =(PUg(D1,j1)-PUg(D2,indx(D2,D1,j1))) +(Sum mid(upper_volume(g,D1),j,j) -Sum mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,len D1))) by A129,Lm5 .=0+(Sum mid(upper_volume(g,D1),j,j)- Sum mid(upper_volume(g,D2),(indx(D2,D1,j1)+1),indx(D2,D1,len D1))) by A67,XCMPLX_1:14; A140:PUg(D2,indx(D2,D1,j))=Sum(upper_volume(g,D2)|indx(D2,D1,j)) by A14,INTEGRA1:def 22 .=Sum upper_volume(g,D2) by A10,TOPREAL1:2; PUg(D1,j)=Sum(upper_volume(g,D1)|j) by A15,INTEGRA1:def 22 .=Sum upper_volume(g,D1) by A15,TOPREAL1:2; hence thesis by A69,A139,A140; end; Lm10: for A be closed-interval Subset of REAL, f be PartFunc of A,REAL st vol(A)<>0 & y in rng lower_sum_set(f) holds ex D being Element of divs A st D in dom lower_sum_set(f) & y=(lower_sum_set(f)).D & D.1 > inf A proof let A be closed-interval Subset of REAL; let f be PartFunc of A,REAL; assume A1:vol(A)<>0; assume y in rng lower_sum_set(f); then consider D3 being Element of divs A such that A2:D3 in dom lower_sum_set(f) & y=(lower_sum_set(f)).D3 by PARTFUN1:26; len D3 <> 0 by FINSEQ_1:25; then A3:len D3 in Seg len D3 by FINSEQ_1:5; then 1 <= len D3 by FINSEQ_1:3; then A4:1 in Seg len D3 by FINSEQ_1:3; rng D3 <> {}; then A5:1 in dom D3 by FINSEQ_3:34; now per cases; suppose A6:D3.1 <> inf A; D3.1 in A by A5,INTEGRA1:8; then inf A <= D3.1 by INTEGRA2:1; then D3.1 > inf A by A6,REAL_1:def 5; hence ex D being Element of divs A st D in dom lower_sum_set(f) & y=(lower_sum_set(f)).D & D.1 > inf A by A2; suppose A7:D3.1 = inf A; ex D being Element of divs A st D in dom lower_sum_set(f) & y=(lower_sum_set(f)).D & D.1 > inf A proof A8: D3.(len D3) = sup A by INTEGRA1:def 2; A9: len D3 in dom D3 by A3,FINSEQ_1:def 3; vol(A) >= 0 by INTEGRA1:11; then sup A - inf A > 0 by A1,INTEGRA1:def 6; then sup A > inf A by REAL_2:106; then A10: len D3 > 1 by A5,A7,A8,A9,GOBOARD2:18; then reconsider D=D3/^1 as increasing FinSequence of REAL by INTEGRA1: 36; A11: len D = len D3 - 1 & for m be Nat st m in dom D holds D.m = D3.(m+1) by A10,RFINSEQ:def 2; then len D <> 0 by A10,SQUARE_1:11; then reconsider D as non empty increasing FinSequence of REAL by FINSEQ_1:25; rng D c= rng D3 & rng D3 c= A by FINSEQ_5:36,INTEGRA1:def 2; then A12: rng D c= A by XBOOLE_1:1; A13: len D+1=len D3 by A11,XCMPLX_1:27; len D in dom D by SCMFSA_7:12; then D.(len D)=sup A by A8,A10,A13,RFINSEQ:def 2; then D is DivisionPoint of A by A12,INTEGRA1:def 2; then reconsider D as Element of divs A by INTEGRA1:def 4; D in divs A; then A14: D in dom lower_sum_set(f) by INTEGRA1:def 12; A15: y=(lower_sum_set(f)).D proof A16: y=lower_sum(f,D3) by A2,INTEGRA1:def 12 .=Sum(lower_volume(f,D3)) by INTEGRA1:def 10 .=Sum((lower_volume(f,D3)|1)^(lower_volume(f,D3)/^1)) by RFINSEQ:21; A17: lower_volume(f,D3)|1 = <*lower_volume(f,D3).1*> proof A18: 1 <= len lower_volume(f,D3) by A10,INTEGRA1:def 8; then A19: len (lower_volume(f,D3)|1)=1 by TOPREAL1:3; A20: 1 in dom lower_volume(f,D3) by A18,FINSEQ_3:27; 1 in Seg 1 by FINSEQ_1:3; then (lower_volume(f,D3)|1).1 = lower_volume(f,D3).1 by A20,RFINSEQ: 19; hence thesis by A19,FINSEQ_1:57; end; A21: vol(divset(D3,1))=sup divset(D3,1)-inf divset(D3,1) by INTEGRA1:def 6 .=sup divset(D3,1) - inf A by A5,INTEGRA1:def 5 .=D3.1 - inf A by A5,INTEGRA1:def 5 .=0 by A7,XCMPLX_1:14; A22: lower_volume(f,D3).1=(inf rng(f|divset(D3,1)))*vol(divset(D3,1)) by A4,INTEGRA1:def 8; lower_volume(f,D3)/^1 = lower_volume(f,D) proof A23: 2-'1=2-1 by SCMFSA_7:3 .= 1; A24: len D3 >= 1+1 by A10,NAT_1:38; then len lower_volume(f,D3) >= 2 by INTEGRA1:def 8; then A25: mid(lower_volume(f,D3),2,len lower_volume(f,D3)) =lower_volume(f,D3)/^1 by A23,JORDAN3:26; A26: 2 <= len lower_volume(f,D3) & 1 <= len lower_volume(f,D3) by A10,A24,INTEGRA1:def 8; A27: len(mid(lower_volume(f,D3),2,len lower_volume(f,D3))) =len lower_volume(f,D) proof len(mid(lower_volume(f,D3),2,len lower_volume(f,D3))) =len lower_volume(f,D3)-'2+1 by A26,JORDAN3:27 .=len D3-'2+1 by INTEGRA1:def 8 .=len D3-2+1 by A24,SCMFSA_7:3 .=len D3-(2-1) by XCMPLX_1:37 .=len D3-1; hence thesis by A11,INTEGRA1:def 8; end; for i st 1<=i&i<=len mid(lower_volume(f,D3),2,len lower_volume(f,D3 ) ) holds mid(lower_volume(f,D3),2,len lower_volume(f,D3)).i =lower_volume(f,D).i proof let i; assume A28: 1<=i&i<=len mid(lower_volume(f,D3),2,len lower_volume(f,D3)); then A29: 1 <= i & i <= len D by A27,INTEGRA1:def 8; then 1<=i & i<=len lower_volume(f,D3) - 1 by A11,INTEGRA1:def 8; then 1 <= i & i <= len lower_volume(f,D3)-2+1 by Lm2,XCMPLX_1:37; then A30: mid(lower_volume(f,D3),2,len lower_volume(f,D3)).i =lower_volume(f,D3).(i+2-1) by A26,JORDAN3:31 .=lower_volume(f,D3).(i+(2-1)) by XCMPLX_1:29 .=lower_volume(f,D3).(i+1); 1 <= i+1 & i+1 <= len D3 by A11,A29,NAT_1:37,REAL_1:84; then A31: i+1 in Seg len D3 by FINSEQ_1:3; then A32: mid(lower_volume(f,D3),2,len lower_volume(f,D3)).i =(inf rng(f|divset(D3,i+1)))*vol(divset(D3,i+1)) by A30,INTEGRA1:def 8; A33: divset(D3,i+1)=divset(D,i) proof A34: 1<>i+1 by A28,NAT_1:38; i+1 in dom D3 by A31,FINSEQ_1:def 3; then A35: sup divset(D3,i+1)=D3.(i+1) & inf divset(D3,i+1)=D3.(i+1- 1) by A34,INTEGRA1:def 5; then A36: sup divset(D3,i+1)=D3.(i+1) & inf divset(D3,i+1)=D3.i by XCMPLX_1:26; A37: i in dom D by A29,FINSEQ_3:27; then A38: D.i=D3.(i+1) by A10,RFINSEQ:def 2; per cases; suppose A39:i=1; then A40: inf divset(D,i)=inf A & sup divset(D,i)=D.i by A37,INTEGRA1:def 5; divset(D3,i+1)=[.inf A, D.i.] by A7,A35,A38,A39,INTEGRA1:5; hence thesis by A40,INTEGRA1:5; suppose A41:i<>1; then A42: inf divset(D,i)=D.(i-1) & sup divset(D,i)=D.i by A37,INTEGRA1:def 5; i-1 in dom D & i-1 in NAT by A37,A41,INTEGRA1:9; then D.(i-1)=D3.(i-1+1) by A10,RFINSEQ:def 2 .= D3.(i-(1-1)) by XCMPLX_1:37 .=D3.i; then divset(D3,i+1)=[.inf divset(D,i),sup divset(D,i).] by A36,A38,A42,INTEGRA1:5; hence thesis by INTEGRA1:5; end; i in Seg len D by A29,FINSEQ_1:3; hence thesis by A32,A33,INTEGRA1:def 8; end; hence thesis by A25,A27,FINSEQ_1:18; end; then y=0+Sum(lower_volume(f,D)) by A16,A17,A21,A22,RVSUM_1:106 .=lower_sum(f,D) by INTEGRA1:def 10; hence thesis by A14,INTEGRA1:def 12; end; rng D <> {}; then 1 in dom D by FINSEQ_3:34; then A43: D.1=D3.(1+1) by A10,RFINSEQ:def 2 .=D3.2; 1+1 <= len D3 by A10,NAT_1:38; then 2 in dom D3 by FINSEQ_3:27; then D3.1 < D3.2 by A5,GOBOARD1:def 1; hence thesis by A7,A14,A15,A43; end; hence thesis; end; hence thesis; end; theorem Th11: for A be closed-interval Subset of REAL, D be Element of divs A, r be Real, i,j be Nat st i in dom D & j in dom D & i<=j & r < mid(D,i,j).1 holds ex B be closed-interval Subset of REAL st r=inf B & sup B=mid(D,i,j).(len mid(D,i,j)) & len mid(D,i,j)=j-i+1 & mid(D,i,j) is DivisionPoint of B proof let A be closed-interval Subset of REAL; let D be Element of divs A; let r be Real; let i,j be Nat; assume A1:i in dom D; assume A2:j in dom D; assume A3:i <= j; assume A4:r < mid(D,i,j).1; consider C being closed-interval Subset of REAL such that A5:inf C=mid(D,i,j).1 & sup C=mid(D,i,j).(len mid(D,i,j)) & len mid(D,i,j)=j-i+1 & mid(D,i,j) is DivisionPoint of C by A1,A2,A3,INTEGRA1:38; consider a,b such that A6:a <= b & a = inf C & b = sup C by INTEGRA1:4; r <= sup C by A4,A5,A6,AXIOMS:22; then reconsider B=[.r,sup C.] as closed-interval Subset of REAL by INTEGRA1: def 1; reconsider MD=mid(D,i,j) as non empty increasing FinSequence of REAL by A5; B=[.inf B,sup B.] by INTEGRA1:5; then A7:inf B = r & sup B = sup C by INTEGRA1:6; A8:rng mid(D,i,j) c= C&mid(D,i,j).(len mid(D,i,j))=sup C by A5,INTEGRA1:def 2; x in C implies x in B proof assume x in C; then inf C <= x & x <= sup C by INTEGRA2:1; then r <= x & x <= sup C by A4,A5,AXIOMS:22; hence thesis by A7,INTEGRA2:1; end; then C c= B by SUBSET_1:7; then rng mid(D,i,j) c= B by A8,XBOOLE_1:1; then MD is DivisionPoint of B by A5,A7,INTEGRA1:def 2; hence thesis by A5,A7; end; Lm11:for A be closed-interval Subset of REAL, D1 be Element of divs A st vol(A)<>0 & len D1 = 1 holds <*inf A*>^D1 is non empty increasing FinSequence of REAL proof let A be closed-interval Subset of REAL, D1 be Element of divs A; assume A1:vol(A)<>0; assume len D1 = 1; then A2: D1.1=sup A by INTEGRA1:def 2; vol(A) >= 0 by INTEGRA1:11; then D1.1 - inf A > 0 by A1,A2,INTEGRA1:def 6; then A3:inf A < D1.1 by REAL_2:106; set MD1 = <*inf A*>^D1; A4:len MD1 = len <*inf A*> + len D1 by FINSEQ_1:35; len <*inf A*> = 1 by FINSEQ_1:56; then A5: len MD1 <> 0 by A4,NAT_1:21; n in dom MD1 & m in dom MD1 & n<m implies MD1.n < MD1.m proof assume A6:n in dom MD1 & m in dom MD1 & n<m; A7: not m in dom <*inf A*> proof assume m in dom <*inf A*>; then m in Seg len <*inf A*> by FINSEQ_1:def 3; then m in {1} by FINSEQ_1:4,56; then A8: n < 1 by A6,TARSKI:def 1; n in Seg len MD1 by A6,FINSEQ_1:def 3; hence contradiction by A8,FINSEQ_1:3; end; MD1.m in rng MD1 by A6,FUNCT_1:def 5; then A9: MD1.m in rng <*inf A*> \/ rng D1 by FINSEQ_1:44; not(MD1.m in rng <*inf A*>) proof assume MD1.m in rng <*inf A*>; then MD1.m in {inf A} by FINSEQ_1:55; then A10: MD1.m = inf A by TARSKI:def 1; consider n such that A11: n in dom D1 & m=len <*inf A*>+n by A6,A7,FINSEQ_1:38; n in Seg len D1 by A11,FINSEQ_1:def 3; then A12: 1 <= n & n <= len D1 by FINSEQ_1:3; rng D1 <> {}; then A13: 1 in dom D1 by FINSEQ_3:34; D1.n=MD1.m by A11,FINSEQ_1:def 7; hence contradiction by A3,A10,A11,A12,A13,GOBOARD2:18; end; then A14: MD1.m in rng D1 by A9,XBOOLE_0:def 2; now per cases by A6,FINSEQ_1:38; suppose A15:n in dom <*inf A*>; then n in Seg len <*inf A*> by FINSEQ_1:def 3; then n in {1} by FINSEQ_1:4,56; then A16: n = 1 by TARSKI:def 1; consider k such that A17: k in dom D1 & MD1.m = D1.k by A14,PARTFUN1:26; A18: 1 <= k & k <= len D1 by A17,FINSEQ_3:27; rng D1 <> {}; then 1 in dom D1 by FINSEQ_3:34; then A19: D1.1 <= MD1.m by A17,A18,GOBOARD2:18; MD1.n = <*inf A*>.n by A15,FINSEQ_1:def 7 .= inf A by A16,FINSEQ_1:def 8 ; hence MD1.n < MD1.m by A3,A19,AXIOMS:22; suppose ex i st i in dom D1 & n = len <*inf A*> + i; then consider i such that A20: i in dom D1 & n=len <*inf A*> + i; consider j such that A21: j in dom D1 & m=len <*inf A*> + j by A6,A7,FINSEQ_1:38; A22: i < j by A6,A20,A21,AXIOMS:24; A23: D1.i=MD1.n by A20,FINSEQ_1:def 7; D1.j=MD1.m by A21,FINSEQ_1:def 7; hence thesis by A20,A21,A22,A23,GOBOARD1:def 1; end; hence thesis; end; hence thesis by A5,FINSEQ_1:25,GOBOARD1:def 1; end; Lm12:for A be closed-interval Subset of REAL, D2 be Element of divs A st inf A < D2.1 holds <*inf A*>^D2 is non empty increasing FinSequence of REAL proof let A be closed-interval Subset of REAL, D2 be Element of divs A; assume A1: inf A < D2.1; set MD2=<*inf A*>^D2; A2: len MD2 = 1+len D2 by FINSEQ_5:8; n in dom MD2 & m in dom MD2 & n<m implies MD2.n < MD2.m proof assume A3:n in dom MD2 & m in dom MD2 & n<m; A4: not m in dom <*inf A*> proof assume m in dom <*inf A*>; then m in Seg len <*inf A*> by FINSEQ_1:def 3; then m in {1} by FINSEQ_1:4,56; then A5: n < 1 by A3,TARSKI:def 1; n in Seg len MD2 by A3,FINSEQ_1:def 3; hence contradiction by A5,FINSEQ_1:3; end; MD2.m in rng MD2 by A3,FUNCT_1:def 5; then MD2.m in rng <*inf A*> \/ rng D2 by FINSEQ_1:44; then A6: MD2.m in rng <*inf A*> or MD2.m in rng D2 by XBOOLE_0:def 2; A7: not MD2.m in rng <*inf A*> proof assume MD2.m in rng <*inf A*>; then MD2.m in {inf A} by FINSEQ_1:55; then A8: MD2.m = inf A by TARSKI:def 1; consider n such that A9: n in dom D2 & m=len <*inf A*>+n by A3,A4,FINSEQ_1:38; n in Seg len D2 by A9,FINSEQ_1:def 3; then A10: 1 <= n & n <= len D2 by FINSEQ_1:3; rng D2 <> {}; then A11: 1 in dom D2 by FINSEQ_3:34; D2.n=MD2.m by A9,FINSEQ_1:def 7; hence contradiction by A1,A8,A9,A10,A11,GOBOARD2:18; end; now per cases by A3,FINSEQ_1:38; suppose A12:n in dom <*inf A*>; then n in Seg len <*inf A*> by FINSEQ_1:def 3; then n in {1} by FINSEQ_1:4,56; then A13: n = 1 by TARSKI:def 1; consider k such that A14: k in dom D2 & MD2.m = D2.k by A6,A7,PARTFUN1:26; k in Seg len D2 by A14,FINSEQ_1:def 3; then A15: 1 <= k & k <= len D2 by FINSEQ_1:3; rng D2 <> {}; then 1 in dom D2 by FINSEQ_3:34; then A16: D2.1 <= MD2.m by A14,A15,GOBOARD2:18; MD2.n = <*inf A*>.n by A12,FINSEQ_1:def 7 .= inf A by A13,FINSEQ_1:def 8 ; hence MD2.n < MD2.m by A1,A16,AXIOMS:22; suppose ex i st i in dom D2 & n = len <*inf A*> + i; then consider i such that A17: i in dom D2 & n=len <*inf A*> + i; consider j such that A18: j in dom D2 & m=len <*inf A*> + j by A3,A4,FINSEQ_1:38; A19: i < j by A3,A17,A18,AXIOMS:24; A20: D2.i=MD2.n by A17,FINSEQ_1:def 7; D2.j=MD2.m by A18,FINSEQ_1:def 7; hence thesis by A17,A18,A19,A20,GOBOARD1:def 1; end; hence thesis; end; hence thesis by A2,FINSEQ_1:25,GOBOARD1:def 1; end; Lm13:for A be closed-interval Subset of REAL, f being PartFunc of A,REAL, D1, MD1 being Element of divs A st MD1 = <*inf A*>^D1 holds (for i st i in Seg len D1 holds divset(MD1,i+1)=divset(D1,i)) & upper_volume(f,D1)=upper_volume(f,MD1)/^1 & lower_volume(f,D1)=lower_volume(f,MD1)/^1 proof let A be closed-interval Subset of REAL, f being PartFunc of A,REAL, D1, MD1 being Element of divs A; assume A1:MD1 = <*inf A*>^D1; then A2: len MD1=len <*inf A*> + len D1 by FINSEQ_1:35 .= 1+len D1 by FINSEQ_1:56; thus A3:for i st i in Seg len D1 holds divset(MD1,i+1)=divset(D1,i) proof let i; assume A4:i in Seg len D1; then A5: 1 <= i & i <= len D1 by FINSEQ_1:3; A6: i in dom D1 by A4,FINSEQ_1:def 3; A7: 1 <= i+1 by NAT_1:29; i+1 <= len D1+1 by A5,AXIOMS:24; then i+1 <= len D1 + len<*inf A*> by FINSEQ_1:56; then i+1 <= len MD1 by A1,FINSEQ_1:35; then A8: i+1 in dom MD1 by A7,FINSEQ_3:27; A9: divset(D1,i)=[.inf divset(D1,i), sup divset(D1,i).] by INTEGRA1:5; inf divset(D1,i)=inf divset(MD1,i+1) & sup divset(D1,i)=sup divset(MD1,i+1) proof per cases; suppose A10:i=1; i+1 > 1 by A5,NAT_1:38; then A11: inf divset(MD1,i+1)=MD1.(i+1-1) & sup divset(MD1,i+1)=MD1.(i+1) by A8,INTEGRA1:def 5; then A12: inf divset(MD1,i+1) = inf A by A1,A10,FINSEQ_1:58; MD1.(i+1) = MD1.(i+len <*inf A*>) by FINSEQ_1:57 .= D1.i by A1,A6,FINSEQ_1:def 7; hence thesis by A6,A10,A11,A12,INTEGRA1:def 5; suppose A13:i<>1; A14: i+1 > 1 by A5,NAT_1:38; i-1 in dom D1 & i-1 in NAT by A6,A13,INTEGRA1:9; then D1.(i-1) = MD1.(i-1+len <*inf A*>) by A1,FINSEQ_1:def 7 .=MD1.(i-1+1) by FINSEQ_1:56 .=MD1.(i-(1-1)) by XCMPLX_1:37 .=MD1.(i+1-1) by XCMPLX_1:26; then A15: inf divset(D1,i)=MD1.(i+1-1) by A6,A13,INTEGRA1:def 5 .=inf divset(MD1,i+1) by A8,A14,INTEGRA1:def 5; MD1.(i+1) = MD1.(i+len <*inf A*>) by FINSEQ_1:57 .= D1.i by A1,A6,FINSEQ_1:def 7; then sup divset(MD1,i+1)=D1.i by A8,A14,INTEGRA1:def 5 .= sup divset(D1,i) by A6,A13,INTEGRA1:def 5; hence thesis by A15; end; hence thesis by A9,INTEGRA1:5; end; thus upper_volume(f,D1)=upper_volume(f,MD1)/^1 proof set D2 = D1, MD2 = MD1; A16: len upper_volume(f,D2)=len (upper_volume(f,MD2)/^1) proof rng upper_volume(f,MD2) <> {}; then 1 in dom upper_volume(f,MD2) by FINSEQ_3:34; then 1 <= len upper_volume(f,MD2) by FINSEQ_3:27; then len (upper_volume(f,MD2)/^1)=len upper_volume(f,MD2)-1 by RFINSEQ: def 2 .=len MD2 -1 by INTEGRA1:def 7 .=len D2 by A2,XCMPLX_1:26; hence thesis by INTEGRA1:def 7; end; 1 <= k & k <= len upper_volume(f,D2) implies upper_volume(f,D2).k = (upper_volume(f,MD2)/^1).k proof assume A17:1 <= k & k <= len upper_volume(f,D2); then k in Seg len upper_volume(f,D2) by FINSEQ_1:3; then A18: k in Seg len D2 by INTEGRA1:def 7; then A19: upper_volume(f,D2).k=(sup rng(f|divset(D2,k)))*vol(divset(D2,k)) by INTEGRA1:def 7 .=(sup rng(f|divset(MD2,k+1)))*vol(divset(D2,k)) by A3,A18 .=(sup rng(f|divset(MD2,k+1)))*vol(divset(MD2,k+1)) by A3,A18; A20: 1 <= k+1 by NAT_1:29; k+1 <= len upper_volume(f,D2)+1 by A17,AXIOMS:24; then k+1 <= len D2+1 by INTEGRA1:def 7; then A21: k+1 in Seg len MD2 by A2,A20,FINSEQ_1:3; A22: k in dom (upper_volume(f,MD2)/^1) by A16,A17,FINSEQ_3:27; A23: len (upper_volume(f,MD2)/^1) <= len upper_volume(f,MD2) by FINSEQ_5:28; 1 <= len upper_volume(f,D2) by A17,AXIOMS:22; then 1 <= len upper_volume(f,MD2) by A16,A23,AXIOMS:22; then (upper_volume(f,MD2)/^1).k=upper_volume(f,MD2).(k+1) by A22,RFINSEQ: def 2 .=(sup rng(f|divset(MD2,k+1)))*vol(divset(MD2,k+1)) by A21,INTEGRA1:def 7; hence thesis by A19; end; hence thesis by A16,FINSEQ_1:18; end; A24: len lower_volume(f,D1)=len (lower_volume(f,MD1)/^1) proof rng lower_volume(f,MD1) <> {}; then 1 in dom lower_volume(f,MD1) by FINSEQ_3:34; then 1 <= len lower_volume(f,MD1) by FINSEQ_3:27; then len (lower_volume(f,MD1)/^1)=len lower_volume(f,MD1)-1 by RFINSEQ: def 2 .=len MD1 -1 by INTEGRA1:def 8 .=len D1 by A2,XCMPLX_1:26; hence thesis by INTEGRA1:def 8; end; 1 <= k & k <= len lower_volume(f,D1) implies lower_volume(f,D1).k = (lower_volume(f,MD1)/^1).k proof assume A25:1 <= k & k <= len lower_volume(f,D1); then k in Seg len lower_volume(f,D1) by FINSEQ_1:3; then A26: k in Seg len D1 by INTEGRA1:def 8; then A27: lower_volume(f,D1).k=(inf rng(f|divset(D1,k)))*vol(divset(D1,k)) by INTEGRA1:def 8 .=(inf rng(f|divset(MD1,k+1)))*vol(divset(D1,k)) by A3,A26 .=(inf rng(f|divset(MD1,k+1)))*vol(divset(MD1,k+1)) by A3,A26; A28: 1 <= k+1 by NAT_1:29; A29: len MD1=len <*inf A*>+len D1 by A1,FINSEQ_1:35 .=len D1 + 1 by FINSEQ_1:56; k+1 <= len lower_volume(f,D1)+1 by A25,AXIOMS:24; then k+1 <= len D1+1 by INTEGRA1:def 8; then A30: k+1 in Seg len MD1 by A28,A29,FINSEQ_1:3; A31: k in dom (lower_volume(f,MD1)/^1) by A24,A25,FINSEQ_3:27; A32: len (lower_volume(f,MD1)/^1) <= len lower_volume(f,MD1) by FINSEQ_5:28; 1 <= len (lower_volume(f,MD1)/^1) by A24,A25,AXIOMS:22; then 1 <= len lower_volume(f,MD1) by A32,AXIOMS:22; then (lower_volume(f,MD1)/^1).k=lower_volume(f,MD1).(k+1) by A31,RFINSEQ: def 2 .=(inf rng(f|divset(MD1,k+1)))*vol(divset(MD1,k+1)) by A30,INTEGRA1:def 8; hence thesis by A27; end; hence thesis by A24,FINSEQ_1:18; end; Lm14:for A be closed-interval Subset of REAL, D2, MD2 being Element of divs A st MD2 = <*inf A*>^D2 holds vol(divset(MD2,1))=0 proof let A be closed-interval Subset of REAL, D2,MD2 be Element of divs A; assume A1:MD2 = <*inf A*>^D2; rng MD2 <> {}; then 1 in dom MD2 by FINSEQ_3:34; then inf divset(MD2,1) = inf A & sup divset(MD2,1) = MD2.1 by INTEGRA1:def 5; then vol(divset(MD2,1)) = MD2.1 - inf A by INTEGRA1:def 6.=inf A-inf A by A1,FINSEQ_1:58; hence thesis by XCMPLX_1:14; end; Lm15: for A be closed-interval Subset of REAL, f be Function of A,REAL, D1,MD1 be Element of divs A st MD1 = <*inf A*>^D1 holds delta(MD1)=delta(D1) proof let A be closed-interval Subset of REAL, f be Function of A,REAL, D1,MD1 be Element of divs A; assume A1:MD1 = <*inf A*>^D1; then A2: vol(divset(MD1,1))=0 by Lm14; A3: delta(D1)=max rng upper_volume(chi(A,A),D1) by INTEGRA1:def 19; then delta(D1) in rng upper_volume(chi(A,A),D1) & for x st x in rng upper_volume(chi(A,A),D1) holds x <= delta(D1) by PRE_CIRC:def 1; then consider i such that A4: i in dom upper_volume(chi(A,A),D1) & upper_volume(chi(A,A),D1).i=delta(D1) by PARTFUN1:26; A5: delta(MD1)=max rng upper_volume(chi(A,A),MD1) by INTEGRA1:def 19; then delta(MD1) in rng upper_volume(chi(A,A),MD1) &for x st x in rng upper_volume(chi(A,A),MD1) holds x <= delta(MD1) by PRE_CIRC:def 1; then consider j such that A6: j in dom upper_volume(chi(A,A),MD1)&upper_volume(chi(A,A),MD1).j=delta(MD1 ) by PARTFUN1:26; i in Seg len upper_volume(chi(A,A),D1) by A4,FINSEQ_1:def 3; then A7: i in Seg len D1 by INTEGRA1:def 7; then A8: delta(D1)=(sup rng (chi(A,A)|divset(D1,i)))*vol(divset(D1,i)) by A4,INTEGRA1:def 7 .=(sup rng(chi(A,A)|divset(MD1,i+1)))*vol(divset(D1,i)) by A1,A7,Lm13 .=(sup rng (chi(A,A)|divset(MD1,i+1)))*vol(divset(MD1,i+1)) by A1,A7,Lm13; i in dom D1 by A7,FINSEQ_1:def 3; then len <*inf A*>+i in dom MD1 by A1,FINSEQ_1:41; then i+1 in dom MD1 by FINSEQ_1:56; then A9: i+1 in Seg len MD1 by FINSEQ_1:def 3; then A10: delta(D1)=upper_volume(chi(A,A),MD1).(i+1) by A8,INTEGRA1:def 7; i+1 in Seg len upper_volume(chi(A,A),MD1) by A9,INTEGRA1:def 7; then i+1 in dom upper_volume(chi(A,A),MD1) by FINSEQ_1:def 3; then delta(D1) in rng upper_volume(chi(A,A),MD1) by A10,FUNCT_1:def 5; then A11: delta(D1) <= delta(MD1) by A5,PRE_CIRC:def 1; j in Seg len upper_volume(chi(A,A),MD1) by A6,FINSEQ_1:def 3; then A12: j in Seg len MD1 by INTEGRA1:def 7; then A13: delta(MD1)=(sup rng(chi(A,A)|divset(MD1,j)))*vol(divset(MD1,j)) by A6,INTEGRA1:def 7; delta(MD1) <= delta(D1) proof per cases; suppose j=1; hence thesis by A2,A13,Th8; suppose j<>1; then not(j in Seg 1) by FINSEQ_1:4,TARSKI:def 1; then not(j in Seg len <*inf A*>) by FINSEQ_1:56; then A14: not(j in dom <*inf A*>) by FINSEQ_1:def 3; j in dom MD1 by A12,FINSEQ_1:def 3; then consider k such that A15: k in dom D1 & j=len <*inf A*>+k by A1,A14,FINSEQ_1:38; A16: k in Seg len D1 by A15,FINSEQ_1:def 3; then divset(D1,k)=divset(MD1,k+1) by A1,Lm13 .=divset(MD1,j) by A15,FINSEQ_1:56; then delta(MD1) = (sup rng(chi(A,A)|divset(D1,k)))*vol(divset(D1,k)) by A6,A12,INTEGRA1:def 7; then A17: delta(MD1)=upper_volume(chi(A,A),D1).k by A16,INTEGRA1:def 7; k in Seg len upper_volume(chi(A,A),D1) by A16,INTEGRA1:def 7; then k in dom upper_volume(chi(A,A),D1) by FINSEQ_1:def 3; then delta(MD1) in rng upper_volume(chi(A,A),D1) by A17,FUNCT_1:def 5; hence thesis by A3,PRE_CIRC:def 1; end; hence thesis by A11,AXIOMS:21; end; theorem Th12: for A be closed-interval Subset of REAL, f be Function of A,REAL, D1,D2 be Element of divs A st x in divset(D1,len D1) & vol(A)<>0 & D1<=D2 & rng D2 = rng D1 \/ {x} & f is_bounded_on A & x > inf A holds Sum lower_volume(f,D2)-Sum lower_volume(f,D1)<=(sup rng f-inf rng f)*delta(D1) proof let A be closed-interval Subset of REAL; let f be Function of A,REAL; let D1,D2 be Element of divs A; assume that A1:x in divset(D1,len D1) and A2:vol(A)<>0 and A3:D1 <= D2 and A4:rng D2 = rng D1 \/ {x} and A5:f is_bounded_on A and A6:x > inf A; len D1 <> 0 by FINSEQ_1:25; then len D1 in Seg len D1 by FINSEQ_1:5; then A7:1 <= len D1 by FINSEQ_1:3; then len D1 = 1 or len D1 > 1 by REAL_1:def 5; then A8:len D1 = 1 or len D1 >= 1+1 by NAT_1:38; now per cases by A8; suppose A9:len D1 = 1; then A10: D1.1=sup A by INTEGRA1:def 2; vol(A) >= 0 by INTEGRA1:11; then D1.1 - inf A > 0 by A2,A10,INTEGRA1:def 6; then A11:inf A < D1.1 by REAL_2:106; reconsider MD1 = <*inf A*>^D1 as non empty increasing FinSequence of REAL by A2,A9,Lm11; MD1 is DivisionPoint of A proof y in rng MD1 implies y in A proof assume y in rng MD1; then A12: y in rng <*inf A*> \/ rng D1 by FINSEQ_1:44; per cases by A12,XBOOLE_0:def 2; suppose y in rng <*inf A*>; then y in {inf A} by FINSEQ_1:55; then A13: y = inf A by TARSKI:def 1; consider a,b such that A14: a <= b & a = inf A & b = sup A by INTEGRA1:4; thus thesis by A13,A14,INTEGRA2:1; suppose A15:y in rng D1; rng D1 c= A by INTEGRA1:def 2; hence thesis by A15; end; then A16: rng MD1 c= A by SUBSET_1:7; MD1.(len MD1) = sup A proof A17: len MD1 = len <*inf A*> + len D1 by FINSEQ_1:35; len <*inf A*> + 1 <= len <*inf A*> + len D1 by A7,AXIOMS:24; then MD1.(len MD1)=D1.(len <*inf A*>+len D1-len <*inf A*>) by A17, FINSEQ_1:36 .=D1.(len D1) by XCMPLX_1:26; hence thesis by INTEGRA1:def 2; end; hence thesis by A16,INTEGRA1:def 2; end; then reconsider MD1 as Element of divs A by INTEGRA1:def 4; rng D2 <> {}; then A18: 1 in dom D2 by FINSEQ_3:34; then A19: 1 <= len D2 by FINSEQ_3:27; A20: D2.1 in rng D2 by A18,FUNCT_1:def 5; A21:inf A < D2.1 proof per cases by A4,A20,XBOOLE_0:def 2; suppose D2.1 in rng D1; then consider k such that A22: k in dom D1 & D1.k = D2.1 by PARTFUN1:26; A23: 1 <= k & k <= len D1 by A22,FINSEQ_3:27; rng D1 <> {}; then 1 in dom D1 by FINSEQ_3:34; then D1.1 <= D2.1 by A22,A23,GOBOARD2:18; hence thesis by A11,AXIOMS:22; suppose D2.1 in {x}; hence thesis by A6,TARSKI:def 1; end; set MD2=<*inf A*>^D2; reconsider MD2 as non empty increasing FinSequence of REAL by A21,Lm12; MD2 is DivisionPoint of A proof y in rng MD2 implies y in A proof assume y in rng MD2; then A24: y in rng <*inf A*> \/ rng D2 by FINSEQ_1:44; per cases by A24,XBOOLE_0:def 2; suppose y in rng <*inf A*>; then y in {inf A} by FINSEQ_1:55; then A25: y = inf A by TARSKI:def 1; consider a,b such that A26: a <= b & a = inf A & b = sup A by INTEGRA1:4; thus thesis by A25,A26,INTEGRA2:1; suppose A27:y in rng D2; rng D2 c= A by INTEGRA1:def 2; hence thesis by A27; end; then A28: rng MD2 c= A by SUBSET_1:7; MD2.(len MD2) = sup A proof A29: len MD2 = len <*inf A*> + len D2 by FINSEQ_1:35; len <*inf A*> + 1 <= len <*inf A*> + len D2 by A19,AXIOMS:24; then MD2.(len MD2)=D2.(len <*inf A*>+len D2-len <*inf A*>) by A29, FINSEQ_1:36 .=D2.(len D2) by XCMPLX_1:26; hence thesis by INTEGRA1:def 2; end; hence thesis by A28,INTEGRA1:def 2; end; then reconsider MD2 as Element of divs A by INTEGRA1:def 4; A30: len MD1=len <*inf A*> + len D1 by FINSEQ_1:35 .= 1+len D1 by FINSEQ_1:56; A31: 1+len D1 >= 1+1 by A7,AXIOMS:24; A32: x in divset(MD1,len MD1) proof A33: inf divset(D1,len D1) <= x & x <= sup divset(D1,len D1) by A1,INTEGRA2:1; A34: len D1 in dom D1 by SCMFSA_7:12; A35: len MD1 in dom MD1 by SCMFSA_7:12; A36: len MD1 <> 1 by A30,A31; len MD1 - 1 =len D1 by A30,XCMPLX_1:26; then inf divset(MD1,len MD1) = MD1.(len D1) by A35,A36,INTEGRA1:def 5 .=inf A by A9,FINSEQ_1:58; then A37: inf divset(D1,len D1) = inf divset(MD1,len MD1) by A9,A34,INTEGRA1: def 5; MD1.(len MD1) = MD1.(len <*inf A*> + len D1) by FINSEQ_1:35 .=D1.(len D1) by A34,FINSEQ_1:def 7; then sup divset(MD1,len MD1)=D1.(len D1) by A35,A36,INTEGRA1:def 5 .= sup divset(D1,len D1) by A9,A34,INTEGRA1:def 5; hence thesis by A33,A37,INTEGRA2:1; end; A38:MD1<=MD2 proof A39: len D1 <= len D2 & rng D1 c= rng D2 by A3,INTEGRA1:def 20; then len D1+len <*inf A*> <= len D2+len <*inf A*> by AXIOMS:24; then len MD1 <= len D2+len <*inf A*> by FINSEQ_1:35; then A40: len MD1 <= len MD2 by FINSEQ_1:35; rng D1 \/ rng<*inf A*> c= rng D2 \/ rng <*inf A*> by A39,XBOOLE_1:9; then rng MD1 c= rng D2 \/ rng <*inf A*> by FINSEQ_1:44; then rng MD1 c= rng MD2 by FINSEQ_1:44; hence thesis by A40,INTEGRA1:def 20; end; rng MD2 = rng D2 \/ rng <*inf A*> by FINSEQ_1:44 .=rng D1 \/ rng <*inf A*> \/ {x} by A4,XBOOLE_1:4; then rng MD2 = rng MD1 \/ {x} by FINSEQ_1:44; then A41:Sum lower_volume(f,MD2)-Sum lower_volume(f,MD1)<=(sup rng f-inf rng f)*delta(MD1) by A5,A30,A31,A32,A38,Th9; A42:vol(divset(MD1,1))=0 by Lm14; lower_volume(f,D1)=lower_volume(f,MD1)/^1 by Lm13; then lower_volume(f,MD1)=<*(lower_volume(f,MD1))/.1*>^lower_volume(f,D1) by FINSEQ_5:32; then A43:Sum lower_volume(f,MD1)=((lower_volume(f,MD1))/.1)+Sum lower_volume(f, D1) by RVSUM_1:106; rng MD1 <> {}; then 1 in dom MD1 by FINSEQ_3:34; then A44:1 in Seg len MD1 by FINSEQ_1:def 3; then A45:lower_volume(f,MD1).1 = (inf rng(f|divset(MD1,1)))*vol(divset(MD1,1)) by INTEGRA1:def 8; 1 in Seg len lower_volume(f,MD1) by A44,INTEGRA1:def 8; then 1 in dom lower_volume(f,MD1) by FINSEQ_1:def 3; then A46: lower_volume(f,MD1)/.1 = 0 by A42,A45,FINSEQ_4:def 4; A47:vol(divset(MD2,1))=0 by Lm14; lower_volume(f,D2)=lower_volume(f,MD2)/^1 by Lm13; then lower_volume(f,MD2)=<*(lower_volume(f,MD2))/.1*>^lower_volume(f,D2) by FINSEQ_5:32; then A48:Sum lower_volume(f,MD2)=(lower_volume(f,MD2))/.1+Sum lower_volume(f,D2 ) by RVSUM_1:106; rng MD2 <> {}; then 1 in dom MD2 by FINSEQ_3:34; then A49:1 in Seg len MD2 by FINSEQ_1:def 3; then A50: lower_volume(f,MD2).1 = (inf rng(f|divset(MD2,1)))*vol(divset(MD2 ,1)) by INTEGRA1:def 8; 1 in Seg len lower_volume(f,MD2) by A49,INTEGRA1:def 8; then 1 in dom lower_volume(f,MD2) by FINSEQ_1:def 3; then lower_volume(f,MD2)/.1 = 0 by A47,A50,FINSEQ_4:def 4; hence Sum lower_volume(f,D2)-Sum lower_volume(f,D1)<= (sup rng f-inf rng f)*delta(D1) by A41,A43,A46,A48,Lm15; suppose len D1 >= 2; hence thesis by A1,A3,A4,A5,Th9; end; hence thesis; end; theorem Th13: for A be closed-interval Subset of REAL, f be Function of A,REAL, D1,D2 be Element of divs A st x in divset(D1,len D1) & vol(A)<>0 & D1<=D2 & rng D2 = rng D1 \/ {x} & f is_bounded_on A & x > inf A holds Sum upper_volume(f,D1)-Sum upper_volume(f,D2)<=(sup rng f-inf rng f)*delta(D1) proof let A be closed-interval Subset of REAL; let f be Function of A,REAL; let D1,D2 be Element of divs A; assume that A1:x in divset(D1,len D1) and A2:vol(A)<>0 and A3:D1 <= D2 and A4:rng D2 = rng D1 \/ {x} and A5:f is_bounded_on A and A6:x > inf A; len D1 <> 0 by FINSEQ_1:25; then len D1 in Seg len D1 by FINSEQ_1:5; then A7:1 <= len D1 by FINSEQ_1:3; then len D1 = 1 or len D1 > 1 by REAL_1:def 5; then A8:len D1 = 1 or len D1 >= 1+1 by NAT_1:38; now per cases by A8; suppose A9:len D1 = 1; then A10: D1.1=sup A by INTEGRA1:def 2; vol(A) >= 0 by INTEGRA1:11; then D1.1 - inf A > 0 by A2,A10,INTEGRA1:def 6; then A11:inf A < D1.1 by REAL_2:106; reconsider MD1 = <*inf A*>^D1 as non empty increasing FinSequence of REAL by A2,A9,Lm11; MD1 is DivisionPoint of A proof y in rng MD1 implies y in A proof assume y in rng MD1; then A12: y in rng <*inf A*> \/ rng D1 by FINSEQ_1:44; per cases by A12,XBOOLE_0:def 2; suppose y in rng <*inf A*>; then y in {inf A} by FINSEQ_1:55; then A13: y = inf A by TARSKI:def 1; consider a,b such that A14: a <= b & a = inf A & b = sup A by INTEGRA1:4; thus thesis by A13,A14,INTEGRA2:1; suppose A15:y in rng D1; rng D1 c= A by INTEGRA1:def 2; hence thesis by A15; end; then A16: rng MD1 c= A by SUBSET_1:7; MD1.(len MD1) = sup A proof A17: len MD1 = len <*inf A*> + len D1 by FINSEQ_1:35; len <*inf A*> + 1 <= len <*inf A*> + len D1 by A7,AXIOMS:24; then MD1.(len MD1)=D1.(len <*inf A*>+len D1-len <*inf A*>) by A17, FINSEQ_1:36 .=D1.(len D1) by XCMPLX_1:26; hence thesis by INTEGRA1:def 2; end; hence thesis by A16,INTEGRA1:def 2; end; then reconsider MD1 as Element of divs A by INTEGRA1:def 4; rng D2 <> {}; then A18: 1 in dom D2 by FINSEQ_3:34; then A19: 1 <= len D2 by FINSEQ_3:27; A20:D2.1 in rng D2 by A18,FUNCT_1:def 5; A21:inf A < D2.1 proof per cases by A4,A20,XBOOLE_0:def 2; suppose D2.1 in rng D1; then consider k such that A22: k in dom D1 & D1.k = D2.1 by PARTFUN1:26; A23: 1 <= k & k <= len D1 by A22,FINSEQ_3:27; rng D1 <> {}; then 1 in dom D1 by FINSEQ_3:34; then D1.1 <= D2.1 by A22,A23,GOBOARD2:18; hence thesis by A11,AXIOMS:22; suppose D2.1 in {x}; hence thesis by A6,TARSKI:def 1; end; set MD2=<*inf A*>^D2; reconsider MD2 as non empty increasing FinSequence of REAL by A21,Lm12; MD2 is DivisionPoint of A proof y in rng MD2 implies y in A proof assume y in rng MD2; then A24: y in rng <*inf A*> \/ rng D2 by FINSEQ_1:44; per cases by A24,XBOOLE_0:def 2; suppose y in rng <*inf A*>; then y in {inf A} by FINSEQ_1:55; then A25: y = inf A by TARSKI:def 1; consider a,b such that A26: a <= b & a = inf A & b = sup A by INTEGRA1:4; thus thesis by A25,A26,INTEGRA2:1; suppose A27:y in rng D2; rng D2 c= A by INTEGRA1:def 2; hence thesis by A27; end; then A28: rng MD2 c= A by SUBSET_1:7; MD2.(len MD2) = sup A proof A29: len MD2 = len <*inf A*> + len D2 by FINSEQ_1:35; len <*inf A*> + 1 <= len <*inf A*> + len D2 by A19,AXIOMS:24; then MD2.(len MD2)=D2.(len <*inf A*>+len D2-len <*inf A*>) by A29, FINSEQ_1:36 .=D2.(len D2) by XCMPLX_1:26; hence thesis by INTEGRA1:def 2; end; hence thesis by A28,INTEGRA1:def 2; end; then reconsider MD2 as Element of divs A by INTEGRA1:def 4; A30: len MD1=len <*inf A*> + len D1 by FINSEQ_1:35 .= 1+len D1 by FINSEQ_1:56; A31:1+len D1 >= 1+1 by A7,AXIOMS:24; A32: x in divset(MD1,len MD1) proof A33: inf divset(D1,len D1) <= x & x <= sup divset(D1,len D1) by A1,INTEGRA2:1; A34: len D1 in dom D1 by SCMFSA_7:12; A35: len MD1 in dom MD1 by SCMFSA_7:12; A36: len MD1 <> 1 by A30,A31; len MD1 - 1 = len D1 by A30,XCMPLX_1:26; then inf divset(MD1,len MD1) = MD1.(len D1) by A35,A36,INTEGRA1:def 5 .=inf A by A9,FINSEQ_1:58; then A37: inf divset(D1,len D1) = inf divset(MD1,len MD1) by A9,A34,INTEGRA1: def 5; MD1.(len MD1) = MD1.(len <*inf A*> + len D1) by FINSEQ_1:35 .=D1.(len D1) by A34,FINSEQ_1:def 7; then sup divset(MD1,len MD1)=D1.(len D1) by A35,A36,INTEGRA1:def 5 .= sup divset(D1,len D1) by A9,A34,INTEGRA1:def 5; hence thesis by A33,A37,INTEGRA2:1; end; A38:MD1<=MD2 proof A39: len D1 <= len D2 & rng D1 c= rng D2 by A3,INTEGRA1:def 20; then len D1+len <*inf A*> <= len D2+len <*inf A*> by AXIOMS:24; then len MD1 <= len D2+len <*inf A*> by FINSEQ_1:35; then A40: len MD1 <= len MD2 by FINSEQ_1:35; rng D1 \/ rng<*inf A*> c= rng D2 \/ rng <*inf A*> by A39,XBOOLE_1:9; then rng MD1 c= rng D2 \/ rng <*inf A*> by FINSEQ_1:44; then rng MD1 c= rng MD2 by FINSEQ_1:44; hence thesis by A40,INTEGRA1:def 20; end; rng MD2 = rng MD1 \/ {x} proof rng MD2 = rng D2 \/ rng <*inf A*> by FINSEQ_1:44 .=rng D1 \/ rng <*inf A*> \/ {x} by A4,XBOOLE_1:4; hence thesis by FINSEQ_1:44; end; then A41:Sum upper_volume(f,MD1)-Sum upper_volume(f,MD2) <=(sup rng f-inf rng f)*delta(MD1) by A5,A30,A31,A32,A38,Th10; A42:vol(divset(MD1,1))=0 by Lm14; upper_volume(f,D1)=upper_volume(f,MD1)/^1 by Lm13; then upper_volume(f,MD1)=<*upper_volume(f,MD1)/.1*>^upper_volume(f,D1) by FINSEQ_5:32; then A43:Sum upper_volume(f,MD1)=(upper_volume(f,MD1)/.1)+Sum upper_volume(f,D1 ) by RVSUM_1:106; rng MD1 <> {}; then 1 in dom MD1 by FINSEQ_3:34; then A44:1 in Seg len MD1 by FINSEQ_1:def 3; then A45:upper_volume(f,MD1).1 = (sup rng(f|divset(MD1,1)))*vol(divset(MD1,1)) by INTEGRA1:def 7; 1 in Seg len upper_volume(f,MD1) by A44,INTEGRA1:def 7; then 1 in dom upper_volume(f,MD1) by FINSEQ_1:def 3; then A46: upper_volume(f,MD1)/.1 = 0 by A42,A45,FINSEQ_4:def 4; A47:vol(divset(MD2,1))=0 by Lm14; upper_volume(f,D2)=upper_volume(f,MD2)/^1 by Lm13; then upper_volume(f,MD2)=<*upper_volume(f,MD2)/.1*>^upper_volume(f,D2) by FINSEQ_5:32; then A48:Sum upper_volume(f,MD2)=(upper_volume(f,MD2)/.1)+Sum upper_volume(f,D2 ) by RVSUM_1:106; rng MD2 <> {}; then 1 in dom MD2 by FINSEQ_3:34; then A49:1 in Seg len MD2 by FINSEQ_1:def 3; then A50:upper_volume(f,MD2).1 = (sup rng(f|divset(MD2,1)))*vol(divset(MD2,1)) by INTEGRA1:def 7; 1 in Seg len upper_volume(f,MD2) by A49,INTEGRA1:def 7; then 1 in dom upper_volume(f,MD2) by FINSEQ_1:def 3; then upper_volume(f,MD2)/.1 = 0 by A47,A50,FINSEQ_4:def 4; hence Sum upper_volume(f,D1)-Sum upper_volume(f,D2)<= (sup rng f-inf rng f)*delta(D1) by A41,A43,A46,A48,Lm15; suppose len D1 >= 2; hence thesis by A1,A3,A4,A5,Th10; end; hence thesis; end; theorem Th14: for A be closed-interval Subset of REAL, D1,D2 be Element of divs A, r be Real, i,j be Nat st i in dom D1 & j in dom D1 & i<=j & D1 <= D2 & r < mid(D2,indx(D2,D1,i),indx(D2,D1,j)).1 holds ex B be closed-interval Subset of REAL, MD1,MD2 be Element of divs B st r=inf B & sup B=MD2.(len MD2) & sup B=MD1.(len MD1) & MD1 <= MD2 & MD1=mid(D1,i,j) & MD2=mid(D2,indx(D2,D1,i),indx(D2,D1,j)) proof let A be closed-interval Subset of REAL; let D1,D2 be Element of divs A; let r be Real; let i,j; assume A1:i in dom D1; assume A2:j in dom D1; assume A3:i <= j; assume A4:D1 <= D2; assume A5:r < mid(D2,indx(D2,D1,i),indx(D2,D1,j)).1; set MD1=mid(D1,i,j); set MD2=mid(D2,indx(D2,D1,i),indx(D2,D1,j)); A6: indx(D2,D1,i) in dom D2 & D2.indx(D2,D1,i)=D1.i by A1,A4,INTEGRA1:def 21; A7: indx(D2,D1,j) in dom D2 & D2.indx(D2,D1,j)=D1.j by A2,A4,INTEGRA1:def 21; D1.i <= D1.j by A1,A2,A3,GOBOARD2:18; then A8:indx(D2,D1,i) <= indx(D2,D1,j) by A6,A7,GOBOARD1:def 1; then consider B being closed-interval Subset of REAL such that A9: r = inf B & sup B=MD2.(len MD2) & len MD2=indx(D2,D1,j)-indx(D2,D1,i)+1 &MD2 is DivisionPoint of B by A5,A6,A7,Th11; reconsider MD2 as Element of divs B by A9,INTEGRA1:def 3; indx(D2,D1,j)-indx(D2,D1,i) >= 0 by A8,SQUARE_1:12; then A10:indx(D2,D1,j)-indx(D2,D1,i)+1 >= 0+1 by AXIOMS:24; A11:1 <= indx(D2,D1,i) & indx(D2,D1,j) <= len D2 by A6,A7,FINSEQ_3:27; then A12:MD2.1=D2.(1+indx(D2,D1,i)-1) by A8,A10,JORDAN3:31 .=D2.indx(D2,D1,i) by XCMPLX_1:26 .=D1.i by A1,A4,INTEGRA1:def 21; j-i >= 0 by A3,SQUARE_1:12; then A13:j-i+1 >= 0+1 by AXIOMS:24; A14:1 <= i & j <= len D1 by A1,A2,FINSEQ_3:27; then A15:MD1.1 = D1.(1+i-1) by A3,A13,JORDAN3:31 .= D1.i by XCMPLX_1:26; then consider C being closed-interval Subset of REAL such that A16:r = inf C & sup C=MD1.(len MD1) & len MD1=j-i+1 &MD1 is DivisionPoint of C by A1,A2,A3,A5,A12,Th11; A17:MD1.(len MD1)=D1.(j-i+1-1+i) by A3,A13,A14,A16,JORDAN3:31 .=D1.(j-i+i) by XCMPLX_1:26 .=D1.(j-(i-i)) by XCMPLX_1:37 .=D1.j by XCMPLX_1:17; A18: MD2.(len MD2)=D2.(indx(D2,D1,j)-indx(D2,D1,i)+1-1+indx(D2,D1,i)) by A8,A9,A10,A11,JORDAN3:31 .=D2.(indx(D2,D1,j)-indx(D2,D1,i)+indx(D2,D1,i)) by XCMPLX_1:26 .=D2.(indx(D2,D1,j)-(indx(D2,D1,i)-indx(D2,D1,i))) by XCMPLX_1:37 .=D2.indx(D2,D1,j) by XCMPLX_1:17 .= D1.j by A2,A4,INTEGRA1:def 21; A19:B=[.inf B,sup B.] by INTEGRA1:5 .=C by A9,A16,A17,A18,INTEGRA1:5; then reconsider MD1 as Element of divs B by A16,INTEGRA1:def 3; A20:rng MD1 c= rng MD2 proof A21: rng MD1 c= rng D1 by JORDAN3:28; rng D1 c= rng D2 by A4,INTEGRA1:def 20; then A22: rng MD1 c= rng D2 by A21,XBOOLE_1:1; let x1;assume A23:x1 in rng MD1; then consider k2 being Nat such that A24: k2 in dom D2 & D2.k2=x1 by A22,PARTFUN1:26; consider k1 being Nat such that A25: k1 in dom MD1 & MD1.k1=x1 by A23,PARTFUN1:26; A26: 1 <= k1 & k1 <= len MD1 by A25,FINSEQ_3:27; then 1 <= len MD1 by AXIOMS:22; then A27: 1 in dom MD1 by FINSEQ_3:27; len MD1 in dom MD1 by SCMFSA_7:12; then MD1.1<=MD1.k1 & MD1.k1<=MD1.(len MD1) by A25,A26,A27,GOBOARD2:18; then A28: indx(D2,D1,i) <= k2 & k2 <= indx(D2,D1,j) by A6,A7,A15,A17,A24,A25,GOBOARD1:def 1; then indx(D2,D1,i)+1<=k2+1 & k2+1<=indx(D2,D1,j)+1 by AXIOMS:24; then A29: 1<=k2+1-indx(D2,D1,i) & k2+1-indx(D2,D1,i)<=indx(D2,D1,j)+1-indx(D2, D1,i) by REAL_1:49,84; indx(D2,D1,i) <= k2+1 by A28,NAT_1:37; then consider k3 being Nat such that A30: k2+1 = indx(D2,D1,i)+k3 by NAT_1:28; A31: 1 <= k3 & k3 <= indx(D2,D1,j)+1-indx(D2,D1,i) by A29,A30,XCMPLX_1:26; then MD2.k3 = D2.(k3+indx(D2,D1,i)-1) by A8,A11,JORDAN3:31; then A32: MD2.k3=D2.k2 by A30,XCMPLX_1:26; 1 <= k3 & k3 <= indx(D2,D1,j)-indx(D2,D1,i)+1 by A31,XCMPLX_1:29; then k3 in dom MD2 by A9,FINSEQ_3:27; hence thesis by A24,A32,FUNCT_1:def 5; end; len MD1 <= len MD2 proof MD1 is one-to-one & MD2 is one-to-one by JORDAN7:17; then card(rng MD1) = len MD1 & card(rng MD2) = len MD2 by FINSEQ_4:77; hence thesis by A20,CARD_1:80; end; then MD1 <= MD2 by A20,INTEGRA1:def 20; hence thesis by A9,A16,A19; end; theorem Th15: for A be closed-interval Subset of REAL, D be Element of divs A st x in rng D holds D.1 <= x & x <= D.(len D) proof let A be closed-interval Subset of REAL; let D be Element of divs A; assume x in rng D; then consider i such that A1:i in dom D & x=D.i by PARTFUN1:26; A2:1 <= i & i <= len D by A1,FINSEQ_3:27; then 1 <= len D by AXIOMS:22; then 1 in dom D & len D in dom D by FINSEQ_3:27; hence thesis by A1,A2,GOBOARD2:18; end; theorem Th16: for p be FinSequence of REAL, i,j,k st p is increasing & i in dom p & j in dom p & k in dom p & p.i <= p.k & p.k <= p.j holds p.k in rng mid(p,i,j) proof let p be FinSequence of REAL; let i,j,k; assume that A1:p is increasing and A2:i in dom p and A3:j in dom p and A4:k in dom p and A5:p.i <= p.k and A6:p.k <= p.j; A7:1 <= i & i <= len p by A2,FINSEQ_3:27; A8:1 <= j & j <= len p by A3,FINSEQ_3:27; A9:i <= k by A1,A2,A4,A5,GOBOARD1:def 1; A10:k <= j by A1,A3,A4,A6,GOBOARD1:def 1; then A11:i <= j by A9,AXIOMS:22; then len mid(p,i,j) = j-'i+1 by A7,A8,JORDAN3:27; then A12:len mid(p,i,j) = j-i+1 by A11,SCMFSA_7:3; i <= k+1 by A9,NAT_1:37; then consider n such that A13:k+1=i+n by NAT_1:28; A14:n=k+1-i by A13,XCMPLX_1:26 .=k-i+1 by XCMPLX_1:29; k-i >= 0 by A9,SQUARE_1:12; then A15:k-i+1 >= 0+1 by AXIOMS:24; k-i <= j-i by A10,REAL_1:49; then A16:k-i+1 <= j-i+1 by AXIOMS:24; then A17:n in dom mid(p,i,j) by A12,A14,A15,FINSEQ_3:27; mid(p,i,j).n = p.(n+i-1) by A7,A8,A11,A14,A15,A16,JORDAN3:31 .=p.k by A13,XCMPLX_1:26; hence thesis by A17,FUNCT_1:def 5; end; theorem Th17: for A be closed-interval Subset of REAL, f be Function of A,REAL, D be Element of divs A st f is_bounded_on A & i in dom D holds inf rng(f|divset(D,i)) <= sup rng f proof let A be closed-interval Subset of REAL; let f be Function of A,REAL; let D be Element of divs A; assume A1:f is_bounded_on A; assume i in dom D; then divset(D,i) c= A by INTEGRA1:10; hence thesis by A1,Lm7; end; theorem Th18: for A be closed-interval Subset of REAL, f be Function of A,REAL, D be Element of divs A st f is_bounded_on A & i in dom D holds sup rng(f|divset(D,i)) >= inf rng f proof let A be closed-interval Subset of REAL; let f be Function of A,REAL; let D be Element of divs A; assume A1:f is_bounded_on A; assume i in dom D; then divset(D,i) c= A by INTEGRA1:10; hence thesis by A1,Lm7; end; begin :: Darboux's Theorem theorem for A be closed-interval Subset of REAL, f be Function of A,REAL, T be DivSequence of A st f is_bounded_on A & delta(T) is convergent_to_0 & vol(A)<>0 holds lower_sum(f,T) is convergent & lim lower_sum(f,T) = lower_integral(f) proof let A be closed-interval Subset of REAL; let f be Function of A,REAL; let T be DivSequence of A; assume A1:f is_bounded_on A; assume A2:delta(T) is convergent_to_0; assume A3:vol(A)<>0; A4:delta(T) is_not_0 & delta(T) is convergent & lim delta(T) = 0 by A2,FDIFF_1:def 1; A5:for e st e>0 ex n st for m st n<=m holds 0 < (delta(T)).m & (delta(T)).m < e proof let e; assume e>0; then consider n such that A6: for m st n<=m holds abs((delta(T)).m-0)<e by A4,SEQ_2:def 7; for m st n<=m holds 0 < (delta(T)).m & (delta(T)).m < e proof let m; assume n<=m; then A7: abs((delta(T)).m-0)<e by A6; (delta(T)).m-0<=abs((delta(T)).m-0) by ABSVALUE:11; then A8: (delta(T)).m+abs((delta(T)).m-0)<e+abs((delta(T)).m-0) by A7,REAL_1 :67; A9: (delta(T)).m<>0 by A4,SEQ_1:7; A10: (delta(T)).m = delta(T.m) by INTEGRA2:def 3; delta(T.m) = max rng upper_volume(chi(A,A),T.m) by INTEGRA1:def 19; then delta(T.m) in rng upper_volume(chi(A,A),T.m) by PRE_CIRC:def 1; then consider i such that A11: i in dom upper_volume(chi(A,A),T.m) & delta(T.m)=upper_volume(chi(A,A),T.m).i by PARTFUN1:26; A12: i in Seg len upper_volume(chi(A,A),T.m) by A11,FINSEQ_1:def 3; consider D being Element of divs A such that A13: D = T.m; i in Seg len D by A12,A13,INTEGRA1:def 7; then delta(T.m)=vol(divset(T.m,i)) by A11,A13,INTEGRA1:22; hence thesis by A8,A9,A10,AXIOMS:24,INTEGRA1:11; end; hence thesis; end; A14:for D,D1 be Element of divs A holds ex D2 be Element of divs A st D<=D2 & D1<=D2 & rng D2=rng D1 \/ rng D & 0<=lower_sum(f,D2)-lower_sum(f,D) & 0<=lower_sum(f,D2)-lower_sum(f,D1) proof let D,D1 be Element of divs A; consider D2 be Element of divs A such that A15: D<=D2 & D1<=D2 & rng D2=rng D1 \/ rng D by Th3; A16: f is_bounded_below_on A by A1,RFUNCT_1:def 11; then lower_sum(f,D2)>=lower_sum(f,D) by A15,INTEGRA1:48; then A17: lower_sum(f,D2)-lower_sum(f,D)>=0 by SQUARE_1:12; lower_sum(f,D2)>=lower_sum(f,D1) by A15,A16,INTEGRA1:48; then lower_sum(f,D2)-lower_sum(f,D1)>=0 by SQUARE_1:12; hence thesis by A15,A17; end; A18:sup rng f >= inf rng f by A1,Lm6; A19:for D,D1 be Element of divs A st delta(D1)<min rng upper_volume(chi(A,A),D) holds ex D2 be Element of divs A st D<=D2 & D1<=D2 & rng D2=rng D1 \/ rng D & lower_sum(f,D2)-lower_sum(f,D1) <= (len D)*(sup(rng f)-inf(rng f))*delta(D1) proof let D,D1 be Element of divs A; assume A20:delta(D1)<min rng upper_volume(chi(A,A),D); ex D2 be Element of divs A st D<=D2 & D1<=D2 & rng D2=rng D1 \/ rng D & lower_sum(f,D2)-lower_sum(f,D1)<=(len D)*(sup(rng f)-inf(rng f))*delta(D1) proof consider D2 be Element of divs A such that A21: D<=D2 & D1<=D2 & rng D2=rng D1 \/ rng D & 0<=lower_sum(f,D2)-lower_sum(f,D) & 0<=lower_sum(f,D2)-lower_sum(f,D1) by A14; lower_sum(f,D2)-lower_sum(f,D1) <= (len D)*(sup(rng f)-inf(rng f))*delta(D1) proof deffunc PLf(Element of divs A,Nat) = (PartSums(lower_volume(f,$1))).$2; deffunc LVf(Element of divs A) = lower_volume(f,$1); A22: for i st i in dom D holds ex j st j in dom D1 & D.i in divset(D1,j) & PLf(D2,indx(D2,D1,j))-PLf(D1,j)<=i*(sup(rng f)-inf(rng f))*delta(D1) proof let i; assume A23:i in dom D; A24: for i,j st i in dom D & j in dom D1 & D.i in divset(D1,j) holds j >= 2 proof let i,j; assume A25:i in dom D; assume A26:j in dom D1 & D.i in divset(D1,j); assume j<2; then j<1+1; then A27: j <= 1 by NAT_1:38; A28: inf divset(D1,j)<=D.i & D.i<=sup divset(D1,j) by A26,INTEGRA2:1; j in Seg len D1 by A26,FINSEQ_1:def 3; then j >= 1 by FINSEQ_1:3; then j = 1 by A27,AXIOMS:21; then A29: inf divset(D1,j)=inf A & sup divset(D1,j)=D1.j by A26,INTEGRA1:def 5; delta(D1) >= min rng upper_volume(chi(A,A),D) proof per cases; suppose A30:i=1; len D <> 0 by FINSEQ_1:25; then len D in Seg len D by FINSEQ_1:5; then 1 <= len D by FINSEQ_1:3; then A31: 1 in Seg len D by FINSEQ_1:3; then 1 in dom D by FINSEQ_1:def 3; then A32: inf divset(D,1)=inf A & sup divset(D,1)=D.1 by INTEGRA1:def 5; sup divset(D1,j)-inf A >= D.1-inf A by A28,A30,REAL_1:49; then vol(divset(D1,j)) >= sup divset(D,1)-inf divset(D,1) by A29,A32,INTEGRA1:def 6; then A33: vol(divset(D1,j)) >= vol(divset(D,1)) by INTEGRA1:def 6; vol(divset(D1,j)) <= delta(D1) by A26,Lm8; then A34: delta(D1) >= vol(divset(D,1)) by A33,AXIOMS:22; 1 in Seg len upper_volume(chi(A,A),D) by A31,INTEGRA1:def 7; then A35: 1 in dom upper_volume(chi(A,A),D) by FINSEQ_1:def 3; vol(divset(D,1)) = upper_volume(chi(A,A),D).1 by A31,INTEGRA1:22; then vol(divset(D,1)) in rng upper_volume(chi(A,A),D) by A35,FUNCT_1:def 5; then vol(divset(D,1))>=min rng upper_volume(chi(A,A),D) by SFMASTR3: def 1; hence thesis by A34,AXIOMS:22; suppose A36:i<>1; then A37: inf divset(D,i)=D.(i-1) & sup divset(D,i)=D.i by A25,INTEGRA1:def 5; then A38: sup divset(D1,j)-inf divset(D1,j) >= sup divset(D,i)-inf A by A28,A29,REAL_1:49; D.(i-1) in A by A25,A36,INTEGRA1:9; then inf A <= D.(i-1) by INTEGRA2:1; then sup divset(D,i)-inf A >= sup divset(D,i)-inf divset(D,i) by A37,REAL_2:106; then sup divset(D1,j)-inf divset(D1,j) >= sup divset(D,i)-inf divset( D,i) by A38,AXIOMS:22; then vol(divset(D1,j)) >= sup divset(D,i)-inf divset(D,i) by INTEGRA1:def 6; then A39: vol(divset(D1,j)) >= vol(divset(D,i)) by INTEGRA1:def 6; vol(divset(D1,j)) <= delta(D1) by A26,Lm8; then A40: delta(D1) >= vol(divset(D,i)) by A39,AXIOMS:22; A41: i in Seg len D by A25,FINSEQ_1:def 3; then i in Seg len upper_volume(chi(A,A),D) by INTEGRA1:def 7; then A42: i in dom upper_volume(chi(A,A),D) by FINSEQ_1:def 3; vol(divset(D,i)) = upper_volume(chi(A,A),D).i by A41,INTEGRA1:22; then vol(divset(D,i)) in rng upper_volume(chi(A,A),D) by A42,FUNCT_1:def 5; then vol(divset(D,i))>=min rng upper_volume(chi(A,A),D) by SFMASTR3: def 1; hence thesis by A40,AXIOMS:22; end; hence contradiction by A20; end; defpred P[non empty Nat] means $1 in dom D implies ex j st j in dom D1 & D.$1 in divset(D1,j) & PLf(D2,indx(D2,D1,j))-PLf(D1,j)<=$1*(sup rng f-inf rng f)*delta(D1); A43: P[1] proof len D <> 0 by FINSEQ_1:25; then len D in Seg len D by FINSEQ_1:5; then 1 <= len D by FINSEQ_1:3; then A44: 1 in dom D by FINSEQ_3:27; then D.1 in A by INTEGRA1:8; then consider j such that A45: j in dom D1 & D.1 in divset(D1,j) by Th2; PLf(D2,indx(D2,D1,j))-PLf(D1,j)<=1*(sup rng f-inf rng f)*delta(D1) proof A46: j <> 1 by A24,A44,A45; then A47: j-1 in NAT & j-1 in dom D1 by A45,INTEGRA1:9; reconsider j1=j-1 as Element of NAT by A45,A46,INTEGRA1:9; inf divset(D1,j) <= D.1 & D.1 <= sup divset(D1,j) by A45,INTEGRA2:1; then A48: D1.j1 <= D.1 & D.1 <= D1.j by A45,A46,INTEGRA1:def 5; A49: indx(D2,D1,j1) in dom D2 & 1 <= indx(D2,D1,j1) & indx(D2,D1,j1) <= len D2 proof thus indx(D2,D1,j1) in dom D2 by A21,A47,INTEGRA1:def 21; then indx(D2,D1,j1) in Seg len D2 by FINSEQ_1:def 3; hence thesis by FINSEQ_1:3; end; then mid(D2,1,indx(D2,D1,j1)) is increasing by INTEGRA1:37; then A50: D2|indx(D2,D1,j1) is increasing by A49,JORDAN3:25; A51: j1 in dom D1 & 1 <= j1 & j1 <= len D1 proof thus j1 in dom D1 by A45,A46,INTEGRA1:9; j1 in Seg len D1 by A47,FINSEQ_1:def 3; hence thesis by FINSEQ_1:3; end; then mid(D1,1,j1) is increasing by INTEGRA1:37; then A52: D1|j1 is increasing by A51,JORDAN3:25; A53: rng (D2|indx(D2,D1,j1)) = rng (D1|j1) proof for x1 st x1 in rng(D2|indx(D2,D1,j1)) holds x1 in rng(D1|j1) proof let x1; assume x1 in rng(D2|indx(D2,D1,j1)); then consider k such that A54: k in dom(D2|indx(D2,D1,j1)) & x1=(D2|indx(D2,D1,j1)).k by PARTFUN1:26; k in Seg len(D2|indx(D2,D1,j1)) by A54,FINSEQ_1:def 3; then A55: k in Seg indx(D2,D1,j1) by A49,TOPREAL1:3; then A56: (D2|indx(D2,D1,j1)).k = D2.k & k in dom D2 by A49,RFINSEQ:19; then A57: D2.k in rng D2 by FUNCT_1:def 5; 1 <= k & k <= indx(D2,D1,j1) by A55,FINSEQ_1:3; then D2.k <= D2.indx(D2,D1,j1) by A49,A56,GOBOARD2:18; then A58: D2.k <= D1.j1 by A21,A47,INTEGRA1:def 21; A59: len(D1|j1) = j1 by A51,TOPREAL1:3; A60: D2.k in rng D implies D2.k = D1.j1 proof assume D2.k in rng D; then consider n such that A61: n in dom D & D2.k=D.n by PARTFUN1:26; 1 <= n by A61,FINSEQ_3:27; then D.1 <= D2.k by A44,A61,GOBOARD2:18; then D1.j1 <= D2.k by A48,AXIOMS:22; hence thesis by A58,AXIOMS:21; end; A62: D2.k in rng D implies D2.k in rng(D1|j1) proof assume A63:D2.k in rng D; j1 in Seg len(D1|j1) by A51,A59,FINSEQ_1:3; then j1 in dom(D1|j1) by FINSEQ_1:def 3; then A64: (D1|j1).j1 in rng(D1|j1) by FUNCT_1:def 5; j1 in Seg j1 by A51,FINSEQ_1:3; hence thesis by A51,A60,A63,A64,RFINSEQ:19; end; D2.k in rng D1 implies D2.k in rng(D1|j1) proof assume D2.k in rng D1; then consider m such that A65: m in dom D1 & D2.k = D1.m by PARTFUN1:26; m in Seg len D1 by A65,FINSEQ_1:def 3; then A66: 1 <= m & m <= j1 by A51,A58,A65,FINSEQ_1:3,GOBOARD1:def 1; then A67: m in dom (D1|j1) by A59,FINSEQ_3:27; m in Seg j1 by A66,FINSEQ_1:3; then D2.k = (D1|j1).m by A51,A65,RFINSEQ:19; hence thesis by A67,FUNCT_1:def 5; end; hence thesis by A21,A49,A54,A55,A57,A62,RFINSEQ:19,XBOOLE_0:def 2; end; then A68: rng(D2|indx(D2,D1,j1)) c= rng (D1|j1) by TARSKI:def 3; for x1 st x1 in rng(D1|j1) holds x1 in rng(D2|indx(D2,D1,j1)) proof let x1; assume x1 in rng(D1|j1); then consider k such that A69: k in dom(D1|j1) & x1=(D1|j1).k by PARTFUN1:26; k in Seg len(D1|j1) by A69,FINSEQ_1:def 3; then A70: k in Seg j1 by A51,TOPREAL1:3; then A71: (D1|j1).k = D1.k & k in dom D1 by A51,RFINSEQ:19; then D1.k in rng D1 by FUNCT_1:def 5; then x1 in rng D2 by A21,A69,A71,XBOOLE_0:def 2; then consider n such that A72: n in dom D2 & x1=D2.n by PARTFUN1:26; A73: indx(D2,D1,k) in dom D2 & D2.indx(D2,D1,k)=D2.n by A21,A69,A71,A72,INTEGRA1:def 21; k <= j1 by A70,FINSEQ_1:3; then D1.k <= D1.j1 by A47,A71,GOBOARD2:18; then D2.indx(D2,D1,k) <= D1.j1 by A21,A71,INTEGRA1:def 21; then D2.indx(D2,D1,k)<=D2.indx(D2,D1,j1) by A21,A47,INTEGRA1:def 21; then A74: n <= indx(D2,D1,j1) by A49,A72,A73,GOBOARD1:def 1; 1 <= n by A72,FINSEQ_3:27; then A75: n in Seg indx(D2,D1,j1) by A74,FINSEQ_1:3; then A76: D2.n = (D2|indx(D2,D1,j1)).n by A49,RFINSEQ:19; n in Seg len(D2|indx(D2,D1,j1)) by A49,A75,TOPREAL1:3; then n in dom(D2|indx(D2,D1,j1)) by FINSEQ_1:def 3; hence thesis by A72,A76,FUNCT_1:def 5; end; then rng(D1|j1) c= rng(D2|indx(D2,D1,j1)) by TARSKI:def 3; hence thesis by A68,XBOOLE_0:def 10; end; then A77: D2|indx(D2,D1,j1)=D1|j1 by A50,A52,Th5; A78: for k st 1 <= k & k <= j1 holds k=indx(D2,D1,k) proof let k; assume A79:1 <= k & k <= j1; assume A80:k<>indx(D2,D1,k); now per cases by A80,AXIOMS:21; suppose A81:k > indx(D2,D1,k); 1 <= k & k <= len D1 by A51,A79,AXIOMS:22; then A82: k in dom D1 by FINSEQ_3:27; then A83: indx(D2,D1,k) in dom D2 & D1.k=D2.indx(D2,D1,k) by A21,INTEGRA1:def 21; then indx(D2,D1,k) in Seg len D2 by FINSEQ_1:def 3; then A84: 1<=indx(D2,D1,k)&indx(D2,D1,k)<=indx(D2,D1,j1) by A21,A51,A79,A82,Th6,FINSEQ_1:3; then indx(D2,D1,k) in Seg indx(D2,D1,j1) by FINSEQ_1:3; then A85: (D2|indx(D2,D1,j1)).indx(D2,D1,k)=D2.indx(D2,D1,k) by A49, RFINSEQ:19; A86: indx(D2,D1,k) < j1 by A79,A81,AXIOMS:22; then indx(D2,D1,k) <= len D1 by A51,AXIOMS:22; then indx(D2,D1,k) in dom D1 by A84,FINSEQ_3:27; then A87: D1.k > D1.indx(D2,D1,k) by A81,A82,GOBOARD1:def 1; indx(D2,D1,k) in Seg j1 by A84,A86,FINSEQ_1:3; hence contradiction by A51,A77,A83,A85,A87,RFINSEQ:19; suppose A88:k < indx(D2,D1,k); k in Seg j1 by A79,FINSEQ_1:3; then A89: D1.k = (D1|j1).k by A47,RFINSEQ:19; 1 <= k & k <= len D1 by A51,A79,AXIOMS:22; then A90: k in dom D1 by FINSEQ_3:27; then A91: indx(D2,D1,k) in dom D2 & D1.k=D2.indx(D2,D1,k) by A21,INTEGRA1:def 21; indx(D2,D1,k) <= indx(D2,D1,j1) & indx(D2,D1,k) in dom D2 & indx(D2,D1,j1) in dom D2 by A21,A51,A79,A90,Th6; then A92: k <= indx(D2,D1,j1) by A88,AXIOMS:22; then k <= len D2 by A49,AXIOMS:22; then k in dom D2 by A79,FINSEQ_3:27; then A93: D2.k < D2.indx(D2,D1,k) by A88,A91,GOBOARD1:def 1; k in Seg indx(D2,D1,j1) by A79,A92,FINSEQ_1:3; hence contradiction by A49,A77,A89,A91,A93,RFINSEQ:19; end; hence contradiction; end; A94: len (D2|indx(D2,D1,j1))=len (D1|j1) by A50,A52,A53,Th5; len (D2|indx(D2,D1,j1))=indx(D2,D1,j1) by A49,TOPREAL1:3; then A95: indx(D2,D1,j1) =j1 by A51,A94,TOPREAL1:3; j1 <= len D1 by A47,FINSEQ_3:27; then j1 <= len lower_volume(f,D1) by INTEGRA1:def 8; then A96: len(lower_volume(f,D1)|j1)=indx(D2,D1,j1) by A95,TOPREAL1:3; indx(D2,D1,j1) in dom D2 by A21,A47,INTEGRA1:def 21; then indx(D2,D1,j1) <= len D2 by FINSEQ_3:27; then indx(D2,D1,j1) <= len lower_volume(f,D2) by INTEGRA1:def 8; then A97: len(lower_volume(f,D1)|j1)=len(lower_volume(f,D2)|indx(D2,D1,j1)) by A96,TOPREAL1:3; for k st 1 <= k & k <= len(lower_volume(f,D1)|j1) holds (lower_volume(f,D1)|j1).k = (lower_volume(f,D2)|indx(D2,D1,j1)).k proof let k; assume A98:1 <= k & k <= len(lower_volume(f,D1)|j1); A99: len(lower_volume(f,D1)) = len D1 by INTEGRA1:def 8; then A100: 1 <= k & k <= j1 by A51,A98,TOPREAL1:3; then A101: k in Seg j1 by FINSEQ_1:3; 1 <= k & k <= len D1 by A51,A100,AXIOMS:22; then A102: k in Seg len D1 by FINSEQ_1:3; A103: divset(D1,k)=divset(D2,indx(D2,D1,k)) proof A104: divset(D2,indx(D2,D1,k))= [. inf divset(D2,indx(D2,D1,k)), sup divset(D2,indx(D2,D1,k)).] by INTEGRA1:5; A105: k in dom D1 by A102,FINSEQ_1:def 3; then A106: indx(D2,D1,k) in dom D2 & D1.k=D2.indx(D2,D1,k) by A21, INTEGRA1:def 21; inf divset(D1,k)=inf divset(D2,indx(D2,D1,k)) & sup divset(D1,k)=sup divset(D2,indx(D2,D1,k)) proof per cases; suppose A107:k=1; then A108: inf divset(D1,k)=inf A&sup divset(D1,k)=D1.k by A105,INTEGRA1 :def 5; indx(D2,D1,k)=1 by A51,A78,A107; hence thesis by A106,A108,INTEGRA1:def 5; suppose A109:k<>1; then A110: inf divset(D1,k)=D1.(k-1) & sup divset(D1,k)=D1.k by A105,INTEGRA1:def 5; A111: k-1 in dom D1 & D1.(k-1) in A & k-1 in NAT by A105,A109,INTEGRA1:9; reconsider k1=k-1 as Nat by A105,A109,INTEGRA1:9; k <= k+1 by NAT_1:29; then k1 <= k by REAL_1:86; then A112: k1 <= j1 by A100,AXIOMS:22; k1 in Seg len D1 by A111,FINSEQ_1:def 3; then 1 <= k1 by FINSEQ_1:3; then A113: k1=indx(D2,D1,k1) by A78,A112; indx(D2,D1,k)<>1 by A78,A100,A109; then A114: inf divset(D2,indx(D2,D1,k))=D2.(indx(D2,D1,k)-1) & sup divset(D2,indx(D2,D1,k))=D2.indx(D2,D1,k) by A106,INTEGRA1:def 5; D2.(indx(D2,D1,k)-1)= D2.indx(D2,D1,k1) by A78,A100,A113; hence thesis by A21,A105,A110,A111,A114,INTEGRA1:def 21; end; hence thesis by A104,INTEGRA1:5; end; j1 in Seg len(lower_volume(f,D1)) by A51,A99,FINSEQ_1:def 3; then j1 in dom(lower_volume(f,D1)) by FINSEQ_1:def 3; then A115: (lower_volume(f,D1)|j1).k = lower_volume(f,D1).k by A101, RFINSEQ:19 .=(inf(rng(f|divset(D2,indx(D2,D1,k)))))*vol(divset(D2,indx(D2,D1,k))) by A102,A103,INTEGRA1:def 8; indx(D2,D1,k) in Seg j1 by A78,A100,A101; then A116: indx(D2,D1,k) in Seg indx(D2,D1,j1) by A51,A78; then 1<=indx(D2,D1,k) & indx(D2,D1,k)<=indx(D2,D1,j1) by FINSEQ_1:3; then 1<=indx(D2,D1,k) & indx(D2,D1,k)<=len D2 by A49,AXIOMS:22; then A117: indx(D2,D1,k) in Seg len D2 by FINSEQ_1:3; indx(D2,D1,j1) in Seg len D2 by A49,FINSEQ_1:def 3; then indx(D2,D1,j1) in Seg len lower_volume(f,D2) by INTEGRA1:def 8; then A118: indx(D2,D1,j1) in dom lower_volume(f,D2) by FINSEQ_1:def 3; (lower_volume(f,D2)|indx(D2,D1,j1)).k =(lower_volume(f,D2)|indx(D2,D1,j1)).indx(D2,D1,k) by A78,A100 .=lower_volume(f,D2).indx(D2,D1,k) by A116,A118,RFINSEQ:19 .=(inf(rng(f|divset(D2,indx(D2,D1,k)))))*vol(divset(D2,indx(D2,D1,k))) by A117,INTEGRA1:def 8; hence thesis by A115; end; then A119: lower_volume(f,D2)|indx(D2,D1,j1)=lower_volume(f,D1)|j1 by A97,FINSEQ_1:18; indx(D2,D1,j1) in Seg len D2 by A49,FINSEQ_1:def 3; then A120: indx(D2,D1,j1) in Seg len lower_volume(f,D2) by INTEGRA1:def 8; j1 in Seg len D1 by A51,FINSEQ_1:def 3; then A121: j1 in Seg len lower_volume(f,D1) by INTEGRA1:def 8; A122: PLf(D2,indx(D2,D1,j1))=Sum(lower_volume(f,D2)|indx(D2,D1,j1)) by A120,INTEGRA1:def 22 .=PLf(D1,j1) by A119,A121,INTEGRA1:def 22; A123: indx(D2,D1,j1)+1 <= indx(D2,D1,j) proof j < j+1 by NAT_1:38; then j1 < j by REAL_1:84; then indx(D2,D1,j1) < indx(D2,D1,j) by A21,A45,A51,Th7; hence thesis by NAT_1:38; end; A124: Sum mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) -Sum mid(lower_volume(f,D1),j,j) <= (sup rng f-inf rng f)*delta(D1) proof A125: 1 <= indx(D2,D1,j) & indx(D2,D1,j) <= len lower_volume(f,D2) proof indx(D2,D1,j) in dom D2 by A21,A45,INTEGRA1:def 21; then indx(D2,D1,j) in Seg len D2 by FINSEQ_1:def 3; then indx(D2,D1,j) in Seg len lower_volume(f,D2) by INTEGRA1:def 8; hence thesis by FINSEQ_1:3; end; then A126: 1 <= indx(D2,D1,j1)+1 & indx(D2,D1,j1)+1 <= len lower_volume(f, D2) by A49,A123,AXIOMS:22,NAT_1:38; A127: indx(D2,D1,j)-indx(D2,D1,j1) <= 2 proof assume A128:indx(D2,D1,j)-indx(D2,D1,j1) > 2; reconsider ID1=indx(D2,D1,j1)+1 as Nat; reconsider ID2=ID1+1 as Nat; A129: indx(D2,D1,j1) < ID1 & ID1 < ID2 & ID2 < indx(D2,D1,j) proof thus indx(D2,D1,j1) < ID1 by NAT_1:38; thus ID1 < ID2 by NAT_1:38; indx(D2,D1,j1)+(1+1)<indx(D2,D1,j) by A128,REAL_1:86; hence thesis by XCMPLX_1:1; end; A130: indx(D2,D1,j) in dom D2 by A21,A45,INTEGRA1:def 21; then A131: indx(D2,D1,j) <= len D2 by FINSEQ_3:27; A132: ID1 in dom D2 proof 1 <= ID1 & ID1 <= indx(D2,D1,j) by A49,A129,AXIOMS:22; then 1 <= ID1 & ID1 <= len D2 by A131,AXIOMS:22; then ID1 in Seg len D2 by FINSEQ_1:3; hence thesis by FINSEQ_1:def 3; end; A133: ID2 in dom D2 proof indx(D2,D1,j1) <= ID2 & ID2 <= len D2 by A129,A131,AXIOMS:22; then 1 <= ID2 & ID2 <= len D2 by A49,AXIOMS:22; then ID2 in Seg len D2 by FINSEQ_1:3; hence thesis by FINSEQ_1:def 3; end; then A134: D2.indx(D2,D1,j1)<D2.ID1 & D2.ID1<D2.ID2 & D2.ID2<D2.indx(D2,D1, j) by A49,A129,A130,A132,GOBOARD1:def 1; A135: D2.ID1 in rng D & D2.ID2 in rng D proof A136: D2.ID1 in rng D \/ rng D1 & D2.ID2 in rng D \/ rng D1 by A21,A132,A133,FUNCT_1:def 5; A137: D1.j1 = D2.indx(D2,D1,j1) & D1.j = D2.indx(D2,D1,j) by A21,A45,A51,INTEGRA1:def 21; not(D2.ID1 in rng D1) & not D2.ID2 in rng D1 proof assume A138:D2.ID1 in rng D1 or D2.ID2 in rng D1; per cases by A138; suppose D2.ID1 in rng D1; then consider n such that A139: n in dom D1 & D1.n=D2.ID1 by PARTFUN1:26; D2.indx(D2,D1,j1)<D2.ID1 & D2.ID1<D2.indx(D2,D1,j) by A134,AXIOMS:22; then j1<n & n<j by A45,A51,A137,A139,GOBOARD2:18; then j<n+1 & n<j by REAL_1:84; hence contradiction by NAT_1:38; suppose D2.ID2 in rng D1; then consider n such that A140: n in dom D1 & D1.n=D2.ID2 by PARTFUN1:26; D2.indx(D2,D1,j1)<D2.ID2 & D2.ID2<D2.indx(D2,D1,j) by A134,AXIOMS:22; then j1<n & n<j by A45,A51,A137,A140,GOBOARD2:18; then j<n+1 & n<j by REAL_1:84; hence contradiction by NAT_1:38; end; hence thesis by A136,XBOOLE_0:def 2; end; A141: inf divset(D1,j)=D2.indx(D2,D1,j1) & sup divset(D1,j)=D2.indx(D2,D1,j) proof inf divset(D1,j)=D1.j1 & sup divset(D1,j)=D1.j by A45,A46,INTEGRA1:def 5; hence thesis by A21,A45,A51,INTEGRA1:def 21; end; A142: D2.ID1 in rng D /\ divset(D1,j) proof D2.indx(D2,D1,j1)<=D2.ID1&D2.ID1<=D2.indx(D2,D1,j) by A134,AXIOMS:22; then D2.ID1 in divset(D1,j) by A141,INTEGRA2:1; hence thesis by A135,XBOOLE_0:def 3; end; D2.ID2 in rng D /\ divset(D1,j) proof D2.indx(D2,D1,j1)<=D2.ID2&D2.ID2<=D2.indx(D2,D1,j) by A134,AXIOMS:22; then D2.ID2 in divset(D1,j) by A141,INTEGRA2:1; hence thesis by A135,XBOOLE_0:def 3; end; then D2.ID1=D2.ID2 by A20,A45,A142,Th4; hence contradiction by A129,A132,A133,GOBOARD2:19; end; 1 <= indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 & indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 <= 2 proof A143: indx(D2,D1,j)-'(indx(D2,D1,j1)+1) =indx(D2,D1,j)-(indx(D2,D1,j1)+1) by A123,SCMFSA_7:3; then indx(D2,D1,j)-'(indx(D2,D1,j1)+1) >= 0 by A123,SQUARE_1:12; then indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 >= 0+1 by AXIOMS:24; hence indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 >= 1; indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 =indx(D2,D1,j)-indx(D2,D1,j1)-1+1 by A143,XCMPLX_1:36 .=indx(D2,D1,j)-indx(D2,D1,j1)+1-1 by XCMPLX_1:29 .=indx(D2,D1,j)-indx(D2,D1,j1) by XCMPLX_1:26; hence thesis by A127; end; then A144: 1 <= len mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j )) & len mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j))<=2 by A123,A125,A126,JORDAN3:27; A145: 1 <= j & j <= len lower_volume(f,D1) proof 1 <= j & j <= len D1 by A45,FINSEQ_3:27; hence thesis by INTEGRA1:def 8; end; j-'j+1 = 1 by Lm3; then A146: len mid(lower_volume(f,D1),j,j)= 1 by A145,JORDAN3:27; mid(lower_volume(f,D1),j,j).1 = lower_volume(f,D1).j by A145,JORDAN3 :27; then mid(lower_volume(f,D1),j,j) =<*lower_volume(f,D1).j*> by A146,FINSEQ_1:57; then A147: Sum mid(lower_volume(f,D1),j,j)=lower_volume(f,D1).j by RVSUM_1: 103; A148: indx(D2,D1,j1)+1 in Seg len D2 proof indx(D2,D1,j1)+1 in Seg len lower_volume(f,D2) by A126,FINSEQ_1:3; hence thesis by INTEGRA1:def 8; end; A149: j in Seg len D1 by A45,FINSEQ_1:def 3; now per cases by A144,Lm4; suppose A150: len mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j))=1; A151: indx(D2,D1,j1)+1=indx(D2,D1,j) proof indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1=1 by A123,A125,A126,A150,JORDAN3:27; then indx(D2,D1,j)-'(indx(D2,D1,j1)+1)=1-1 by XCMPLX_1:26; then indx(D2,D1,j)-(indx(D2,D1,j1)+1)=0 by A123,SCMFSA_7:3; then indx(D2,D1,j)=0+(indx(D2,D1,j1)+1) by XCMPLX_1:27; hence thesis; end; A152: divset(D2,indx(D2,D1,j))=divset(D1,j) proof inf divset(D1,j)=D2.indx(D2,D1,j1) & sup divset(D1,j)=D2.indx(D2,D1,j) proof inf divset(D1,j)=D1.j1 & sup divset(D1,j)=D1.j by A45,A46,INTEGRA1:def 5; hence thesis by A21,A45,A51,INTEGRA1:def 21; end; then A153: divset(D1,j)=[. D2.indx(D2,D1,j1),D2.indx(D2,D1,j).] by INTEGRA1:5; A154: indx(D2,D1,j)-1=indx(D2,D1,j1) by A151,XCMPLX_1:26; A155: indx(D2,D1,j) in dom D2 by A21,A45,INTEGRA1:def 21; indx(D2,D1,j)<>1 by A49,A151,NAT_1:38; then inf divset(D2,indx(D2,D1,j))=D2.indx(D2,D1,j1) & sup divset(D2,indx(D2,D1,j))=D2.indx(D2,D1,j) by A154,A155,INTEGRA1:def 5; hence thesis by A153,INTEGRA1:5; end; mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)).1 =lower_volume(f,D2).(indx(D2,D1,j1)+1) by A125,A126,JORDAN3:27; then mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =<*lower_volume(f,D2).(indx(D2,D1,j1)+1)*> by A150,FINSEQ_1:57; then A156: Sum mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j )) =lower_volume(f,D2).(indx(D2,D1,j1)+1) by RVSUM_1:103 .=(inf(rng(f|divset(D2,(indx(D2,D1,j1)+1))))) *vol(divset(D2,(indx(D2,D1,j1)+1))) by A148,INTEGRA1:def 8 .=Sum mid(lower_volume(f,D1),j,j) by A147,A149,A151,A152,INTEGRA1:def 8; A157: delta(D1) >= 0 by Th8; sup rng f - inf rng f >= 0 by A18,SQUARE_1:12; then (sup rng f - inf rng f)*delta(D1) >= 0*delta(D1) by A157,AXIOMS: 25; hence Sum mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) -Sum mid(lower_volume(f,D1),j,j) <= (sup rng f-inf rng f)*delta(D1) by A156,XCMPLX_1:14; suppose A158: len mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j))=2; A159: mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)).1 =lower_volume(f,D2).(indx(D2,D1,j1)+1) by A125,A126,JORDAN3:27; mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)).2 =lower_volume(f,D2).(indx(D2,D1,j1)+2) proof A160: 2+(indx(D2,D1,j1)+1)>=0+1 by A126,REAL_1:55; mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)).2 =LVf(D2).(2+(indx(D2,D1,j1)+1)-'1) by A123,A125,A126,A158,JORDAN3: 27 .=LVf(D2).(2+(indx(D2,D1,j1)+1)-1) by A160,SCMFSA_7:3 .=LVf(D2).((indx(D2,D1,j1)+1)+(2-1)) by XCMPLX_1:29 .=LVf(D2).(indx(D2,D1,j1)+(1+1)) by XCMPLX_1:1; hence thesis; end; then mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =<*lower_volume(f,D2).(indx(D2,D1,j1)+1), lower_volume(f,D2).(indx(D2,D1,j1)+2)*> by A158,A159,FINSEQ_1:61; then A161: Sum mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =lower_volume(f,D2).(indx(D2,D1,j1)+1) +lower_volume(f,D2).(indx(D2,D1,j1)+2) by RVSUM_1:107; A162: vol(divset(D1,j)) =vol(divset(D2,indx(D2,D1,j1)+1))+vol(divset(D2,indx(D2,D1,j1)+2)) proof A163: inf divset(D1,j)=D2.indx(D2,D1,j1)&sup divset(D1,j)=D2.indx(D2,D1,j) proof inf divset(D1,j)=D1.j1 & sup divset(D1,j)=D1.j by A45,A46,INTEGRA1:def 5; hence thesis by A21,A45,A51,INTEGRA1:def 21; end; A164: indx(D2,D1,j)=indx(D2,D1,j1)+2 proof indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1=2 by A123,A125,A126,A158,JORDAN3:27; then indx(D2,D1,j)-(indx(D2,D1,j1)+1)+1=2 by A123,SCMFSA_7:3; then indx(D2,D1,j)-indx(D2,D1,j1)-1+1=2 by XCMPLX_1:36; then indx(D2,D1,j)-indx(D2,D1,j1)-(1-1)=2 by XCMPLX_1:37; hence thesis by XCMPLX_1:27; end; A165: inf divset(D2,(indx(D2,D1,j1)+2))=D2.(indx(D2,D1,j1)+1) & sup divset(D2,(indx(D2,D1,j1)+2))=D2.(indx(D2,D1,j1)+2) & inf divset(D2,(indx(D2,D1,j1)+1))=D2.indx(D2,D1,j1) & sup divset(D2,(indx(D2,D1,j1)+1))=D2.(indx(D2,D1,j1)+1) proof A166: indx(D2,D1,j1)+2 in dom D2 by A21,A45,A164,INTEGRA1:def 21; indx(D2,D1,j1)+1 in Seg len lower_volume(f,D2) by A126,FINSEQ_1:3 ; then indx(D2,D1,j1)+1 in Seg len D2 by INTEGRA1:def 8; then A167: indx(D2,D1,j1)+1 in dom D2 by FINSEQ_1:def 3; A168: indx(D2,D1,j1)+1 <> 1 by A49,NAT_1:38; indx(D2,D1,j1)+1+1 > 1 by A126,NAT_1:38; then A169: indx(D2,D1,j1)+(1+1) > 1 by XCMPLX_1:1; A170: indx(D2,D1,j1)+2-1=indx(D2,D1,j1)+1 by Lm2,XCMPLX_1:29; A171: indx(D2,D1,j1)+1-1=indx(D2,D1,j1)+(1-1) by XCMPLX_1:29 .=indx(D2,D1,j1)+0; thus inf divset(D2,(indx(D2,D1,j1)+2))=D2.(indx(D2,D1,j1)+1) & sup divset(D2,(indx(D2,D1,j1)+2))=D2.(indx(D2,D1,j1)+2) by A166,A169,A170,INTEGRA1:def 5; thus thesis by A167,A168,A171,INTEGRA1:def 5; end; vol(divset(D1,j)) =sup divset(D1,j)-inf divset(D1,j) by INTEGRA1:def 6 .=D2.indx(D2,D1,j)-(D2.(indx(D2,D1,j1)+1) -D2.(indx(D2,D1,j1)+1))-D2.indx(D2,D1,j1) by A163,XCMPLX_1:17 .=D2.(indx(D2,D1,j1)+2)-D2.(indx(D2,D1,j1)+1) +D2.(indx(D2,D1,j1)+1)-D2.indx(D2,D1,j1) by A164,XCMPLX_1:37; then vol(divset(D1,j)) =vol(divset(D2,indx(D2,D1,j1)+2)) +D2.(indx(D2,D1,j1)+1)-D2.indx(D2,D1,j1) by A165,INTEGRA1:def 6 .=vol(divset(D2,indx(D2,D1,j1)+2)) +(sup divset(D2,indx(D2,D1,j1)+1)-inf divset(D2,indx(D2,D1,j1)+1)) by A165,XCMPLX_1:29; hence thesis by INTEGRA1:def 6; end; then A172: lower_volume(f,D1).j=(inf(rng(f|divset(D1,j))))* (vol(divset(D2,indx(D2,D1,j1)+1))+vol(divset(D2,indx(D2,D1,j1)+2))) by A149,INTEGRA1:def 8; A173: vol(divset(D2,indx(D2,D1,j1)+1))>=0 by INTEGRA1:11; A174: vol(divset(D2,indx(D2,D1,j1)+2))>=0 by INTEGRA1:11; A175: Sum mid(LVf(D2),indx(D2,D1,j1)+1,indx(D2,D1,j))-Sum mid(LVf(D1) ,j,j) <=(sup rng f - inf rng f)*(vol(divset(D2,indx(D2,D1,j1)+2)) +vol(divset(D2,indx(D2,D1,j1)+1))) proof A176: indx(D2,D1,j) in dom D2 by A21,A45,INTEGRA1:def 21; A177: indx(D2,D1,j)=indx(D2,D1,j1)+2 proof indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1=2 by A123,A125,A126,A158,JORDAN3:27; then indx(D2,D1,j)-(indx(D2,D1,j1)+1)+1=2 by A123,SCMFSA_7:3; then indx(D2,D1,j)-indx(D2,D1,j1)-1+1=2 by XCMPLX_1:36; then indx(D2,D1,j)-indx(D2,D1,j1)-(1-1)=2 by XCMPLX_1:37; hence thesis by XCMPLX_1:27; end; then A178: indx(D2,D1,j1)+2 in Seg len D2 by A176,FINSEQ_1:def 3; set ID1=indx(D2,D1,j1)+1; set ID2=indx(D2,D1,j1)+2; A179: Sum mid(LVf(D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =(inf rng(f|divset(D2,indx(D2,D1,j1)+2))) *vol(divset(D2,indx(D2,D1,j1)+2)) +LVf(D2).(indx(D2,D1,j1)+1) by A161,A178,INTEGRA1:def 8 .=(inf rng(f|divset(D2,indx(D2,D1,j1)+2))) *vol(divset(D2,indx(D2,D1,j1)+2)) +(inf rng(f|divset(D2,indx(D2,D1,j1)+1))) *vol(divset(D2,indx(D2,D1,j1)+1)) by A148,INTEGRA1:def 8; divset(D2,ID2)c=A by A176,A177,INTEGRA1:10; then inf rng(f|divset(D2,ID2)) <= sup rng f by A1,Lm7; then A180: (inf rng(f|divset(D2,ID2)))*vol(divset(D2,ID2)) <=(sup rng f)*vol(divset(D2,ID2)) by A174,AXIOMS:25; ID1 in dom D2 by A148,FINSEQ_1:def 3; then divset(D2,ID1)c=A by INTEGRA1:10; then inf rng(f|divset(D2,ID1)) <= sup rng f by A1,Lm7; then A181: (inf rng(f|divset(D2,ID1)))*vol(divset(D2,ID1)) <=(sup rng f)*vol(divset(D2,ID1)) by A173,AXIOMS:25; divset(D1,j) c=A by A45,INTEGRA1:10; then inf rng(f|divset(D1,j)) >= inf rng f by A1,Lm7; then A182: (inf rng(f|divset(D1,j)))*vol(divset(D2,ID2)) >=(inf rng f)*vol(divset(D2,ID2)) & (inf rng(f|divset(D1,j)))*vol(divset(D2,ID1)) >=(inf rng f)*vol(divset(D2,ID1)) by A173,A174,AXIOMS:25; Sum mid(LVf(D2),ID1,indx(D2,D1,j)) -(inf rng(f|divset(D2,ID1)))*vol(divset(D2,ID1)) <=(sup rng f)*vol(divset(D2,ID2)) by A179,A180,XCMPLX_1:26; then Sum mid(LVf(D2),ID1,indx(D2,D1,j)) <=(sup rng f)*vol(divset(D2,ID2)) +(inf rng(f|divset(D2,ID1)))*vol(divset(D2,ID1)) by REAL_1:86; then Sum mid(LVf(D2),ID1,indx(D2,D1,j))-(sup rng f)*vol(divset(D2, ID2)) <=(inf rng(f|divset(D2,ID1)))*vol(divset(D2,ID1)) by REAL_1:86; then Sum mid(LVf(D2),ID1,indx(D2,D1,j))-(sup rng f)*vol(divset(D2, ID2)) <=(sup rng f)*vol(divset(D2,ID1)) by A181,AXIOMS:22; then A183: Sum mid(LVf(D2),ID1,indx(D2,D1,j))<=(sup rng f)*vol(divset(D2,ID2 ))+ (sup rng f)*vol(divset(D2,ID1)) by REAL_1:86; Sum mid(LVf(D1),j,j) =(inf rng(f|divset(D1,j)))*vol(divset(D2,indx(D2,D1,j1)+2)) +(inf rng(f|divset(D1,j)))*vol(divset(D2,indx(D2,D1,j1)+1)) by A147,A172,XCMPLX_1:8; then Sum mid(LVf(D1),j,j)-(inf rng(f|divset(D1,j)))*vol(divset(D2, ID1)) >=(inf rng f)*vol(divset(D2,ID2)) by A182,XCMPLX_1:26; then Sum mid(LVf(D1),j,j)>=(inf rng(f|divset(D1,j)))*vol(divset(D2, ID1)) +(inf rng f)*vol(divset(D2,ID2)) by REAL_1:84; then Sum mid(LVf(D1),j,j)-(inf rng f)*vol(divset(D2,ID2)) >=(inf rng(f|divset(D1,j)))*vol(divset(D2,ID1)) by REAL_1:84; then Sum mid(LVf(D1),j,j)-(inf rng f)*vol(divset(D2,ID2)) >=(inf rng f)*vol(divset(D2,ID1)) by A182,AXIOMS:22; then Sum mid(LVf(D1),j,j) >=(inf rng f)*vol(divset(D2,ID2))+(inf rng f)*vol(divset(D2,ID1)) by REAL_1:84; then Sum mid(LVf(D2),ID1,indx(D2,D1,j))-Sum mid(LVf(D1),j,j) <=(sup rng f)*vol(divset(D2,ID2))+(sup rng f)*vol(divset(D2,ID1)) -((inf rng f)*vol(divset(D2,ID2))+(inf rng f)*vol(divset(D2,ID1))) by A183,REAL_1:92; then Sum mid(LVf(D2),ID1,indx(D2,D1,j))-Sum mid(LVf(D1),j,j) <=((sup rng f)*vol(divset(D2,ID2))-(inf rng f)*vol(divset(D2,ID2))) +((sup rng f)*vol(divset(D2,ID1))-(inf rng f)*vol(divset(D2,ID1))) by Lm5; then Sum mid(LVf(D2),ID1,indx(D2,D1,j))-Sum mid(LVf(D1),j,j) <=(sup rng f - inf rng f)*vol(divset(D2,ID2)) +((sup rng f)*vol(divset(D2,ID1))-(inf rng f)*vol(divset(D2,ID1))) by XCMPLX_1:40; then Sum mid(LVf(D2),ID1,indx(D2,D1,j))-Sum mid(LVf(D1),j,j) <=(sup rng f - inf rng f)*vol(divset(D2,ID2)) +(sup rng f - inf rng f)*vol(divset(D2,ID1)) by XCMPLX_1:40; hence thesis by XCMPLX_1:8; end; A184: vol(divset(D1,j)) <= delta(D1) by A45,Lm8; sup rng f - inf rng f >= 0 by A18,SQUARE_1:12; then (sup rng f - inf rng f)*(vol(divset(D1,j))) <=(sup rng f - inf rng f)*delta(D1) by A184,AXIOMS:25; hence thesis by A162,A175,AXIOMS:22; end; hence thesis; end; A185: PLf(D2,indx(D2,D1,j1)) +Sum mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =PLf(D2,indx(D2,D1,j)) proof indx(D2,D1,j) in dom D2 by A21,A45,INTEGRA1:def 21; then A186: indx(D2,D1,j) in Seg len D2 by FINSEQ_1:def 3; then A187: 1 <= indx(D2,D1,j) & indx(D2,D1,j) <= len D2 by FINSEQ_1:3; then A188: 1 <= indx(D2,D1,j) & indx(D2,D1,j) <= len LVf(D2) by INTEGRA1: def 8; A189: indx(D2,D1,j) in Seg len LVf(D2) by A186,INTEGRA1:def 8; A190: indx(D2,D1,j1) < indx(D2,D1,j) by A123,NAT_1:38; indx(D2,D1,j1) in Seg len D2 by A49,FINSEQ_1:def 3; then indx(D2,D1,j1) in Seg len LVf(D2) by INTEGRA1:def 8; then PLf(D2,indx(D2,D1,j1)) =Sum(LVf(D2)|indx(D2,D1,j1)) by INTEGRA1:def 22; then PLf(D2,indx(D2,D1,j1)) +Sum mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =Sum(LVf(D2)|indx(D2,D1,j1) ^mid(LVf(D2),(indx(D2,D1,j1)+1),indx(D2,D1,j))) by RVSUM_1:105 .=Sum(mid(LVf(D2),1,indx(D2,D1,j1)) ^mid(LVf(D2),indx(D2,D1,j1)+1,indx(D2,D1,j))) by A49,JORDAN3:25 .=Sum(mid(LVf(D2),1,indx(D2,D1,j))) by A49,A188,A190,INTEGRA2:4 .=Sum(LVf(D2)|indx(D2,D1,j)) by A187,JORDAN3:25; hence thesis by A189,INTEGRA1:def 22; end; PLf(D1,j1)+Sum mid(lower_volume(f,D1),j,j)=PLf(D1,j) proof A191: j in Seg len D1 by A45,FINSEQ_1:def 3; then A192: 1 <= j & j <= len D1 by FINSEQ_1:3; then A193: 1 <= j & j <= len LVf(D1) by INTEGRA1:def 8; A194: j in Seg len LVf(D1) by A191,INTEGRA1:def 8; A195: j1+1=j by XCMPLX_1:27; j < j+1 by NAT_1:38; then A196: j1 < j by REAL_1:84; j1 in Seg len D1 by A51,FINSEQ_1:def 3; then j1 in Seg len LVf(D1) by INTEGRA1:def 8; then PLf(D1,j1)=Sum(LVf(D1)|j1) by INTEGRA1:def 22; then PLf(D1,j1)+Sum mid(LVf(D1),j,j) =Sum((LVf(D1)|j1)^mid(LVf(D1),j,j)) by RVSUM_1:105 .=Sum(mid(LVf(D1),1,j1)^mid(LVf(D1),j1+1,j)) by A51,A195,JORDAN3:25 .=Sum(mid(LVf(D1),1,j)) by A51,A193,A196,INTEGRA2:4 .=Sum(LVf(D1)|j) by A192,JORDAN3:25; hence thesis by A194,INTEGRA1:def 22; end; then PLf(D2,indx(D2,D1,j))-PLf(D1,j) =(PLf(D2,indx(D2,D1,j1))-PLf(D1,j1)) +(Sum mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) -Sum mid(lower_volume(f,D1),j,j)) by A185,Lm5 .=0+(Sum mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) -Sum mid(lower_volume(f,D1),j,j)) by A122,XCMPLX_1:14; hence thesis by A124; end; hence thesis by A45; end; A197: for i being non empty Nat st P[i] holds P[i+1] proof let i be non empty Nat; assume A198:P[i]; A199: i>=1 by RLVECT_1:99; P[i+1] proof assume A200:i+1 in dom D; then D.(i+1) in A by INTEGRA1:8; then consider j such that A201: j in dom D1 & D.(i+1) in divset(D1,j) by Th2; A202: 1 <= i+1 & i+1 <= len D by A200,FINSEQ_3:27; i <= i+1 by NAT_1:29; then i <= len D by A202,AXIOMS:22; then A203: i in Seg len D by A199,FINSEQ_1:3; then A204: i in dom D by FINSEQ_1:def 3; consider n1 being Nat such that A205: n1 in dom D1 & D.i in divset(D1,n1) & PLf(D2,indx(D2,D1,n1))-PLf(D1,n1)<=i*(sup rng f-inf rng f)*delta(D1) by A198,A203,FINSEQ_1:def 3; A206: indx(D2,D1,n1) in dom D2 & D2.indx(D2,D1,n1) = D1.n1 by A21,A205,INTEGRA1:def 21; then A207: 1 <= indx(D2,D1,n1) & indx(D2,D1,n1) <= len D2 by FINSEQ_3:27; A208: n1 < j proof assume A209:n1 >= j; now per cases by A209,REAL_1:def 5; suppose n1=j; then D.i in divset(D1,j) & D.i in rng D by A204,A205,FUNCT_1:def 5; then A210: D.i in rng D /\ divset(D1,j) by XBOOLE_0:def 3; D.(i+1) in rng D by A200,FUNCT_1:def 5; then D.(i+1) in rng D /\ divset(D1,j) by A201,XBOOLE_0:def 3; then A211: D.(i+1)=D.i by A20,A201,A210,Th4; i+1 > i by REAL_1:69; hence contradiction by A200,A204,A211,GOBOARD2:19; suppose n1>j; then A212: n1>=j+1 by NAT_1:38; then A213: n1-1 >= j by REAL_1:84; A214: inf divset(D1,n1) <= D.i by A205,INTEGRA2:1; A215: D.(i+1)<=sup divset(D1,j) by A201,INTEGRA2:1; 1 <= j by A201,FINSEQ_3:27; then 1+1 <= j+1 by AXIOMS:24; then A216: n1 <> 1 by A212,AXIOMS:22; then n1-1 in dom D1 by A205,INTEGRA1:9; then A217: D1.j <= D1.(n1-1) by A201,A213,GOBOARD2:18; A218: sup divset(D1,j)=D1.j proof per cases; suppose j=1; hence thesis by A201,INTEGRA1:def 5; suppose j<>1; hence thesis by A201,INTEGRA1:def 5; end; D.i >= D1.(n1-1) by A205,A214,A216,INTEGRA1:def 5; then D.i >= D1.j by A217,AXIOMS:22; then A219: D.i >= D.(i+1) by A215,A218,AXIOMS:22; i < i+1 by REAL_1:69; hence contradiction by A200,A204,A219,GOBOARD1:def 1; end; hence thesis; end; then A220: n1+1 <= j by NAT_1:38; then A221: j-n1 >= 1 by REAL_1:84; A222: n1 >= 1 by A205,FINSEQ_3:27; A223: 1 <= n1 & 1 <= j & j <= len D1 by A201,A205,FINSEQ_3:27; then A224: 1 <= n1+1 & n1+1 <= len D1 by A220,AXIOMS:22,NAT_1:37; then A225: n1+1 in dom D1 by FINSEQ_3:27; then A226: indx(D2,D1,n1+1) in dom D2 & D2.indx(D2,D1,n1+1)=D1.(n1+1) by A21,INTEGRA1:def 21; A227: indx(D2,D1,j) in dom D2 & D2.indx(D2,D1,j)=D1.j by A21,A201,INTEGRA1:def 21; then D2.indx(D2,D1,n1+1) <= D2.indx(D2,D1,j) by A201,A220,A225,A226,GOBOARD2:18; then A228: indx(D2,D1,n1+1) <= indx(D2,D1,j) by A226,A227,GOBOARD1:def 1; A229: 1 <= indx(D2,D1,n1+1) & indx(D2,D1,n1+1) <= len D2 by A226,FINSEQ_3:27 ; A230: 1 <= indx(D2,D1,j) & indx(D2,D1,j) <= len D2 by A227,FINSEQ_3:27; n1 < n1+1 by NAT_1:38; then D1.n1 < D1.(n1+1) by A205,A225,GOBOARD1:def 1; then A231: indx(D2,D1,n1) < indx(D2,D1,n1+1) by A206,A226,GOBOARD2:18; 1+indx(D2,D1,n1+1) <= indx(D2,D1,j)+1 by A228,AXIOMS:24; then 1 <= indx(D2,D1,j)+1-indx(D2,D1,n1+1) by REAL_1:84; then A232: mid(D2,indx(D2,D1,n1+1),indx(D2,D1,j)).1 =D2.(1-1+indx(D2,D1,n1+1)) by A228,A229,A230,JORDAN3:31 .=D1.(n1+1) by A21,A225,INTEGRA1:def 21; A233: Sum mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) -Sum mid(LVf(D1),n1+1,j) <= (sup rng f-inf rng f)*delta(D1) proof now per cases by A220,REAL_1:def 5; suppose A234:n1+1=j; then A235: n1=j-1 by XCMPLX_1:26; A236: 1 <= n1+1 & n1+1 <= len LVf(D1) by A224,INTEGRA1:def 8; A237: n1+1 in Seg len D1 by A225,FINSEQ_1:def 3; A238: len mid(LVf(D1),n1+1,j)=j-'(n1+1)+1 by A234,A236,JORDAN3:27 .=j-j+1 by A234,SCMFSA_7:3 .= 1 by XCMPLX_1:25; (n1+1)+1 <= j+1 by A220,AXIOMS:24; then 1 <= j+1-(n1+1) by REAL_1:84; then mid(LVf(D1),n1+1,j).1 =LVf(D1).(1-1+(n1+1)) by A234,A236,JORDAN3:31 .=(inf rng(f|divset(D1,n1+1)))*vol(divset(D1,n1+1)) by A237,INTEGRA1:def 8; then mid(LVf(D1),n1+1,j) =<*(inf rng(f|divset(D1,n1+1)))*vol(divset(D1,n1+1))*> by A238,FINSEQ_1:57; then A239: Sum mid(LVf(D1),n1+1,j) =(inf rng(f|divset(D1,n1+1)))*vol(divset(D1,n1+1)) by RVSUM_1:103; divset(D1,n1+1) c= A by A225,INTEGRA1:10; then A240: inf rng(f|divset(D1,n1+1)) >= inf rng f by A1,Lm7; vol(divset(D1,n1+1)) >= 0 by INTEGRA1:11; then A241: Sum mid(LVf(D1),n1+1,j) >=(inf rng f)*vol(divset(D1,n1+1)) by A239,A240,AXIOMS:25; A242: indx(D2,D1,n1)<indx(D2,D1,j) proof D1.n1 < D1.j by A201,A205,A208,GOBOARD1:def 1; hence thesis by A206,A227,GOBOARD2:18; end; then A243: indx(D2,D1,n1)+1 <= indx(D2,D1,j) by NAT_1:38; then indx(D2,D1,n1)+1 <= len D2 by A230,AXIOMS:22; then A244: indx(D2,D1,n1)+1 <= len LVf(D2) by INTEGRA1:def 8; A245: 1<=indx(D2,D1,n1)+1 by NAT_1:37; A246: indx(D2,D1,j) <= len LVf(D2) by A230,INTEGRA1:def 8; then A247: len mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) =indx(D2,D1,j)-'(indx(D2,D1,n1)+1)+1 by A230,A243,A244,A245,JORDAN3:27 .=indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1 by A243,SCMFSA_7:3 .=indx(D2,D1,j)-indx(D2,D1,n1)-1+1 by XCMPLX_1:36 .=indx(D2,D1,j)-indx(D2,D1,n1)-(1-1) by XCMPLX_1:37 .=indx(D2,D1,j)-indx(D2,D1,n1); A248: indx(D2,D1,j)-indx(D2,D1,n1)<=2 proof assume indx(D2,D1,j) - indx(D2,D1,n1) > 2; then A249: indx(D2,D1,n1)+2 < indx(D2,D1,j) by REAL_1:86; A250: indx(D2,D1,n1) < indx(D2,D1,n1)+1 by NAT_1:38; A251: indx(D2,D1,n1)+1 < indx(D2,D1,n1)+2 by REAL_1:53; A252: indx(D2,D1,n1)+1 < indx(D2,D1,j) & indx(D2,D1,n1)+1 in dom D2 proof thus indx(D2,D1,n1)+1 < indx(D2,D1,j) by A249,A251,AXIOMS:22; then 1 <= indx(D2,D1,n1)+1 & indx(D2,D1,n1)+1 <= len D2 by A207,A230,AXIOMS:22,NAT_1:38; hence thesis by FINSEQ_3:27; end; A253: D2.(indx(D2,D1,n1)+1) in rng D proof A254: D2.(indx(D2,D1,n1)+1) in rng D2 by A252,FUNCT_1:def 5; not( D2.(indx(D2,D1,n1)+1) in rng D1 ) proof assume D2.(indx(D2,D1,n1)+1) in rng D1; then consider k1 being Nat such that A255: k1 in dom D1 & D2.(indx(D2,D1,n1)+1) = D1.k1 by PARTFUN1:26; D2.indx(D2,D1,n1) < D2.(indx(D2,D1,n1)+1) & D2.(indx(D2,D1,n1)+1) < D2.indx(D2,D1,j) by A206,A227,A250,A252,GOBOARD1:def 1; then n1 < k1 & k1 < j by A201,A205,A206,A227,A255,GOBOARD2:18; hence contradiction by A234,NAT_1:38; end; hence thesis by A21,A254,XBOOLE_0:def 2; end; A256: D2.(indx(D2,D1,n1)+1) in rng D /\ divset(D1,j) proof A257: inf divset(D1,j) = D1.(j-1) & sup divset(D1,j) = D1.j by A201,A208,A222,INTEGRA1:def 5; D2.(indx(D2,D1,n1)+1) >= D2.indx(D2,D1,n1) & D2.indx(D2,D1,j) >= D2.(indx(D2,D1,n1)+1) by A206,A227,A250,A252,GOBOARD2:18; then D2.(indx(D2,D1,n1)+1) in divset(D1,j) by A206,A227,A235,A257,INTEGRA2:1; hence thesis by A253,XBOOLE_0:def 3; end; A258: indx(D2,D1,n1) < indx(D2,D1,n1)+2 & indx(D2,D1,n1)+2 in dom D2 proof thus indx(D2,D1,n1) < indx(D2,D1,n1)+2 by A251,NAT_1:38; then 1 <= indx(D2,D1,n1)+2 & indx(D2,D1,n1)+2 <= len D2 by A207,A230,A249,AXIOMS:22; hence thesis by FINSEQ_3:27; end; A259: D2.(indx(D2,D1,n1)+2) in rng D proof A260: D2.(indx(D2,D1,n1)+2) in rng D2 by A258,FUNCT_1:def 5; not D2.(indx(D2,D1,n1)+2) in rng D1 proof assume D2.(indx(D2,D1,n1)+2) in rng D1; then consider k1 being Nat such that A261: k1 in dom D1 & D2.(indx(D2,D1,n1)+2) = D1.k1 by PARTFUN1:26; D2.indx(D2,D1,n1) < D2.(indx(D2,D1,n1)+2) & D2.(indx(D2,D1,n1)+2) < D2.indx(D2,D1,j) by A206,A227,A249,A258,GOBOARD1:def 1; then n1 < k1 & k1 < j by A201,A205,A206,A227,A261,GOBOARD2:18; hence contradiction by A234,NAT_1:38; end; hence thesis by A21,A260,XBOOLE_0:def 2; end; D2.(indx(D2,D1,n1)+2) in rng D /\ divset(D1,j) proof A262: inf divset(D1,j) = D1.(j-1) & sup divset(D1,j) = D1.j by A201,A208,A222,INTEGRA1:def 5; D2.(indx(D2,D1,n1)+2) >= D2.indx(D2,D1,n1) & D2.indx(D2,D1,j) >= D2.(indx(D2,D1,n1)+2) by A206,A227,A249,A258,GOBOARD2:18; then D2.(indx(D2,D1,n1)+2) in divset(D1,j) by A206,A227,A235,A262,INTEGRA2:1; hence thesis by A259,XBOOLE_0:def 3; end; then D2.(indx(D2,D1,n1)+1) = D2.(indx(D2,D1,n1)+2) by A20,A201,A256, Th4; hence contradiction by A251,A252,A258,GOBOARD1:def 1; end; A263: indx(D2,D1,j)-indx(D2,D1,n1)=1 or indx(D2,D1,j)-indx(D2,D1,n1)=2 proof indx(D2,D1,n1)+1 <= indx(D2,D1,j) by A242,NAT_1:38; then A264: indx(D2,D1,n1)+1 = indx(D2,D1,j) or indx(D2,D1,n1)+1 < indx(D2,D1,j) by REAL_1:def 5; indx(D2,D1,n1)+1<indx(D2,D1,j) implies indx(D2,D1,n1)+2=indx(D2,D1,j ) proof assume indx(D2,D1,n1)+1 < indx(D2,D1,j); then indx(D2,D1,n1)+1+1 <= indx(D2,D1,j) by NAT_1:38; then A265: indx(D2,D1,n1)+(1+1) <= indx(D2,D1,j) by XCMPLX_1:1; indx(D2,D1,n1)+2 >= indx(D2,D1,j) by A248,REAL_1:86; hence thesis by A265,AXIOMS:21; end; hence thesis by A264,XCMPLX_1:26; end; Sum mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) <=(sup rng f)*vol(divset(D1,n1+1)) proof per cases by A263; suppose A266:indx(D2,D1,j)-indx(D2,D1,n1)=1; then 1=indx(D2,D1,j)-indx(D2,D1,n1)-1+1 .=indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1 by XCMPLX_1:36; then A267: mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)).1 =LVf(D2).(1+(indx(D2,D1,n1)+1)-1) by A243,A245,A246,JORDAN3:31 .=LVf(D2).((indx(D2,D1,n1)+1)-1+1) by XCMPLX_1:29 .=LVf(D2).((indx(D2,D1,n1)+1)-(1-1)) by XCMPLX_1:37 .=LVf(D2).(indx(D2,D1,n1)+1); A268: indx(D2,D1,n1)+1=indx(D2,D1,j) by A266,XCMPLX_1:27; then indx(D2,D1,n1)+1 in Seg len D2 by A227,FINSEQ_1:def 3; then A269: LVf(D2).(indx(D2,D1,n1)+1) =(inf rng(f|divset(D2,indx(D2,D1,n1)+1))) *vol(divset(D2,indx(D2,D1,n1)+1)) by INTEGRA1:def 8; A270: divset(D2,indx(D2,D1,n1)+1)=divset(D1,n1+1) proof A271: divset(D2,indx(D2,D1,n1)+1) =[.inf divset(D2,indx(D2,D1,n1)+1), sup divset(D2,indx(D2,D1,n1)+1).] by INTEGRA1:5; A272: divset(D1,n1+1)=[.inf divset(D1,n1+1), sup divset(D1,n1+1).] by INTEGRA1:5; indx(D2,D1,n1)+1 > 1 by A207,NAT_1:38; then inf divset(D2,indx(D2,D1,n1)+1)=D2.(indx(D2,D1,n1)+1-1) & sup divset(D2,indx(D2,D1,n1)+1)=D2.(indx(D2,D1,n1)+1) by A227,A268,INTEGRA1:def 5; then A273: inf divset(D2,indx(D2,D1,n1)+1)=D1.n1 & sup divset(D2,indx(D2,D1,n1)+1)=D1.j by A21,A201,A206,A268, INTEGRA1:def 21,XCMPLX_1:26; inf divset(D1,n1+1)=D1.(n1+1-1) & sup divset(D1,n1+1)=D1.(n1+1) by A208,A222,A225,A234,INTEGRA1:def 5; hence thesis by A234,A271,A272,A273,XCMPLX_1:26; end; A274: inf rng(f|divset(D2,indx(D2,D1,n1)+1))<=sup rng f by A1,A227,A268, Th17; vol(divset(D2,indx(D2,D1,n1)+1))>=0 by INTEGRA1:11; then A275: LVf(D2).(indx(D2,D1,n1)+1) <=(sup rng f)*vol(divset(D1,n1+1)) by A269,A270,A274,AXIOMS:25; mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) =<*LVf(D2).(indx(D2,D1,n1)+1)*> by A247,A266,A267,FINSEQ_1:57; hence thesis by A275,RVSUM_1:103; suppose A276:indx(D2,D1,j)-indx(D2,D1,n1)=2; then A277: indx(D2,D1,j)=indx(D2,D1,n1)+2 by XCMPLX_1:27; A278: indx(D2,D1,n1)+2-1=indx(D2,D1,n1)+(2-1) by XCMPLX_1:29 .=indx(D2,D1,n1)+1; then indx(D2,D1,j)-(indx(D2,D1,n1)+1)=1 by A277,XCMPLX_1:18; then A279: indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1>=1; A280: indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1=1+1 by A277,A278,XCMPLX_1:18; A281: 1 <= indx(D2,D1,n1)+1 & indx(D2,D1,n1)+1 <= indx(D2,D1,n1)+2 by NAT_1:37,REAL_1:53; then indx(D2,D1,n1)+1 <= len D2 by A230,A277,AXIOMS:22; then A282: indx(D2,D1,n1)+1 in dom D2 by A281,FINSEQ_3:27; A283: indx(D2,D1,j) <= len LVf(D2) by A230,INTEGRA1:def 8; then A284: mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)).1 =LVf(D2).(1+(indx(D2,D1,n1)+1)-1) by A277,A279,A281,JORDAN3:31 .=LVf(D2).((indx(D2,D1,n1)+1)-1+1) by XCMPLX_1:29 .=LVf(D2).((indx(D2,D1,n1)+1)-(1-1)) by XCMPLX_1:37 .=LVf(D2).(indx(D2,D1,n1)+1); mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)).2 =LVf(D2).(2+(indx(D2,D1,n1)+1)-1) by A277,A280,A281,A283,JORDAN3:31 .=LVf(D2).((indx(D2,D1,n1)+1)-1+2) by XCMPLX_1:29 .=LVf(D2).(indx(D2,D1,n1)+(1-1)+2) by XCMPLX_1:29 .=LVf(D2).(indx(D2,D1,n1)+0+2); then mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) =<*LVf(D2).(indx(D2,D1,n1)+1),LVf(D2).(indx(D2,D1,n1)+2)*> by A247,A276,A284,FINSEQ_1:61; then A285: Sum mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) =LVf(D2).(indx(D2,D1,n1)+1)+LVf(D2).(indx(D2,D1,n1)+2) by RVSUM_1:107; A286: LVf(D2).(indx(D2,D1,n1)+1) <= (sup rng f)*vol(divset(D2,indx(D2,D1,n1)+1)) proof indx(D2,D1,n1)+1 in Seg len D2 by A282,FINSEQ_1:def 3; then A287: LVf(D2).(indx(D2,D1,n1)+1) =(inf rng(f|divset(D2,indx(D2,D1,n1)+1))) *vol(divset(D2,indx(D2,D1,n1)+1)) by INTEGRA1:def 8; A288: inf rng(f|divset(D2,indx(D2,D1,n1)+1)) <=sup rng f by A1,A282,Th17; vol(divset(D2,indx(D2,D1,n1)+1)) >= 0 by INTEGRA1:11; hence thesis by A287,A288,AXIOMS:25; end; LVf(D2).(indx(D2,D1,n1)+2) <= (sup rng f)*vol(divset(D2,indx(D2,D1,n1)+2)) proof indx(D2,D1,n1)+2 in Seg len D2 by A227,A277,FINSEQ_1:def 3; then A289: LVf(D2).(indx(D2,D1,n1)+2) =(inf rng(f|divset(D2,indx(D2,D1,n1)+2))) *vol(divset(D2,indx(D2,D1,n1)+2)) by INTEGRA1:def 8; A290: inf rng(f|divset(D2,indx(D2,D1,n1)+2)) <=sup rng f by A1,A227,A277,Th17; vol(divset(D2,indx(D2,D1,n1)+2)) >= 0 by INTEGRA1:11; hence thesis by A289,A290,AXIOMS:25; end; then A291: Sum mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) <= (sup rng f)*vol(divset(D2,indx(D2,D1,n1)+1)) + (sup rng f)*vol(divset(D2,indx(D2,D1,n1)+2)) by A285,A286,REAL_1:55 ; vol(divset(D2,indx(D2,D1,n1)+1))+vol(divset(D2,indx(D2,D1,n1)+2)) =vol(divset(D1,n1+1)) proof indx(D2,D1,n1)+1 > 1 by A207,NAT_1:38; then A292: inf divset(D2,indx(D2,D1,n1)+1) = D2.(indx(D2,D1,n1)+1-1) & sup divset(D2,indx(D2,D1,n1)+1)=D2.(indx(D2,D1,n1)+1) by A282,INTEGRA1:def 5; then A293: inf divset(D2,indx(D2,D1,n1)+1) = D2.indx(D2,D1,n1) by XCMPLX_1:26; indx(D2,D1,n1)+2 >= 2+1 by A207,AXIOMS:24; then indx(D2,D1,n1)+2 <> 1; then A294: inf divset(D2,indx(D2,D1,n1)+2) = D2.(indx(D2,D1,n1)+1) & sup divset(D2,indx(D2,D1,n1)+2)=D2.indx(D2,D1,j) by A227,A277,A278,INTEGRA1:def 5; A295: vol(divset(D2,indx(D2,D1,n1)+1)) =D2.(indx(D2,D1,n1)+1)-D1.n1 by A206,A292,A293,INTEGRA1:def 6; A296: vol(divset(D2,indx(D2,D1,n1)+2)) =D1.j-D2.(indx(D2,D1,n1)+1) by A227,A294,INTEGRA1:def 6; vol(divset(D1,n1+1))=D1.(n1+1)-D1.n1 proof A297: inf divset(D1,n1+1)=D1.(n1+1-1) & sup divset(D1,n1+1)=D1.(n1+1) by A208,A222,A225,A234,INTEGRA1:def 5; vol(divset(D1,n1+1))=sup divset(D1,n1+1)-inf divset(D1,n1+1) by INTEGRA1:def 6; hence thesis by A297,XCMPLX_1:26; end; hence thesis by A234,A295,A296,XCMPLX_1:39; end; hence thesis by A291,XCMPLX_1:8; end; then Sum mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j))-Sum mid(LVf(D1), n1+1,j) <=(sup rng f)*vol(divset(D1,n1+1))-(inf rng f)*vol(divset(D1,n1+1)) by A241,REAL_1:92; then A298: Sum mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j))-Sum mid(LVf(D1),n1+ 1,j) <=(sup rng f-inf rng f)*vol(divset(D1,n1+1)) by XCMPLX_1:40; sup rng f >= inf rng f by A1,Lm6; then A299: sup rng f-inf rng f >= 0 by SQUARE_1:12; vol(divset(D1,n1+1))<=delta(D1) proof A300: n1+1 in Seg len D1 by A225,FINSEQ_1:def 3; then A301: vol(divset(D1,n1+1))=upper_volume(chi(A,A),D1).(n1+1) by INTEGRA1:22; n1+1 in Seg len upper_volume(chi(A,A),D1) by A300,INTEGRA1:def 7; then n1+1 in dom upper_volume(chi(A,A),D1) by FINSEQ_1:def 3; then A302: vol(divset(D1,n1+1)) in rng upper_volume(chi(A,A),D1) by A301,FUNCT_1:def 5; delta(D1)=max rng upper_volume(chi(A,A),D1) by INTEGRA1:def 19; hence thesis by A302,PRE_CIRC:def 1; end; then (sup rng f-inf rng f)*vol(divset(D1,n1+1)) <=(sup rng f-inf rng f)*delta(D1) by A299,AXIOMS:25; hence Sum mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) -Sum mid(LVf(D1),n1+1,j) <= (sup rng f-inf rng f)*delta(D1) by A298,AXIOMS:22; suppose A303:n1+1 < j; A304: n1 < n1+1 by NAT_1:38; then A305: D1.n1 < D1.(n1+1) by A205,A225,GOBOARD1:def 1; then consider B being closed-interval Subset of REAL, MD1,MD2 being Element of divs B such that A306: D1.n1=inf B & sup B=MD2.(len MD2) & sup B=MD1.(len MD1) & MD1 <= MD2 & MD1=mid(D1,n1+1,j) & MD2=mid(D2,indx(D2,D1,n1+1),indx(D2,D1,j)) by A21,A201,A220,A225,A232,Th14; A307: len MD1 = j-'(n1+1)+1 by A220,A223,A224,A306,JORDAN3:27; j-'(n1+1) = j-(n1+1) by A220,SCMFSA_7:3; then A308: j-'(n1+1)+1=j-n1-1+1 by XCMPLX_1:36 .= j-n1-(1-1) by XCMPLX_1:37 .=j-n1; then A309: len MD1 = j-n1 by A220,A223,A224,A306,JORDAN3:27; A310: len MD1+(n1+1)-1 = j-(n1+1)+1+(n1+1)-1 by A220,A307,SCMFSA_7:3 .=j-((n1+1)-1)+(n1+1)-1 by XCMPLX_1:37 .=j-((n1+1)-1)+((n1+1)-1) by XCMPLX_1:29 .=j-(((n1+1)-1)-((n1+1)-1)) by XCMPLX_1:37 .=j by XCMPLX_1:17; A311: B c= A proof let x1; assume A312:x1 in B; then reconsider x1 as Real; A313: rng D1 c= A by INTEGRA1:def 2; A314: D1.n1 <= x1 & x1 <= MD1.(len MD1) by A306,A312,INTEGRA2:1; D1.n1 in rng D1 & D1.j in rng D1 by A201,A205,FUNCT_1:def 5; then A315: inf A <= D1.n1 & D1.j <= sup A by A313,INTEGRA2:1; MD1.(len MD1)=D1.(j-n1-1+(n1+1)) by A220,A221,A223,A224,A306,A308,A309,JORDAN3:31 .=D1.(j-(n1+1)+(n1+1)) by XCMPLX_1:36 .=D1.(j-((n1+1)-(n1+1))) by XCMPLX_1:37 .=D1.j by XCMPLX_1:17; then inf A <= x1 & x1 <= sup A by A314,A315,AXIOMS:22; hence thesis by INTEGRA2:1; end; then reconsider g=f|B as Function of B,REAL by FUNCT_2:38; A316: g is_bounded_on B proof A317: f is_bounded_above_on A&f is_bounded_below_on A by A1,RFUNCT_1:def 11 ; then consider a be real number such that A318: for x being Element of A st x in A /\ dom f holds a<=f.x by RFUNCT_1:def 10; for x being Element of B st x in B /\ dom g holds a<=g.x proof let x be Element of B; assume x in B /\ dom g; then A319: x in dom g by XBOOLE_0:def 3; then A320: x in dom f /\ B by FUNCT_1:68; reconsider x as Element of A by A311,TARSKI:def 3; dom f /\ B c= dom f /\ A by A311,XBOOLE_1:26; then a <= f.x by A318,A320; hence thesis by A319,FUNCT_1:68; end; then A321: g is_bounded_below_on B by RFUNCT_1:def 10; consider a be real number such that A322: for x being Element of A st x in A /\ dom f holds f.x<=a by A317,RFUNCT_1:def 9; for x being Element of B st x in B /\ dom g holds g.x<=a proof let x be Element of B; assume x in B /\ dom g; then A323: x in dom g by XBOOLE_0:def 3; then A324: x in dom f /\ B by FUNCT_1:68; reconsider x as Element of A by A311,TARSKI:def 3; dom f /\ B c= dom f /\ A by A311,XBOOLE_1:26; then a >= f.x by A322,A324; hence thesis by A323,FUNCT_1:68; end; then g is_bounded_above_on B by RFUNCT_1:def 9; hence thesis by A321,RFUNCT_1:def 11; end; rng MD2 <> {}; then 1 in dom MD2 by FINSEQ_3:34; then A325: 1 <= len MD2 by FINSEQ_3:27; A326: len MD2=indx(D2,D1,j)-'indx(D2,D1,n1+1)+1 by A228,A229,A230,A306,JORDAN3:27; A327: 1 <= indx(D2,D1,j)-'indx(D2,D1,n1+1)+1 by A228,A229,A230,A306,A325,JORDAN3:27; A328: len MD2 = indx(D2,D1,j)-indx(D2,D1,n1+1)+1 by A228,A326,SCMFSA_7:3; then len MD2-1=indx(D2,D1,j)-indx(D2,D1,n1+1) by XCMPLX_1:26; then A329: len MD2-1+indx(D2,D1,n1+1)=indx(D2,D1,j) by XCMPLX_1:27; A330: rng MD2 = rng MD1 \/ {D.(i+1)} proof x1 in rng MD2 implies x1 in (rng MD1 \/ {D.(i+1)}) proof assume A331:x1 in rng MD2; then reconsider x1 as Real; A332: MD2.1=D2.(1+indx(D2,D1,n1+1)-1) by A228,A229,A230,A306,A327,JORDAN3: 31 .=D2.indx(D2,D1,n1+1) by XCMPLX_1:26; MD2.(len MD2)=D2.indx(D2,D1,j) by A228,A229,A230,A306,A325,A326,A329,JORDAN3:31; then A333: D1.(n1+1) <= x1 & x1 <= D1.j by A226,A227,A331,A332,Th15; A334: rng MD2 c= rng D2 by A306,JORDAN3:28; now per cases by A21,A331,A334,XBOOLE_0:def 2; suppose x1 in rng D1; then consider k such that A335: k in dom D1 & D1.k = x1 by PARTFUN1:26; A336: n1+1 <= k & k <= j by A201,A225,A333,A335,GOBOARD1:def 1; then A337: 1 <= k-n1 & k-n1 <= len MD1 by A309,REAL_1:49,84; j-(n1+1)+1=j-n1-1+1 by XCMPLX_1:36 .=j-n1-(1-1) by XCMPLX_1:37 .=j-n1; then A338: k-n1 <= j-(n1+1)+1 by A336,REAL_1:49; n1 <= n1+1 by NAT_1:29; then n1 <= k by A336,AXIOMS:22; then consider n such that A339: k=n1+n by NAT_1:28; A340: n = k-n1 by A339,XCMPLX_1:26; then n in dom MD1 by A337,FINSEQ_3:27; then A341: MD1.n in rng MD1 by FUNCT_1:def 5; MD1.n = D1.(k-n1-1+(n1+1)) by A220,A223,A224,A306,A337,A338,A340, JORDAN3:31 .= D1.(k-(n1+1)+(n1+1)) by XCMPLX_1:36 .= D1.(k-((n1+1)-(n1+1))) by XCMPLX_1:37 .= D1.k by XCMPLX_1:17; hence x1 in (rng MD1 \/ {D.(i+1)}) by A335,A341,XBOOLE_0:def 2; suppose x1 in rng D; then consider n such that A342: n in dom D & D.n = x1 by PARTFUN1:26; A343: D.i <= sup divset(D1,n1) by A205,INTEGRA2:1; sup divset(D1,n1)=D1.n1 proof per cases; suppose n1=1; hence thesis by A205,INTEGRA1:def 5; suppose n1<>1; hence thesis by A205,INTEGRA1:def 5; end; then D.i < D1.(n1+1) by A305,A343,AXIOMS:22; then D.i < D.n by A333,A342,AXIOMS:22; then i < n by A204,A342,GOBOARD2:18; then i+1 <= n by NAT_1:38; then A344: i+1 = n or i+1 < n by REAL_1:def 5; not i+1 < n proof assume i+1 < n; then A345: D.(i+1) < D.n by A200,A342,GOBOARD1:def 1; inf divset(D1,j) <= D.(i+1) by A201,INTEGRA2:1; then A346: inf divset(D1,j) <= D.n by A345,AXIOMS:22; sup divset(D1,j)=D1.j proof per cases; suppose j=1; hence thesis by A201,INTEGRA1:def 5; suppose j<>1; hence thesis by A201,INTEGRA1:def 5; end; then D.n in rng D&D.n in divset(D1,j) by A333,A342,A346,FUNCT_1:def 5,INTEGRA2:1; then A347: x1 in rng D /\ divset(D1,j) by A342,XBOOLE_0:def 3; A348: D.(i+1) in rng D by A200,FUNCT_1:def 5; consider y1 being Real such that A349: y1=D.(i+1); y1 in rng D /\ divset(D1,j) by A201,A348,A349,XBOOLE_0:def 3; hence contradiction by A20,A201,A342,A345,A347,A349,Th4; end; then x1 in {D.(i+1)} by A342,A344,TARSKI:def 1; hence x1 in (rng MD1 \/ {D.(i+1)}) by XBOOLE_0:def 2; end; hence thesis; end; then A350: rng MD2 c= rng MD1 \/ {D.(i+1)} by TARSKI:def 3; x1 in (rng MD1 \/ {D.(i+1)}) implies x1 in rng MD2 proof assume A351:x1 in rng MD1 \/ {D.(i+1)}; then reconsider x1 as Real; now per cases by A351,XBOOLE_0:def 2; suppose A352:x1 in rng MD1; rng MD1 c= rng D1 by A306,JORDAN3:28; then A353: x1 in rng D1 by A352; rng MD1 <> {}; then 1 in dom MD1 by FINSEQ_3:34; then A354: 1 <= len MD1 by FINSEQ_3:27; A355: len MD1=j-'(n1+1)+1 by A220,A223,A224,A306,JORDAN3:27; then A356: MD1.1=D1.(1+(n1+1)-1) by A220,A223,A224,A306,A354,JORDAN3:31 .=D1.((n1+1)-1+1) by XCMPLX_1:29 .=D1.((n1+1)-(1-1)) by XCMPLX_1:37 .=D1.(n1+1); len MD1+(n1+1)-1=j-(n1+1)+1+(n1+1)-1 by A220,A355,SCMFSA_7:3 .=j-((n1+1)-1)+(n1+1)-1 by XCMPLX_1:37 .=j-((n1+1)-1)+((n1+1)-1) by XCMPLX_1:29 .=j-(((n1+1)-1)-((n1+1)-1)) by XCMPLX_1:37 .=j by XCMPLX_1:17; then MD1.(len MD1) =D1.j by A220,A223,A224,A306,A354,A355,JORDAN3:31; then A357: D2.indx(D2,D1,n1+1) <= x1 & x1 <= D2.indx(D2,D1,j) by A226,A227,A352,A356,Th15; rng D1 c= rng D2 by A21,INTEGRA1:def 20; then consider k such that A358: k in dom D2 & D2.k = x1 by A353,PARTFUN1:26; A359: indx(D2,D1,n1+1)<=k & k<=indx(D2,D1,j) by A226,A227,A357,A358,GOBOARD1:def 1; then indx(D2,D1,n1+1)+1<=k+1 by AXIOMS:24; then 1 <= k+1-indx(D2,D1,n1+1) by REAL_1:84; then A360: 1 <= k-indx(D2,D1,n1+1)+1 by XCMPLX_1:29; k-indx(D2,D1,n1+1)<=indx(D2,D1,j)-indx(D2,D1,n1+1) by A359,REAL_1:49; then A361: k-indx(D2,D1,n1+1)+1<=indx(D2,D1,j)-indx(D2,D1,n1+1)+1 by AXIOMS:24; indx(D2,D1,n1+1)<=k+1 by A359,NAT_1:37; then consider n such that A362: k+1 = indx(D2,D1,n1+1)+n by NAT_1:28; A363: n=k+1-indx(D2,D1,n1+1) by A362,XCMPLX_1:26 .=k-indx(D2,D1,n1+1)+1 by XCMPLX_1:29; then A364: n in dom MD2 by A328,A360,A361,FINSEQ_3:27; MD2.n=D2.(n+indx(D2,D1,n1+1)-1) by A228,A229,A230,A306,A360,A361,A363,JORDAN3:31 .=D2.k by A362,XCMPLX_1:26; hence x1 in rng MD2 by A358,A364,FUNCT_1:def 5; suppose x1 in {D.(i+1)}; then A365: x1 = D.(i+1) by TARSKI:def 1; A366: D.(i+1) in rng D by A200,FUNCT_1:def 5; rng D c= rng D2 by A21,INTEGRA1:def 20; then consider k such that A367: k in dom D2 & x1 = D2.k by A365,A366,PARTFUN1:26; inf divset(D1,j) <= D.(i+1) & D.(i+1) <= sup divset(D1,j) by A201,INTEGRA2:1; then A368: D1.(j-1) <= x1 & x1 <= D1.j by A201,A208,A222,A365,INTEGRA1: def 5; A369: j-1 in dom D1 & j-1 in NAT by A201,A208,A222,INTEGRA1:9; reconsider j1 = j-1 as Nat by A201,A208,A222,INTEGRA1:9; n1<j1 by A303,REAL_1:86; then n1+1 <= j1 by NAT_1:38; then D1.(n1+1) <= D1.(j-1) by A225,A369,GOBOARD2:18; then D2.indx(D2,D1,n1+1) <= D2.k & D2.k <= D2.indx(D2,D1,j) by A21,A201,A226,A367,A368,AXIOMS:22,INTEGRA1:def 21; hence x1 in rng MD2 by A226,A227,A306,A367,Th16; end; hence thesis; end; then rng MD1 \/ {D.(i+1)} c= rng MD2 by TARSKI:def 3; hence thesis by A350,XBOOLE_0:def 10; end; A370: len MD1 in dom MD1 by SCMFSA_7:12; then A371: 1 <= len MD1 by FINSEQ_3:27; A372: inf divset(MD1,len MD1)=inf divset(D1,j) & sup divset(MD1,len MD1)=sup divset(D1,j) proof per cases; suppose A373:len MD1=1; then A374: inf divset(MD1,len MD1)=inf B & sup divset(MD1,len MD1)=MD1.(len MD1) by A370,INTEGRA1:def 5; inf divset(D1,j)=D1.(j-1) & sup divset(D1,j)=D1.j by A201,A208,A222,INTEGRA1:def 5; hence thesis by A220,A223,A224,A306,A307,A308,A310,A373,A374,JORDAN3: 31,XCMPLX_1:18; suppose A375:len MD1<>1; then A376: inf divset(MD1,len MD1)=MD1.(len MD1-1) & sup divset(MD1,len MD1)=MD1.(len MD1) by A370,INTEGRA1:def 5; A377: len MD1-1+(n1+1)-1=j-(n1+1)+(n1+1)-1 by A309,XCMPLX_1:36 .=j-((n1+1)-(n1+1))-1 by XCMPLX_1:37 .=j-1 by XCMPLX_1:17; A378: len MD1-1 in dom MD1 & len MD1-1 in NAT by A370,A375,INTEGRA1:9; then A379: len MD1-1 >= 1 by FINSEQ_3:27; len MD1 <= len MD1+1 by NAT_1:29; then len MD1 -1 <= len MD1 by REAL_1:86; then A380: inf divset(MD1,len MD1)=D1.(j-1) by A220,A223,A224,A306,A307, A376,A377,A378,A379,JORDAN3:31; sup divset(MD1,len MD1) =D1.j by A220,A223,A224,A306,A307,A310,A371,A376,JORDAN3:31; hence thesis by A201,A208,A222,A380,INTEGRA1:def 5; end; inf divset(D1,j)<=D.(i+1) & D.(i+1)<=sup divset(D1,j) by A201,INTEGRA2:1; then A381: D.(i+1) in divset(MD1,len MD1) by A372,INTEGRA2:1; A382: Sum lower_volume(g,MD2)-Sum lower_volume(g,MD1)<= (sup rng g-inf rng g)*delta(MD1) proof vol(B)<>0 & D.(i+1) > inf B proof A383: vol(B)=sup B-D1.n1 by A306,INTEGRA1:def 6; A384: len MD1 in dom MD1 by SCMFSA_7:12; sup divset(MD1,len MD1)=MD1.(len MD1) proof per cases; suppose len MD1 = 1; hence thesis by A384,INTEGRA1:def 5; suppose len MD1 <> 1; hence thesis by A384,INTEGRA1:def 5; end; then A385: vol(B)=D1.j-D1.n1 by A201,A208,A222,A306,A372,A383,INTEGRA1:def 5; D1.j>D1.n1 by A201,A205,A208,GOBOARD1:def 1; hence vol(B)<>0 by A385,SQUARE_1:11; inf divset(D1,j) <= D.(i+1) by A201,INTEGRA2:1; then A386: D1.(j-1) <= D.(i+1) by A201,A208,A222,INTEGRA1:def 5; A387: n1 < j-1 by A303,REAL_1:86; j-1 in dom D1 by A201,A208,A222,INTEGRA1:9; then D1.n1 <D1.(j-1) by A205,A387,GOBOARD1:def 1; hence thesis by A306,A386,AXIOMS:22; end; hence thesis by A306,A316,A330,A381,Th12; end; dom g = B by FUNCT_2:def 1; then A388: rng g <> {} by RELAT_1:65; A389: rng g c= rng f by FUNCT_1:76; f is_bounded_above_on A & f is_bounded_below_on A by A1,RFUNCT_1:def 11; then rng f is bounded_above & rng f is bounded_below by INTEGRA1:13,15; then sup rng f >= sup rng g & inf rng f <= inf rng g by A388,A389,PSCOMP_1:12,13; then A390: sup rng f-inf rng f>=sup rng g - inf rng g by REAL_1:92; delta(MD1) >= 0 by Th8; then (sup rng f-inf rng f)*delta(MD1)>=(sup rng g-inf rng g)*delta(MD1 ) by A390,AXIOMS:25; then A391: Sum lower_volume(g,MD2)-Sum lower_volume(g,MD1)<= (sup rng f-inf rng f)*delta(MD1) by A382,AXIOMS:22; delta(MD1)=max rng upper_volume(chi(B,B),MD1) by INTEGRA1:def 19; then delta(MD1) in rng upper_volume(chi(B,B),MD1) by PRE_CIRC:def 1; then consider k such that A392: k in dom upper_volume(chi(B,B),MD1) & upper_volume(chi(B,B),MD1).k = delta(MD1) by PARTFUN1:26; k in Seg len upper_volume(chi(B,B),MD1) by A392,FINSEQ_1:def 3; then A393: k in Seg len MD1 by INTEGRA1:def 7; then A394: delta(MD1) = vol(divset(MD1,k)) by A392,INTEGRA1:22; n1+1>1 by A222,NAT_1:38; then n1>1-1 by REAL_1:84; then A395: 1 <= k & k <= len MD1 & k < k+n1 by A393,FINSEQ_1:3,REAL_1:69; then A396: 1 < k+n1 by AXIOMS:22; A397: k in dom MD1 by A393,FINSEQ_1:def 3; k+n1 <= j by A309,A395,REAL_1:84; then k+n1 <= len D1 by A223,AXIOMS:22; then A398: k+n1 in dom D1 by A396,FINSEQ_3:27; A399: divset(MD1,k)=divset(D1,k+n1) proof inf divset(MD1,k)=inf divset(D1,k+n1) & sup divset(MD1,k)=sup divset(D1,k+n1) proof per cases; suppose A400:k=1; then inf divset(MD1,k)=inf B & sup divset(MD1,k)=MD1.k by A397,INTEGRA1:def 5; then A401: inf divset(MD1,k)=D1.n1 & sup divset(MD1,k)=D1.(k+(n1+1)-1) by A220,A223,A224,A306,A307,A395,JORDAN3:31; inf divset(D1,k+n1)=D1.(k+n1-1) & sup divset(D1,k+n1)=D1.(k+n1) by A395,A398,INTEGRA1:def 5; hence thesis by A400,A401,XCMPLX_1:26; suppose A402:k<>1; then A403: k-1 in dom MD1 & k-1 in NAT by A397,INTEGRA1:9; then A404: 1 <= k-1 & k-1 <= len MD1 by FINSEQ_3:27; inf divset(MD1,k)=MD1.(k-1) & sup divset(MD1,k)=MD1.k by A397,A402,INTEGRA1:def 5; then inf divset(MD1,k)=D1.(k-1+(n1+1)-1) & sup divset(MD1,k)=D1.(k+(n1+1)-1) by A220,A223,A224,A306,A307,A395,A403,A404,JORDAN3:31; then inf divset(MD1,k)=D1.(k-1+n1+1-1) & sup divset(MD1,k)=D1.(k+n1+1-1) by XCMPLX_1:1; then A405: inf divset(MD1,k)=D1.(k-1+n1) & sup divset(MD1,k)=D1.(k+n1) by XCMPLX_1:26; inf divset(D1,k+n1)=D1.(k+n1-1) & sup divset(D1,k+n1)=D1.(k+n1) by A395,A398,INTEGRA1:def 5; hence thesis by A405,XCMPLX_1:29; end; then divset(MD1,k)=[.inf divset(D1,k+n1),sup divset(D1,k+n1).] by INTEGRA1:5; hence thesis by INTEGRA1:5; end; A406: k+n1 in Seg len D1 by A398,FINSEQ_1:def 3; then A407: delta(MD1)=upper_volume(chi(A,A),D1).(k+n1) by A394,A399,INTEGRA1 :22; k+n1 in Seg len upper_volume(chi(A,A),D1) by A406,INTEGRA1:def 7; then k+n1 in dom upper_volume(chi(A,A),D1) by FINSEQ_1:def 3; then delta(MD1) in rng upper_volume(chi(A,A),D1) by A407,FUNCT_1:def 5 ; then delta(MD1) <= max rng upper_volume(chi(A,A),D1) by PRE_CIRC:def 1 ; then A408: delta(MD1) <= delta(D1) by INTEGRA1:def 19; inf rng f <= sup rng f by A1,Lm6; then sup rng f - inf rng f >= 0 by SQUARE_1:12; then A409: (sup rng f-inf rng f)*delta(MD1) <= (sup rng f-inf rng f)* delta(D1) by A408,AXIOMS:25; A410: indx(D2,D1,n1) < indx(D2,D1,n1)+1 by NAT_1:38; A411: 1 <= indx(D2,D1,n1)+1 by A207,NAT_1:38; D1.n1 < D1.(n1+1) by A205,A225,A304,GOBOARD1:def 1; then indx(D2,D1,n1) < indx(D2,D1,n1+1) by A206,A226,GOBOARD2:18; then A412: indx(D2,D1,n1)+1 <= indx(D2,D1,n1+1) by NAT_1:38; then A413: indx(D2,D1,n1)+1 <= len D2 by A229,AXIOMS:22; then A414: indx(D2,D1,n1)+1 <= len LVf(D2) by INTEGRA1:def 8; A415: indx(D2,D1,n1+1)=indx(D2,D1,n1)+1 proof assume indx(D2,D1,n1+1)<> indx(D2,D1,n1)+1; then A416: indx(D2,D1,n1+1)>indx(D2,D1,n1)+1 by A412,REAL_1:def 5; A417: indx(D2,D1,n1)+1 in dom D2 by A411,A413,FINSEQ_3:27; then A418: D2.(indx(D2,D1,n1)+1) in rng D2 by FUNCT_1:def 5; now per cases by A21,A418,XBOOLE_0:def 2; suppose D2.(indx(D2,D1,n1)+1) in rng D1; then consider n2 being Nat such that A419: n2 in dom D1 & D2.(indx(D2,D1,n1)+1) = D1.n2 by PARTFUN1:26; D2.indx(D2,D1,n1) < D2.(indx(D2,D1,n1)+1) by A206,A410,A417,GOBOARD1:def 1; then n1 < n2 by A205,A206,A419,GOBOARD2:18; then A420: n1+1 <= n2 by NAT_1:38; D1.n2 < D1.(n1+1) by A226,A416,A417,A419,GOBOARD1:def 1; hence contradiction by A225,A419,A420,GOBOARD2:18; suppose D2.(indx(D2,D1,n1)+1) in rng D; then consider n2 being Nat such that A421: n2 in dom D & D2.(indx(D2,D1,n1)+1) = D.n2 by PARTFUN1:26; A422: D1.n1 < D.n2 by A206,A410,A417,A421,GOBOARD1:def 1; A423: D.i <= sup divset(D1,n1) by A205,INTEGRA2:1; sup divset(D1,n1)=D1.n1 proof per cases; suppose n1=1; hence thesis by A205,INTEGRA1:def 5; suppose n1<>1; hence thesis by A205,INTEGRA1:def 5; end; then D.i < D.n2 by A422,A423,AXIOMS:22; then i < n2 by A204,A421,GOBOARD2:18; then A424: i+1 <= n2 by NAT_1:38; A425: D.n2 < D1.(n1+1) by A226,A416,A417,A421,GOBOARD1:def 1; A426: j-1 in dom D1 & j-1 in NAT by A201,A208,A222,INTEGRA1:9; n1+1 +1 <= j by A303,NAT_1:38; then n1+1 <= j-1 by REAL_1:84; then A427: D1.(n1+1) <= D1.(j-1) by A225,A426,GOBOARD2:18; A428: inf divset(D1,j) <= D.(i+1) by A201,INTEGRA2:1; inf divset(D1,j) = D1.(j-1) by A201,A208,A222,INTEGRA1:def 5; then D1.(n1+1) <= D.(i+1) by A427,A428,AXIOMS:22; then D.n2 < D.(i+1) by A425,AXIOMS:22; hence contradiction by A200,A421,A424,GOBOARD2:18; end; hence contradiction; end; A429: Sum lower_volume(g,MD2)=Sum mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) proof lower_volume(g,MD2)=mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) proof A430: len lower_volume(g,MD2)=indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1 by A328,A415,INTEGRA1:def 8; A431: indx(D2,D1,j) <= len LVf(D2) by A230,INTEGRA1:def 8; A432: indx(D2,D1,n1)+1 <= indx(D2,D1,j) by A228,A412,AXIOMS:22; then len mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) = indx(D2,D1,j)-'(indx(D2,D1,n1)+1)+1 by A230,A411,A414,A431,JORDAN3: 27; then A433: len lower_volume(g,MD2) =len mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) by A430,A432,SCMFSA_7:3; for k st 1<=k & k<=len lower_volume(g,MD2) holds lower_volume(g,MD2).k=mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)).k proof let k; assume A434:1 <= k & k <= len lower_volume(g,MD2); then k in Seg len lower_volume(g,MD2) by FINSEQ_1:3; then A435: k in Seg len MD2 by INTEGRA1:def 8; A436: divset(MD2,k)=[.inf divset(MD2,k),sup divset(MD2,k).] by INTEGRA1:5; A437: mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)).k =LVf(D2).(k+(indx(D2,D1,n1)+1)-1) by A411,A430,A431,A432,A434,JORDAN3:31; k <= indx(D2,D1,j)-((indx(D2,D1,n1)+1)-1) by A430,A434,XCMPLX_1:37 ; then k+((indx(D2,D1,n1)+1)-1) <= indx(D2,D1,j) by REAL_1:84; then k+(indx(D2,D1,n1)+1)-1 <= indx(D2,D1,j) by XCMPLX_1:29; then A438: k+(indx(D2,D1,n1)+1)-1 <= len LVf(D2) by A431,AXIOMS:22; 1<=indx(D2,D1,n1)+1 by NAT_1:37; then 1+1<=k+(indx(D2,D1,n1)+1) by A434,REAL_1:55; then A439: 1 <= k+(indx(D2,D1,n1)+1)-1 by REAL_1:84; consider k2 being Nat such that A440: indx(D2,D1,n1)+1=1+k2; A441: k2=(indx(D2,D1,n1)+1)-1 by A440,XCMPLX_1:26; then A442: k+k2=k+(indx(D2,D1,n1)+1)-1 by XCMPLX_1:29; then k+k2 in Seg len LVf(D2) by A438,A439,FINSEQ_1:3; then A443: k+k2 in Seg len D2 by INTEGRA1:def 8; then A444: mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)).k =(inf rng(f|divset(D2,k+k2)))*vol(divset(D2,k+k2)) by A437,A442,INTEGRA1:def 8; inf divset(MD2,k)=inf divset(D2,k+k2) & sup divset(MD2,k)=sup divset(D2,k+k2) proof k2=indx(D2,D1,n1) by A441,XCMPLX_1:26; then k+k2>=1+1 by A207,A434,REAL_1:55; then A445: k+k2>1 by NAT_1:38; A446: k in dom MD2 by A435,FINSEQ_1:def 3; A447: k+k2 in dom D2 by A443,FINSEQ_1:def 3; per cases; suppose A448:k=1; then inf divset(MD2,k)=inf B & sup divset(MD2,k)=MD2.k by A446,INTEGRA1:def 5; then A449: inf divset(MD2,k)=D1.n1 & sup divset(MD2,k)=D2.(k+(indx(D2,D1,n1)+1)-1) by A228,A230,A306,A411,A415,A430,A434,JORDAN3:31; inf divset(D2,k+k2)=D2.(k+k2-1) &sup divset(D2,k+k2)=D2.(k+k2) by A445,A447,INTEGRA1:def 5; hence thesis by A206,A440,A448,A449,XCMPLX_1:26; suppose A450:k<>1; then A451: k-1 in dom MD2 & k-1 in NAT by A446,INTEGRA1:9; then A452: 1 <= k-1 by FINSEQ_3:27; k-1 <= k by Lm1; then A453: k-1 <= indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1 by A430,A434,AXIOMS :22; inf divset(MD2,k)=MD2.(k-1) & sup divset(MD2,k)=MD2.k by A446,A450,INTEGRA1:def 5; then A454: inf divset(MD2,k)=D2.(k-1+(indx(D2,D1,n1)+1)-1) & sup divset(MD2,k)=D2.(k+(indx(D2,D1,n1)+1)-1) by A228,A230,A306,A411,A415,A430,A434,A451,A452,A453,JORDAN3:31; inf divset(D2,k+k2)=D2.(k+k2-1) &sup divset(D2,k+k2)=D2.(k+k2) by A445,A447,INTEGRA1:def 5; then inf divset(D2,k+k2)=D2.(k-1+k2) & sup divset(D2,k+k2)=D2.(k+((indx(D2,D1,n1)+1)-1)) by A440,XCMPLX_1:26,29; hence thesis by A441,A454,XCMPLX_1:29; end; then A455: divset(MD2,k)=divset(D2,k+k2) by A436,INTEGRA1:5; rng(f|divset(D2,k+k2))=rng(g|divset(D2,k+k2)) proof k in dom MD2 by A435,FINSEQ_1:def 3; then divset(D2,k+k2) c= B by A455,INTEGRA1:10; hence thesis by FUNCT_1:82; end; hence thesis by A435,A444,A455,INTEGRA1:def 8; end; hence thesis by A433,FINSEQ_1:18; end; hence thesis; end; Sum lower_volume(g,MD1)=Sum mid(LVf(D1),n1+1,j) proof A456: len lower_volume(g,MD1)=len MD1 by INTEGRA1:def 8 .=j-'(n1+1)+1 by A220,A223,A224,A306,JORDAN3:27 .=j-(n1+1)+1 by A220,SCMFSA_7:3; A457: n1+1 <= len LVf(D1) by A224,INTEGRA1:def 8; A458: j <= len LVf(D1) by A223,INTEGRA1:def 8; then A459: len mid(LVf(D1),n1+1,j) =j-'(n1+1)+1 by A220,A223,A224,A457,JORDAN3:27 .=j-(n1+1)+1 by A220,SCMFSA_7:3; lower_volume(g,MD1)=mid(LVf(D1),n1+1,j) proof for k st 1<=k & k<=len lower_volume(g,MD1) holds lower_volume(g,MD1).k=mid(LVf(D1),n1+1,j).k proof let k; assume A460:1 <= k & k <= len lower_volume(g,MD1); then k in Seg len lower_volume(g,MD1) by FINSEQ_1:3; then A461: k in Seg len MD1 by INTEGRA1:def 8; then A462: lower_volume(g,MD1).k =(inf rng(g|divset(MD1,k)))*vol(divset(MD1,k)) by INTEGRA1:def 8; k <= j-((n1+1)-1) by A456,A460,XCMPLX_1:37; then A463: k+((n1+1)-1) <= j by REAL_1:84; consider k2 being Nat such that A464: n1+1=1+k2; A465: k2=(n1+1)-1 by A464,XCMPLX_1:26; A466: 1 <= k+k2 by A460,NAT_1:37; k+k2 <= len D1 by A223,A463,A465,AXIOMS:22; then A467: k+k2 in Seg len D1 by A466,FINSEQ_1:3; A468: mid(LVf(D1),n1+1,j).k =LVf(D1).(k+(n1+1)-1) by A220,A224,A456,A458,A460,JORDAN3:31 .=LVf(D1).(k+((n1+1)-1)) by XCMPLX_1:29 .=(inf rng(f|divset(D1,k+k2)))*vol(divset(D1,k+k2)) by A465,A467,INTEGRA1:def 8; A469: k2=n1 by A465,XCMPLX_1:26; then 1+1 <= k+k2 by A223,A460,REAL_1:55; then A470: 1 < k+k2 by NAT_1:38; A471: divset(MD1,k)=[.inf divset(MD1,k),sup divset(MD1,k).] by INTEGRA1:5; A472: k in dom MD1 by A461,FINSEQ_1:def 3; k+k2 in dom D1 by A467,FINSEQ_1:def 3; then A473: inf divset(D1,k+k2)=D1.(k+k2-1) & sup divset(D1,k+k2)=D1.(k+k2) by A470,INTEGRA1:def 5; A474: divset(D1,k+k2)=divset(MD1,k) proof inf divset(D1,k+k2)=inf divset(MD1,k) & sup divset(D1,k+k2)=sup divset(MD1,k) proof per cases; suppose A475:k=1; then inf divset(MD1,k)=inf B & sup divset(MD1,k)=MD1.k by A472,INTEGRA1:def 5; then inf divset(MD1,k)=D1.n1 & sup divset(MD1,k)=D1.(k+(n1+1)-1) by A220,A223,A224,A306,A456,A460,JORDAN3:31; hence thesis by A469,A473,A475,XCMPLX_1:26; suppose A476:k<>1; then A477: k-1 in dom MD1 & k-1 in NAT by A472,INTEGRA1:9; then A478: 1 <= k-1 by FINSEQ_3:27; k-1 <= k by Lm1; then A479: k-1 <= j-(n1+1)+1 by A456,A460,AXIOMS:22; inf divset(MD1,k)=MD1.(k-1) & sup divset(MD1,k)=MD1.k by A472,A476,INTEGRA1:def 5; then inf divset(MD1,k)=D1.(k-1+(n1+1)-1) & sup divset(MD1,k)=D1.(k+(n1+1)-1) by A220,A223,A224,A306,A456,A460,A477,A478,A479,JORDAN3:31; then inf divset(MD1,k)=D1.(k-1+((n1+1)-1)) & sup divset(MD1,k)=D1.(k+((n1+1)-1)) by XCMPLX_1:29; hence thesis by A465,A473,XCMPLX_1:29; end; hence thesis by A471,INTEGRA1:5; end; k in dom MD1 by A461,FINSEQ_1:def 3; then divset(D1,k+k2) c= B by A474,INTEGRA1:10; hence thesis by A462,A468,A474,FUNCT_1:82; end; hence thesis by A456,A459,FINSEQ_1:18; end; hence thesis; end; hence Sum mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) -Sum mid(LVf(D1),n1+1,j) <= (sup rng f-inf rng f)*delta(D1) by A391,A409,A429,AXIOMS :22; end; hence thesis; end; A480: PLf(D2,indx(D2,D1,j))= PLf(D2,indx(D2,D1,n1))+Sum mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) proof A481: indx(D2,D1,n1)<indx(D2,D1,j) by A228,A231,AXIOMS:22; indx(D2,D1,j) in Seg len D2 by A227,FINSEQ_1:def 3; then A482: indx(D2,D1,j) in Seg len LVf(D2) by INTEGRA1:def 8; A483: indx(D2,D1,j) <= len LVf(D2) by A230,INTEGRA1:def 8; indx(D2,D1,n1) in Seg len D2 by A206,FINSEQ_1:def 3; then indx(D2,D1,n1) in Seg len LVf(D2) by INTEGRA1:def 8; then PLf(D2,indx(D2,D1,n1))=Sum(LVf(D2)|indx(D2,D1,n1)) by INTEGRA1: def 22 .=Sum mid(LVf(D2),1,indx(D2,D1,n1)) by A207,JORDAN3:25; then PLf(D2,indx(D2,D1,n1))+Sum mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2, D1,j)) =Sum(mid(LVf(D2),1,indx(D2,D1,n1))^ mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j))) by RVSUM_1:105 .=Sum mid(LVf(D2),1,indx(D2,D1,j)) by A207,A481,A483,INTEGRA2:4 .=Sum(LVf(D2)|indx(D2,D1,j)) by A230,JORDAN3:25; hence thesis by A482,INTEGRA1:def 22; end; A484: PLf(D1,j)=PLf(D1,n1)+Sum mid(LVf(D1),n1+1,j) proof A485: j <= len LVf(D1) by A223,INTEGRA1:def 8; then A486: j in Seg len LVf(D1) by A223,FINSEQ_1:3; n1 in Seg len D1 by A205,FINSEQ_1:def 3; then n1 in Seg len LVf(D1) by INTEGRA1:def 8; then PLf(D1,n1)=Sum(LVf(D1)|n1) by INTEGRA1:def 22 .=Sum mid(LVf(D1),1,n1) by A223,JORDAN3:25; then PLf(D1,n1)+Sum mid(LVf(D1),n1+1,j) =Sum(mid(LVf(D1),1,n1)^mid(LVf(D1),n1+1,j)) by RVSUM_1:105 .=Sum mid(LVf(D1),1,j) by A208,A223,A485,INTEGRA2:4 .=Sum(LVf(D1)|j) by A223,JORDAN3:25; hence thesis by A486,INTEGRA1:def 22; end; A487: i*(sup rng f-inf rng f)*delta(D1)+(sup rng f-inf rng f)*delta(D1) =i*((sup rng f-inf rng f)*delta(D1))+1* ((sup rng f-inf rng f)*delta(D1)) by XCMPLX_1:4 .=(i+1)*((sup rng f-inf rng f)*delta(D1)) by XCMPLX_1:8 .=(i+1)*(sup rng f-inf rng f)*delta(D1) by XCMPLX_1:4; A488: (PLf(D2,indx(D2,D1,n1))-PLf(D1,n1))+ (Sum mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) -Sum mid(LVf(D1),n1+1,j)) <=i*(sup rng f-inf rng f)*delta(D1)+(sup rng f-inf rng f)*delta(D1) by A205,A233,REAL_1:55; (PLf(D2,indx(D2,D1,n1))-PLf(D1,n1))+ (Sum mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) -Sum mid(LVf(D1),n1+1,j)) =PLf(D2,indx(D2,D1,j))-PLf(D1,j) by A480,A484,Lm5; hence thesis by A201,A487,A488; end; hence thesis; end; A489: for k being non empty Nat holds P[k] from Ind_from_1(A43,A197); i in Seg len D by A23,FINSEQ_1:def 3; then reconsider i as non empty Nat by BINARITH:5; P[i] by A489; hence thesis by A23; end; len D in dom D by SCMFSA_7:12; then consider j such that A490: j in dom D1 & D.(len D) in divset(D1,j) & PLf(D2,indx(D2,D1,j))-PLf(D1,j) <= (len D)*(sup(rng f)-inf(rng f))*delta(D1) by A22; A491: len D1 in dom D1 by SCMFSA_7:12; A492: j = len D1 proof assume A493:j<>len D1; A494: D.(len D) <= sup divset(D1,j) by A490,INTEGRA2:1; j <= len D1 by A490,FINSEQ_3:27; then j < len D1 by A493,REAL_1:def 5; then D1.j < D1.(len D1) by A490,A491,GOBOARD1:def 1; then A495: D1.j < sup A by INTEGRA1:def 2; sup divset(D1,j) < sup A proof per cases; suppose j=1; hence thesis by A490,A495,INTEGRA1:def 5; suppose j<>1; hence thesis by A490,A495,INTEGRA1:def 5; end; hence contradiction by A494,INTEGRA1:def 2; end; A496:len D2 in dom D2 by SCMFSA_7:12; A497:indx(D2,D1,len D1) in dom D2 & D1.(len D1) = D2.indx(D2,D1,len D1) by A21,A491,INTEGRA1:def 21; then sup A = D2.indx(D2,D1,len D1) by INTEGRA1:def 2; then D2.(len D2) = D2.indx(D2,D1,len D1) by INTEGRA1:def 2; then indx(D2,D1,len D1)=len D2 by A496,A497,GOBOARD2:19; then PLf(D2,len D2)-lower_sum(f,D1)<=(len D)*(sup(rng f)-inf(rng f))* delta(D1) by A490,A492,INTEGRA1:45; hence thesis by INTEGRA1:45; end; hence thesis by A21; end; hence thesis; end; A498:for e be real number st e>0 ex n st for m st n<=m holds abs((lower_sum(f,T)).m-lower_integral(f))<e proof let e be real number; assume A499:e>0; then A500:e/2>0 by REAL_2:127; reconsider e as Real by XREAL_0:def 1; A501:lower_integral(f) = sup rng lower_sum_set(f) by INTEGRA1:def 16; A502:rng lower_sum_set(f) is bounded_above by A1,INTEGRA2:36; dom lower_sum_set(f) is non empty by INTEGRA1:def 12; then rng lower_sum_set(f) is non empty by RELAT_1:65; then consider y be real number such that A503: y in rng lower_sum_set(f) & lower_integral(f)-e/2<y by A500,A501,A502,SEQ_4:def 4; consider D being Element of divs A such that A504: D in dom lower_sum_set(f) & y=(lower_sum_set(f)).D & D.1>inf A by A3,A503,Lm10; y=lower_sum(f,D) by A504,INTEGRA1:def 12; then A505: lower_integral(f)-lower_sum(f,D)<e/2 by A503,REAL_2:165; set p=len D; set H=sup rng f; set h=inf rng f; deffunc F(Nat)=vol(divset(D,$1)); consider v being FinSequence of REAL such that A506:len v = len D & for j st j in Seg len D holds v.j=F(j) from SeqLambdaD; consider v1 being non-decreasing FinSequence of REAL such that A507: v,v1 are_fiberwise_equipotent by INTEGRA2:3; defpred P[Nat] means $1 in dom v1 & v1.$1>0; A508:ex k st P[k] proof consider H being Function such that A509: dom H = dom v & rng H = dom v1 & H is one-to-one & v=v1*H by A507,RFINSEQ:3; consider k such that A510: k in dom D & vol(divset(D,k)) > 0 by A3,Th1; dom D = Seg len v by A506,FINSEQ_1:def 3; then k in dom v & v.k > 0 by A506,A510,FINSEQ_1:def 3; then H.k in dom v1 & v1.(H.k) > 0 by A509,FUNCT_1:21,22; hence thesis; end; consider k such that A511: P[k] & for n st P[n] holds k<=n from Min(A508); p <> 0 by FINSEQ_1:25; then A512:p > 0 by NAT_1:19; then A513: 2*p >0 by REAL_2:122; H >= h by A1,Lm6; then H-h >= 0 by SQUARE_1:12; then H-h+1 > 0+0 by REAL_1:67; then A514: 2*p*(H-h+1) > 0 by A513,REAL_2:122; min(v1.k,e/(2*p*(H-h+1))) > 0 proof per cases by SQUARE_1:38; suppose min(v1.k,e/(2*p*(H-h+1))) = v1.k; hence thesis by A511; suppose min(v1.k,e/(2*p*(H-h+1))) = e/(2*p*(H-h+1)); hence thesis by A499,A514,REAL_2:127; end; then consider n such that A515: for m st n<=m holds 0 < (delta(T)).m & (delta(T)).m < min(v1.k,e/(2*p*(H-h+1))) by A5; A516: for m st n<=m holds abs((lower_sum(f,T)).m-lower_integral(f))<e proof let m; assume A517:n<=m; then A518: 0 < (delta(T)).m & (delta(T)).m < min(v1.k,e/(2*p*(H-h+1))) by A515 ; reconsider D1 = T.m as Element of divs A; consider D2 being Element of divs A such that A519: D <= D2 & D1 <= D2 & rng D2 = rng D1 \/ rng D & 0 <= lower_sum(f,D2)-lower_sum(f,D) & 0 <= lower_sum(f,D2)-lower_sum(f,D1) by A14; f is_bounded_below_on A by A1,RFUNCT_1:def 11; then lower_sum(f,D) <= lower_sum(f,D2) by A519,INTEGRA1:48; then A520: lower_sum(f,D)-lower_sum(f,D2)<=0 by REAL_2:106; A521: delta(D1)=(delta(T)).m by INTEGRA2:def 3; A522: delta(D1)<min (v1.k,e/(2*p*(H-h+1))) by A518,INTEGRA2:def 3; min (v1.k,e/(2*p*(H-h+1))) <= v1.k by SQUARE_1:35; then A523: delta(D1)<v1.k by A522,AXIOMS:22; A524: v1.1 > 0 proof A525: for n1 be Element of NAT st n1 in dom D holds vol(divset(D,n1))>0 proof let n1 be Element of NAT; assume A526:n1 in dom D; then A527: 1 <= n1 by FINSEQ_3:27; per cases by A527,REAL_1:def 5; suppose A528:n1=1; then inf divset(D,n1)=inf A & sup divset(D,n1)=D.n1 by A526,INTEGRA1:def 5; then vol(divset(D,n1))=D.n1-inf A by INTEGRA1:def 6; hence thesis by A504,A528,SQUARE_1:11; suppose A529:n1>1; then inf divset(D,n1)=D.(n1-1) & sup divset(D,n1)=D.n1 by A526,INTEGRA1:def 5; then A530: vol(divset(D,n1))=D.n1-D.(n1-1) by INTEGRA1:def 6; A531: n1-1 in dom D by A526,A529,INTEGRA1:9; n1 < n1+1 by REAL_1:69; then n1-1 < n1 by REAL_1:84; then D.(n1-1)<D.n1 by A526,A531,GOBOARD1:def 1; hence thesis by A530,SQUARE_1:11; end; 1 <= k & k <= len v1 by A511,FINSEQ_3:27; then 1 <= len v1 by AXIOMS:22; then 1 in dom v1 by FINSEQ_3:27; then A532: v1.1 in rng v1 by FUNCT_1:def 5; rng v = rng v1 by A507,RFINSEQ:1; then consider n1 being Element of NAT such that A533: n1 in dom v & v1.1 = v.n1 by A532,PARTFUN1:26; A534: n1 in Seg len D by A506,A533,FINSEQ_1:def 3; then A535: v1.1 = vol(divset(D,n1)) by A506,A533; n1 in dom D by A534,FINSEQ_1:def 3; hence thesis by A525,A535; end; v1.k = min rng upper_volume(chi(A,A),D) proof A536: k=1 proof assume A537:k <> 1; A538: len v1 = len v by A507,RFINSEQ:16; A539: k in Seg len v1 by A511,FINSEQ_1:def 3; k in Seg len v by A511,A538,FINSEQ_1:def 3; then A540: 1 <= k & k <= len D by A506,FINSEQ_1:3; then A541: k > 1 by A537,REAL_1:def 5; k <= len v1 by A539,FINSEQ_1:3; then 1 <= len v1 by A540,AXIOMS:22; then 1 in dom v1 by FINSEQ_3:27; hence contradiction by A511,A524,A541; end; min rng upper_volume(chi(A,A),D) in rng upper_volume(chi(A,A),D) & for x st x in rng upper_volume(chi(A,A),D) holds min rng upper_volume(chi(A,A),D) <= x by SFMASTR3:def 1; then consider m such that A542: m in dom upper_volume(chi(A,A),D) & min rng upper_volume(chi(A,A),D)=upper_volume(chi(A,A),D).m by PARTFUN1:26; m in Seg len upper_volume(chi(A,A),D) by A542,FINSEQ_1:def 3; then A543: m in Seg len D by INTEGRA1:def 7; then min rng upper_volume(chi(A,A),D)=vol(divset(D,m)) by A542,INTEGRA1:22; then A544: v.m=min rng upper_volume(chi(A,A),D) by A506,A543; m in dom v by A506,A543,FINSEQ_1:def 3; then A545: min rng upper_volume(chi(A,A),D) in rng v by A544,FUNCT_1:def 5; A546: rng v = rng v1 by A507,RFINSEQ:1; then consider m1 being Element of NAT such that A547: m1 in dom v1 & min rng upper_volume(chi(A,A),D)=v1.m1 by A545,PARTFUN1:26; m1 >= 1 by A547,FINSEQ_3:27; then A548: v1.1 <= min rng upper_volume(chi(A,A),D) by A511,A536,A547, INTEGRA2:2; v1.k in rng v1 by A511,FUNCT_1:def 5; then consider k2 being Element of NAT such that A549: k2 in dom v & v1.k = v.k2 by A546,PARTFUN1:26; A550: k2 in Seg len D by A506,A549,FINSEQ_1:def 3; then v1.k = vol(divset(D,k2)) by A506,A549; then A551: v1.k = upper_volume(chi(A,A),D).k2 by A550,INTEGRA1:22; k2 in Seg len upper_volume(chi(A,A),D) by A550,INTEGRA1:def 7; then k2 in dom upper_volume(chi(A,A),D) by FINSEQ_1:def 3; then v1.k in rng upper_volume(chi(A,A),D) by A551,FUNCT_1:def 5; then v1.k >= min rng upper_volume(chi(A,A),D) by SFMASTR3:def 1; hence thesis by A536,A548,AXIOMS:21; end; then consider D3 being Element of divs A such that A552: D<=D3 & D1<=D3 & rng D3=rng D1 \/ rng D & lower_sum(f,D3)-lower_sum(f,D1) <=(len D)*(sup rng f-inf rng f)*delta(D1) by A19,A523; A553: lower_sum(f,D2)-lower_sum(f,D1)<=(len D)*(sup rng f-inf rng f)*delta(D1) by A519,A552,Th5; H-h <= H-h+1 by REAL_1:69; then p*(H-h)<=p*(H-h+1) by A512,AXIOMS:25; then p*(H-h)*(delta(T)).m <= p*(H-h+1)*(delta(T)).m by A518,AXIOMS:25; then A554: lower_sum(f,D2)-lower_sum(f,T.m) <= p*(H-h+1)*(delta(T)).m by A521,A553,AXIOMS:22; A555: (delta(T)).m < min(v1.k,e/(2*p*(H-h+1))) by A515,A517; min(v1.k,e/(2*p*(H-h+1))) <= e/(2*p*(H-h+1)) by SQUARE_1:35; then (delta(T)).m < e/(2*p*(H-h+1)) by A555,AXIOMS:22; then (delta(T)).m*(2*p*(H-h+1))<e by A514,REAL_2:177; then (delta(T)).m*(2*(p*(H-h+1)))<e by XCMPLX_1:4; then ((delta(T)).m*(p*(H-h+1)))*2<e by XCMPLX_1:4; then A556: p*(H-h+1)*(delta(T)).m < e/2 by REAL_2:178; set s=lower_integral(f); set sD=lower_sum(f,D); set sD1=lower_sum(f,T.m); set sD2=lower_sum(f,D2); s+sD1-sD1-sD < e/2 by A505,XCMPLX_1:26; then s-sD1+sD1-sD < e/2 by XCMPLX_1:29; then s-sD1+sD1 < sD+e/2 by REAL_1:84; then s-sD1 < sD+e/2-sD1 by REAL_1:86; then A557: s-sD1 < sD-sD1+e/2 by XCMPLX_1:29; sD-sD1 = sD+sD2-sD2-sD1 by XCMPLX_1:26 .= sD-sD2+sD2-sD1 by XCMPLX_1:29 .= sD2-sD1+(sD-sD2) by XCMPLX_1:29; then (sD-sD1) - (sD2-sD1) = (sD-sD2) by XCMPLX_1:26; then sD-sD1 <= sD2-sD1 by A520,SQUARE_1:11; then sD-sD1 <= p*(H-h+1)*(delta(T)).m by A554,AXIOMS:22; then sD-sD1 < e/2 by A556,AXIOMS:22; then sD-sD1+e/2 < e/2+e/2 by REAL_1:53; then sD-sD1+e/2 <= e by XCMPLX_1:66; then s-sD1 < e by A557,AXIOMS:22; then A558: lower_integral(f)-(lower_sum(f,T)).m < e by INTEGRA2:def 5; T.m in divs A; then A559: T.m in dom lower_sum_set(f) by INTEGRA1:def 12; (lower_sum(f,T)).m = lower_sum(f,T.m) by INTEGRA2:def 5; then (lower_sum(f,T)).m = (lower_sum_set(f)).(T.m) by A559,INTEGRA1:def 12; then (lower_sum(f,T)).m in rng lower_sum_set(f) by A559,FUNCT_1:def 5; then sup rng lower_sum_set(f)>=(lower_sum(f,T)).m by A502,SEQ_4:def 4; then lower_integral(f)>=(lower_sum(f,T)).m by INTEGRA1:def 16; then lower_integral(f)-(lower_sum(f,T)).m >= 0 by SQUARE_1:12; then abs(lower_integral(f)-(lower_sum(f,T)).m) < e by A558,ABSVALUE:def 1 ; then abs(-(lower_integral(f)-(lower_sum(f,T)).m)) < e by ABSVALUE:17; hence thesis by XCMPLX_1:143; end; take n; thus thesis by A516; end; hence lower_sum(f,T) is convergent by SEQ_2:def 6; hence thesis by A498,SEQ_2:def 7; end; theorem for A be closed-interval Subset of REAL, f be Function of A,REAL, T be DivSequence of A st f is_bounded_on A & delta(T) is convergent_to_0 & vol(A)<>0 holds upper_sum(f,T) is convergent & lim upper_sum(f,T) = upper_integral(f) proof let A be closed-interval Subset of REAL; let f be Function of A,REAL; let T be DivSequence of A; assume A1:f is_bounded_on A; assume A2:delta(T) is convergent_to_0; assume A3:vol(A)<>0; A4:delta(T) is_not_0 & delta(T) is convergent & lim delta(T) = 0 by A2,FDIFF_1:def 1; A5:for e st e>0 ex n st for m st n<=m holds 0 < (delta(T)).m & (delta(T)).m < e proof let e; assume e>0; then consider n such that A6: for m st n<=m holds abs((delta(T)).m-0)<e by A4,SEQ_2:def 7; take n; let m; assume n<=m; then A7: abs((delta(T)).m-0)<e by A6; (delta(T)).m-0<=abs((delta(T)).m-0) by ABSVALUE:11; then A8: (delta(T)).m+abs((delta(T)).m-0)<e+abs((delta(T)).m-0) by A7,REAL_1 :67; A9: (delta(T)).m<>0 by A4,SEQ_1:7; A10: (delta(T)).m = delta(T.m) by INTEGRA2:def 3; delta(T.m) = max rng upper_volume(chi(A,A),T.m) by INTEGRA1:def 19; then delta(T.m) in rng upper_volume(chi(A,A),T.m) by PRE_CIRC:def 1; then consider i such that A11: i in dom upper_volume(chi(A,A),T.m) & delta(T.m)=upper_volume(chi(A,A),T.m).i by PARTFUN1:26; A12: i in Seg len upper_volume(chi(A,A),T.m) by A11,FINSEQ_1:def 3; reconsider D = T.m as Element of divs A; i in Seg len D by A12,INTEGRA1:def 7; then delta(T.m)=vol(divset(T.m,i)) by A11,INTEGRA1:22; hence thesis by A8,A9,A10,AXIOMS:24,INTEGRA1:11; end; A13:for D,D1 be Element of divs A holds ex D2 be Element of divs A st D<=D2 & D1<=D2 & rng D2=rng D1 \/ rng D & 0<=upper_sum(f,D)-upper_sum(f,D2) & 0<=upper_sum(f,D1)-upper_sum(f,D2) proof let D,D1 be Element of divs A; consider D2 be Element of divs A such that A14: D<=D2 & D1<=D2 & rng D2=rng D1 \/ rng D by Th3; A15: f is_bounded_above_on A by A1,RFUNCT_1:def 11; then upper_sum(f,D2)<=upper_sum(f,D) by A14,INTEGRA1:47; then A16: upper_sum(f,D)-upper_sum(f,D2)>=0 by SQUARE_1:12; upper_sum(f,D2)<=upper_sum(f,D1) by A14,A15,INTEGRA1:47; then upper_sum(f,D1)-upper_sum(f,D2)>=0 by SQUARE_1:12; hence thesis by A14,A16; end; A17:sup rng f >= inf rng f by A1,Lm6; A18:for D,D1 be Element of divs A st delta(D1)<min rng upper_volume(chi(A,A),D) holds ex D2 be Element of divs A st D<=D2 & D1<=D2 & rng D2=rng D1 \/ rng D & upper_sum(f,D1)-upper_sum(f,D2) <= (len D)*(sup(rng f)-inf(rng f))*delta(D1) proof let D,D1 be Element of divs A; assume A19:delta(D1)<min rng upper_volume(chi(A,A),D); ex D2 be Element of divs A st D<=D2 & D1<=D2 & rng D2=rng D1 \/ rng D & upper_sum(f,D1)-upper_sum(f,D2)<=(len D)*(sup(rng f)-inf(rng f))*delta(D1) proof consider D2 be Element of divs A such that A20: D<=D2 & D1<=D2 & rng D2=rng D1 \/ rng D & 0<=upper_sum(f,D)-upper_sum(f,D2) & 0<=upper_sum(f,D1)-upper_sum(f,D2) by A13; upper_sum(f,D1)-upper_sum(f,D2) <= (len D)*(sup(rng f)-inf(rng f))*delta(D1) proof deffunc PUf(Element of divs A,Nat) = (PartSums(upper_volume(f,$1))).$2; deffunc UVf(Element of divs A) = upper_volume(f,$1); A21: for i st i in dom D holds ex j st j in dom D1 & D.i in divset(D1,j) & PUf(D1,j)-PUf(D2,indx(D2,D1,j))<=i*(sup(rng f)-inf(rng f))*delta(D1) proof let i; assume A22:i in dom D; A23: for i,j st i in dom D & j in dom D1 & D.i in divset(D1,j) holds j >= 2 proof let i,j; assume A24:i in dom D; assume A25:j in dom D1 & D.i in divset(D1,j); assume j<2; then j<1+1; then A26: j <= 1 by NAT_1:38; A27: inf divset(D1,j)<=D.i & D.i<=sup divset(D1,j) by A25,INTEGRA2:1; j in Seg len D1 by A25,FINSEQ_1:def 3; then j >= 1 by FINSEQ_1:3; then j = 1 by A26,AXIOMS:21; then A28: inf divset(D1,j)=inf A & sup divset(D1,j)=D1.j by A25,INTEGRA1:def 5; delta(D1) >= min rng upper_volume(chi(A,A),D) proof per cases; suppose A29:i=1; len D <> 0 by FINSEQ_1:25; then len D in Seg len D by FINSEQ_1:5; then 1 <= len D by FINSEQ_1:3; then A30: 1 in dom D by FINSEQ_3:27; then A31: inf divset(D,1)=inf A & sup divset(D,1)=D.1 by INTEGRA1:def 5; sup divset(D1,j)-inf A >= D.1-inf A by A27,A29,REAL_1:49; then vol(divset(D1,j)) >= sup divset(D,1)-inf divset(D,1) by A28,A31,INTEGRA1:def 6; then A32: vol(divset(D1,j)) >= vol(divset(D,1)) by INTEGRA1:def 6; vol(divset(D1,j)) <= delta(D1) by A25,Lm8; then A33: delta(D1) >= vol(divset(D,1)) by A32,AXIOMS:22; A34: 1 in Seg len D by A30,FINSEQ_1:def 3; then 1 in Seg len upper_volume(chi(A,A),D) by INTEGRA1:def 7; then A35: 1 in dom upper_volume(chi(A,A),D) by FINSEQ_1:def 3; vol(divset(D,1)) = upper_volume(chi(A,A),D).1 by A34,INTEGRA1:22; then vol(divset(D,1)) in rng upper_volume(chi(A,A),D) by A35,FUNCT_1:def 5; then vol(divset(D,1))>=min rng upper_volume(chi(A,A),D) by SFMASTR3: def 1; hence thesis by A33,AXIOMS:22; suppose A36:i<>1; then A37: inf divset(D,i)=D.(i-1) & sup divset(D,i)=D.i by A24,INTEGRA1:def 5; then A38: sup divset(D1,j)-inf divset(D1,j) >= sup divset(D,i)- inf A by A27,A28,REAL_1:49; D.(i-1) in A by A24,A36,INTEGRA1:9; then inf A <= D.(i-1) by INTEGRA2:1; then sup divset(D,i)-inf A >= sup divset(D,i)-inf divset(D,i) by A37,REAL_2:106; then sup divset(D1,j)-inf divset(D1,j) >= sup divset(D,i)-inf divset( D,i) by A38,AXIOMS:22; then vol(divset(D1,j)) >= sup divset(D,i)-inf divset(D,i) by INTEGRA1:def 6; then A39: vol(divset(D1,j)) >= vol(divset(D,i)) by INTEGRA1:def 6; vol(divset(D1,j)) <= delta(D1) by A25,Lm8; then A40: delta(D1) >= vol(divset(D,i)) by A39,AXIOMS:22; A41: i in Seg len D by A24,FINSEQ_1:def 3; then i in Seg len upper_volume(chi(A,A),D) by INTEGRA1:def 7; then A42: i in dom upper_volume(chi(A,A),D) by FINSEQ_1:def 3; vol(divset(D,i)) = upper_volume(chi(A,A),D).i by A41,INTEGRA1:22; then vol(divset(D,i)) in rng upper_volume(chi(A,A),D) by A42,FUNCT_1:def 5; then vol(divset(D,i))>=min rng upper_volume(chi(A,A),D) by SFMASTR3: def 1; hence thesis by A40,AXIOMS:22; end; hence contradiction by A19; end; defpred P[non empty Nat] means $1 in dom D implies ex j st j in dom D1 & D.$1 in divset(D1,j) & PUf(D1,j)-PUf(D2,indx(D2,D1,j))<=$1*(sup rng f-inf rng f)*delta(D1); A43: P[1] proof len D <> 0 by FINSEQ_1:25; then len D in Seg len D by FINSEQ_1:5; then 1 <= len D by FINSEQ_1:3; then A44: 1 in dom D by FINSEQ_3:27; then D.1 in A by INTEGRA1:8; then consider j such that A45: j in dom D1 & D.1 in divset(D1,j) by Th2; PUf(D1,j)-PUf(D2,indx(D2,D1,j))<=1*(sup rng f-inf rng f)*delta(D1) proof A46: j <> 1 by A23,A44,A45; then reconsider j1=j-1 as Element of NAT by A45,INTEGRA1:9; A47: j-1 in NAT & j-1 in dom D1 by A45,A46,INTEGRA1:9; inf divset(D1,j) <= D.1 & D.1 <= sup divset(D1,j) by A45,INTEGRA2:1; then A48: D1.j1 <= D.1 & D.1 <= D1.j by A45,A46,INTEGRA1:def 5; A49: indx(D2,D1,j1) in dom D2 & 1 <= indx(D2,D1,j1) & indx(D2,D1,j1) <= len D2 proof thus indx(D2,D1,j1) in dom D2 by A20,A47,INTEGRA1:def 21; then indx(D2,D1,j1) in Seg len D2 by FINSEQ_1:def 3; hence thesis by FINSEQ_1:3; end; then mid(D2,1,indx(D2,D1,j1)) is increasing by INTEGRA1:37; then A50: D2|indx(D2,D1,j1) is increasing by A49,JORDAN3:25; A51: j1 in dom D1 & 1 <= j1 & j1 <= len D1 proof thus j1 in dom D1 by A45,A46,INTEGRA1:9; j1 in Seg len D1 by A47,FINSEQ_1:def 3; hence thesis by FINSEQ_1:3; end; then mid(D1,1,j1) is increasing by INTEGRA1:37; then A52: D1|j1 is increasing by A51,JORDAN3:25; A53: rng (D2|indx(D2,D1,j1)) = rng (D1|j1) proof for x1 st x1 in rng(D2|indx(D2,D1,j1)) holds x1 in rng(D1|j1) proof let x1; assume x1 in rng(D2|indx(D2,D1,j1)); then consider k such that A54: k in dom(D2|indx(D2,D1,j1)) & x1=(D2|indx(D2,D1,j1)).k by PARTFUN1:26; k in Seg len(D2|indx(D2,D1,j1)) by A54,FINSEQ_1:def 3; then A55: k in Seg indx(D2,D1,j1) by A49,TOPREAL1:3; then A56: (D2|indx(D2,D1,j1)).k = D2.k & k in dom D2 by A49,RFINSEQ:19; then A57: D2.k in rng D2 by FUNCT_1:def 5; 1 <= k & k <= indx(D2,D1,j1) by A55,FINSEQ_1:3; then D2.k <= D2.indx(D2,D1,j1) by A49,A56,GOBOARD2:18; then A58: D2.k <= D1.j1 by A20,A47,INTEGRA1:def 21; A59: len(D1|j1) = j1 by A51,TOPREAL1:3; A60: D2.k in rng D implies D2.k = D1.j1 proof assume D2.k in rng D; then consider n such that A61: n in dom D & D2.k=D.n by PARTFUN1:26; 1 <= n by A61,FINSEQ_3:27; then D.1 <= D2.k by A44,A61,GOBOARD2:18; then D1.j1 <= D2.k by A48,AXIOMS:22; hence thesis by A58,AXIOMS:21; end; A62: D2.k in rng D implies D2.k in rng(D1|j1) proof assume A63:D2.k in rng D; j1 in Seg len(D1|j1) by A51,A59,FINSEQ_1:3; then j1 in dom(D1|j1) by FINSEQ_1:def 3; then A64: (D1|j1).j1 in rng(D1|j1) by FUNCT_1:def 5; j1 in Seg j1 by A51,FINSEQ_1:3; hence thesis by A51,A60,A63,A64,RFINSEQ:19; end; D2.k in rng D1 implies D2.k in rng(D1|j1) proof assume D2.k in rng D1; then consider m such that A65: m in dom D1 & D2.k = D1.m by PARTFUN1:26; m in Seg len D1 by A65,FINSEQ_1:def 3; then A66: 1 <= m & m <= j1 by A51,A58,A65,FINSEQ_1:3,GOBOARD1:def 1; then m in Seg len(D1|j1) by A59,FINSEQ_1:3; then A67: m in dom (D1|j1) by FINSEQ_1:def 3; m in Seg j1 by A66,FINSEQ_1:3; then D2.k = (D1|j1).m by A51,A65,RFINSEQ:19; hence thesis by A67,FUNCT_1:def 5; end; hence thesis by A20,A49,A54,A55,A57,A62,RFINSEQ:19,XBOOLE_0:def 2; end; then A68: rng(D2|indx(D2,D1,j1)) c= rng (D1|j1) by TARSKI:def 3; for x1 st x1 in rng(D1|j1) holds x1 in rng(D2|indx(D2,D1,j1)) proof let x1; assume x1 in rng(D1|j1); then consider k such that A69: k in dom(D1|j1) & x1=(D1|j1).k by PARTFUN1:26; k in Seg len(D1|j1) by A69,FINSEQ_1:def 3; then A70: k in Seg j1 by A51,TOPREAL1:3; then A71: (D1|j1).k = D1.k & k in dom D1 by A51,RFINSEQ:19; then D1.k in rng D1 by FUNCT_1:def 5; then x1 in rng D2 by A20,A69,A71,XBOOLE_0:def 2; then consider n such that A72: n in dom D2 & x1=D2.n by PARTFUN1:26; A73: indx(D2,D1,k) in dom D2 & D2.indx(D2,D1,k)=D2.n by A20,A69,A71,A72,INTEGRA1:def 21; k <= j1 by A70,FINSEQ_1:3; then D1.k <= D1.j1 by A47,A71,GOBOARD2:18; then D2.indx(D2,D1,k) <= D1.j1 by A20,A71,INTEGRA1:def 21; then D2.indx(D2,D1,k)<=D2.indx(D2,D1,j1) by A20,A47,INTEGRA1:def 21; then A74: n <= indx(D2,D1,j1) by A49,A72,A73,GOBOARD1:def 1; 1 <= n by A72,FINSEQ_3:27; then A75: n in Seg indx(D2,D1,j1) by A74,FINSEQ_1:3; then A76: D2.n = (D2|indx(D2,D1,j1)).n by A49,RFINSEQ:19; n in Seg len(D2|indx(D2,D1,j1)) by A49,A75,TOPREAL1:3; then n in dom(D2|indx(D2,D1,j1)) by FINSEQ_1:def 3; hence thesis by A72,A76,FUNCT_1:def 5; end; then rng(D1|j1) c= rng(D2|indx(D2,D1,j1)) by TARSKI:def 3; hence thesis by A68,XBOOLE_0:def 10; end; then A77: D2|indx(D2,D1,j1)=D1|j1 by A50,A52,Th5; A78: for k st 1 <= k & k <= j1 holds k=indx(D2,D1,k) proof let k; assume A79:1 <= k & k <= j1; assume A80:k<>indx(D2,D1,k); now per cases by A80,AXIOMS:21; suppose A81:k > indx(D2,D1,k); 1 <= k & k <= len D1 by A51,A79,AXIOMS:22; then A82: k in dom D1 by FINSEQ_3:27; then A83: indx(D2,D1,k) in dom D2 & D1.k=D2.indx(D2,D1,k) by A20,INTEGRA1:def 21; then indx(D2,D1,k) in Seg len D2 by FINSEQ_1:def 3; then A84: 1<=indx(D2,D1,k)&indx(D2,D1,k)<=indx(D2,D1,j1) by A20,A51,A79,A82,Th6,FINSEQ_1:3; then indx(D2,D1,k) in Seg indx(D2,D1,j1) by FINSEQ_1:3; then A85: (D2|indx(D2,D1,j1)).indx(D2,D1,k)=D2.indx(D2,D1,k) by A49, RFINSEQ:19; A86: indx(D2,D1,k) < j1 by A79,A81,AXIOMS:22; then indx(D2,D1,k) <= len D1 by A51,AXIOMS:22; then indx(D2,D1,k) in dom D1 by A84,FINSEQ_3:27; then A87: D1.k > D1.indx(D2,D1,k) by A81,A82,GOBOARD1:def 1; indx(D2,D1,k) in Seg j1 by A84,A86,FINSEQ_1:3; hence contradiction by A51,A77,A83,A85,A87,RFINSEQ:19; suppose A88:k < indx(D2,D1,k); k in Seg j1 by A79,FINSEQ_1:3; then A89: D1.k = (D1|j1).k by A47,RFINSEQ:19; 1 <= k & k <= len D1 by A51,A79,AXIOMS:22; then A90: k in dom D1 by FINSEQ_3:27; then A91: indx(D2,D1,k) in dom D2 & D1.k=D2.indx(D2,D1,k) by A20,INTEGRA1:def 21; indx(D2,D1,k) <= indx(D2,D1,j1) & indx(D2,D1,k) in dom D2 & indx(D2,D1,j1) in dom D2 by A20,A51,A79,A90,Th6; then A92: k <= indx(D2,D1,j1) by A88,AXIOMS:22; then k <= len D2 by A49,AXIOMS:22; then k in dom D2 by A79,FINSEQ_3:27; then A93: D2.k < D2.indx(D2,D1,k) by A88,A91,GOBOARD1:def 1; k in Seg indx(D2,D1,j1) by A79,A92,FINSEQ_1:3; hence contradiction by A49,A77,A89,A91,A93,RFINSEQ:19; end; hence contradiction; end; A94: len (D2|indx(D2,D1,j1))=len (D1|j1) by A50,A52,A53,Th5; len (D2|indx(D2,D1,j1))=indx(D2,D1,j1) by A49,TOPREAL1:3; then A95: indx(D2,D1,j1) =j1 by A51,A94,TOPREAL1:3; j1 <= len D1 by A47,FINSEQ_3:27; then j1 <= len upper_volume(f,D1) by INTEGRA1:def 7; then A96: len(upper_volume(f,D1)|j1)=indx(D2,D1,j1) by A95,TOPREAL1:3; indx(D2,D1,j1) in dom D2 by A20,A47,INTEGRA1:def 21; then indx(D2,D1,j1) <= len D2 by FINSEQ_3:27; then indx(D2,D1,j1) <= len upper_volume(f,D2) by INTEGRA1:def 7; then A97: len(upper_volume(f,D1)|j1)=len(upper_volume(f,D2)|indx(D2,D1,j1)) by A96,TOPREAL1:3; for k st 1 <= k & k <= len(upper_volume(f,D1)|j1) holds (upper_volume(f,D1)|j1).k = (upper_volume(f,D2)|indx(D2,D1,j1)).k proof let k; assume A98:1 <= k & k <= len(upper_volume(f,D1)|j1); A99: len(upper_volume(f,D1)) = len D1 by INTEGRA1:def 7; then A100: 1 <= k & k <= j1 by A51,A98,TOPREAL1:3; then A101: k in Seg j1 by FINSEQ_1:3; k <= len D1 by A51,A100,AXIOMS:22; then A102: k in Seg len D1 by A98,FINSEQ_1:3; A103: divset(D1,k)=divset(D2,indx(D2,D1,k)) proof A104: divset(D2,indx(D2,D1,k))= [. inf divset(D2,indx(D2,D1,k)), sup divset(D2,indx(D2,D1,k)).] by INTEGRA1:5; A105: k in dom D1 by A102,FINSEQ_1:def 3; then A106: indx(D2,D1,k) in dom D2 & D1.k=D2.indx(D2,D1,k) by A20,INTEGRA1: def 21; inf divset(D1,k)=inf divset(D2,indx(D2,D1,k)) & sup divset(D1,k)=sup divset(D2,indx(D2,D1,k)) proof per cases; suppose A107:k=1; then A108: inf divset(D1,k)=inf A&sup divset(D1,k)=D1.k by A105,INTEGRA1: def 5; indx(D2,D1,k)=1 by A51,A78,A107; hence thesis by A106,A108,INTEGRA1:def 5; suppose A109:k<>1; then A110: inf divset(D1,k)=D1.(k-1) & sup divset(D1,k)=D1.k by A105,INTEGRA1:def 5; A111: k-1 in dom D1 & D1.(k-1) in A & k-1 in NAT by A105,A109,INTEGRA1:9; reconsider k1=k-1 as Nat by A105,A109,INTEGRA1:9; k <= k+1 by NAT_1:29; then k1 <= k by REAL_1:86; then A112: k1 <= j1 by A100,AXIOMS:22; 1 <= k1 by A111,FINSEQ_3:27; then A113: k1=indx(D2,D1,k1) by A78,A112; indx(D2,D1,k)<>1 by A78,A100,A109; then A114: inf divset(D2,indx(D2,D1,k))=D2.(indx(D2,D1,k)-1) & sup divset(D2,indx(D2,D1,k))=D2.indx(D2,D1,k) by A106,INTEGRA1:def 5; D2.(indx(D2,D1,k)-1)= D2.indx(D2,D1,k1) by A78,A100,A113; hence thesis by A20,A105,A110,A111,A114,INTEGRA1:def 21; end; hence thesis by A104,INTEGRA1:5; end; j1 in Seg len(upper_volume(f,D1)) by A51,A99,FINSEQ_1:def 3; then j1 in dom(upper_volume(f,D1)) by FINSEQ_1:def 3; then A115: (upper_volume(f,D1)|j1).k = upper_volume(f,D1).k by A101,RFINSEQ: 19 .=(sup(rng(f|divset(D2,indx(D2,D1,k)))))*vol(divset(D2,indx(D2,D1,k))) by A102,A103,INTEGRA1:def 7; indx(D2,D1,k) in Seg j1 by A78,A100,A101; then A116: indx(D2,D1,k) in Seg indx(D2,D1,j1) by A51,A78; then 1<=indx(D2,D1,k) & indx(D2,D1,k)<=indx(D2,D1,j1) by FINSEQ_1:3; then 1<=indx(D2,D1,k) & indx(D2,D1,k)<=len D2 by A49,AXIOMS:22; then A117: indx(D2,D1,k) in Seg len D2 by FINSEQ_1:3; indx(D2,D1,j1) in Seg len D2 by A49,FINSEQ_1:def 3; then indx(D2,D1,j1) in Seg len upper_volume(f,D2) by INTEGRA1:def 7; then A118: indx(D2,D1,j1) in dom upper_volume(f,D2) by FINSEQ_1:def 3; (upper_volume(f,D2)|indx(D2,D1,j1)).k =(upper_volume(f,D2)|indx(D2,D1,j1)).indx(D2,D1,k) by A78,A100 .=upper_volume(f,D2).indx(D2,D1,k) by A116,A118,RFINSEQ:19 .=(sup(rng(f|divset(D2,indx(D2,D1,k)))))*vol(divset(D2,indx(D2,D1,k))) by A117,INTEGRA1:def 7; hence thesis by A115; end; then A119: upper_volume(f,D2)|indx(D2,D1,j1)=upper_volume(f,D1)|j1 by A97,FINSEQ_1:18; indx(D2,D1,j1) in Seg len D2 by A49,FINSEQ_1:def 3; then A120: indx(D2,D1,j1) in Seg len upper_volume(f,D2) by INTEGRA1:def 7; j1 in Seg len D1 by A51,FINSEQ_1:def 3; then A121: j1 in Seg len upper_volume(f,D1) by INTEGRA1:def 7; A122: indx(D2,D1,j1)+1 <= indx(D2,D1,j) proof j < j+1 by NAT_1:38; then j1 < j by REAL_1:84; then indx(D2,D1,j1) < indx(D2,D1,j) by A20,A45,A51,Th7; hence thesis by NAT_1:38; end; A123: PUf(D2,indx(D2,D1,j1))=Sum(upper_volume(f,D2)|indx(D2,D1,j1)) by A120,INTEGRA1:def 22 .=PUf(D1,j1) by A119,A121,INTEGRA1:def 22; A124: Sum mid(upper_volume(f,D1),j,j) -Sum mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) <= (sup rng f-inf rng f)*delta(D1) proof A125: 1 <= indx(D2,D1,j) & indx(D2,D1,j) <= len upper_volume(f,D2) proof indx(D2,D1,j) in dom D2 by A20,A45,INTEGRA1:def 21; then indx(D2,D1,j) in Seg len D2 by FINSEQ_1:def 3; then indx(D2,D1,j) in Seg len upper_volume(f,D2) by INTEGRA1:def 7; hence thesis by FINSEQ_1:3; end; then A126: 1 <= indx(D2,D1,j1)+1 & indx(D2,D1,j1)+1 <= len upper_volume(f,D2 ) by A49,A122,AXIOMS:22,NAT_1:38; A127: indx(D2,D1,j)-indx(D2,D1,j1) <= 2 proof assume A128:indx(D2,D1,j)-indx(D2,D1,j1) > 2; reconsider ID1=indx(D2,D1,j1)+1 as Nat; reconsider ID2=ID1+1 as Nat; A129: indx(D2,D1,j1) < ID1 & ID1 < ID2 & ID2 < indx(D2,D1,j) proof thus indx(D2,D1,j1) < ID1 by NAT_1:38; thus ID1 < ID2 by NAT_1:38; indx(D2,D1,j1)+(1+1)<indx(D2,D1,j) by A128,REAL_1:86; hence thesis by XCMPLX_1:1; end; A130: indx(D2,D1,j) in dom D2 by A20,A45,INTEGRA1:def 21; then A131: indx(D2,D1,j) <= len D2 by FINSEQ_3:27; A132: ID1 in dom D2 proof A133: 1 <= ID1 & ID1 <= indx(D2,D1,j) by A49,A129,AXIOMS:22; then ID1 <= len D2 by A131,AXIOMS:22; hence thesis by A133,FINSEQ_3:27; end; A134: ID2 in dom D2 proof A135: indx(D2,D1,j1) <= ID2 & ID2 <= len D2 by A129,A131,AXIOMS:22; then 1 <= ID2 by A49,AXIOMS:22; hence thesis by A135,FINSEQ_3:27; end; then A136: D2.indx(D2,D1,j1)<D2.ID1 & D2.ID1<D2.ID2 & D2.ID2<D2.indx(D2,D1, j) by A49,A129,A130,A132,GOBOARD1:def 1; A137: D2.ID1 in rng D & D2.ID2 in rng D proof A138: D2.ID1 in rng D2 & D2.ID2 in rng D2 by A132,A134,FUNCT_1:def 5 ; A139: D1.j1 = D2.indx(D2,D1,j1) & D1.j = D2.indx(D2,D1,j) by A20,A45,A51,INTEGRA1:def 21; not(D2.ID1 in rng D1) & not D2.ID2 in rng D1 proof assume A140:D2.ID1 in rng D1 or D2.ID2 in rng D1; per cases by A140; suppose D2.ID1 in rng D1; then consider n such that A141: n in dom D1 & D1.n=D2.ID1 by PARTFUN1:26; D2.indx(D2,D1,j1)<D2.ID1 & D2.ID1<D2.indx(D2,D1,j) by A136,AXIOMS:22; then j1<n & n<j by A45,A51,A139,A141,GOBOARD2:18; then j<n+1 & n<j by REAL_1:84; hence contradiction by NAT_1:38; suppose D2.ID2 in rng D1; then consider n such that A142: n in dom D1 & D1.n=D2.ID2 by PARTFUN1:26; D2.indx(D2,D1,j1)<D2.ID2 & D2.ID2<D2.indx(D2,D1,j) by A136,AXIOMS:22; then j1<n & n<j by A45,A51,A139,A142,GOBOARD2:18; then j<n+1 & n<j by REAL_1:84; hence contradiction by NAT_1:38; end; hence thesis by A20,A138,XBOOLE_0:def 2; end; A143: inf divset(D1,j)=D2.indx(D2,D1,j1) & sup divset(D1,j)=D2.indx(D2,D1,j) proof inf divset(D1,j)=D1.j1 & sup divset(D1,j)=D1.j by A45,A46,INTEGRA1:def 5; hence thesis by A20,A45,A51,INTEGRA1:def 21; end; A144: D2.ID1 in rng D /\ divset(D1,j) proof D2.indx(D2,D1,j1)<=D2.ID1&D2.ID1<=D2.indx(D2,D1,j) by A136,AXIOMS:22; then D2.ID1 in divset(D1,j) by A143,INTEGRA2:1; hence thesis by A137,XBOOLE_0:def 3; end; D2.ID2 in rng D /\ divset(D1,j) proof D2.indx(D2,D1,j1)<=D2.ID2&D2.ID2<=D2.indx(D2,D1,j) by A136,AXIOMS:22; then D2.ID2 in divset(D1,j) by A143,INTEGRA2:1; hence thesis by A137,XBOOLE_0:def 3; end; then D2.ID1=D2.ID2 by A19,A45,A144,Th4; hence contradiction by A129,A132,A134,GOBOARD2:19; end; 1 <= indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 & indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 <= 2 proof A145: indx(D2,D1,j)-'(indx(D2,D1,j1)+1) =indx(D2,D1,j)-(indx(D2,D1,j1)+1) by A122,SCMFSA_7:3; then indx(D2,D1,j)-'(indx(D2,D1,j1)+1) >= 0 by A122,SQUARE_1:12; then indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 >= 0+1 by AXIOMS:24; hence indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 >= 1; indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 =indx(D2,D1,j)-indx(D2,D1,j1)-1+1 by A145,XCMPLX_1:36 .=indx(D2,D1,j)-indx(D2,D1,j1)+1-1 by XCMPLX_1:29 .=indx(D2,D1,j)-indx(D2,D1,j1) by XCMPLX_1:26; hence thesis by A127; end; then A146: 1 <= len mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j )) & len mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j))<=2 by A122,A125,A126,JORDAN3:27; 1 <= j & j <= len D1 by A45,FINSEQ_3:27; then A147: 1 <= j & j <= len upper_volume(f,D1) by INTEGRA1:def 7; j-'j+1 = 1 by Lm3; then A148: len mid(upper_volume(f,D1),j,j) = 1 by A147,JORDAN3:27; mid(upper_volume(f,D1),j,j).1 = upper_volume(f,D1).j by A147,JORDAN3:27; then mid(upper_volume(f,D1),j,j) =<*upper_volume(f,D1).j*> by A148,FINSEQ_1:57; then A149: Sum mid(upper_volume(f,D1),j,j)=upper_volume(f,D1).j by RVSUM_1: 103; A150: indx(D2,D1,j1)+1 in Seg len D2 proof indx(D2,D1,j1)+1 in Seg len upper_volume(f,D2) by A126,FINSEQ_1:3; hence thesis by INTEGRA1:def 7; end; A151: j in Seg len D1 by A45,FINSEQ_1:def 3; now per cases by A146,Lm4; suppose A152: len mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j))=1; A153: indx(D2,D1,j1)+1=indx(D2,D1,j) proof len mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) = indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 by A122,A125,A126,JORDAN3:27; then indx(D2,D1,j)-'(indx(D2,D1,j1)+1)=1-1 by A152,XCMPLX_1:26; then indx(D2,D1,j)-(indx(D2,D1,j1)+1)=0 by A122,SCMFSA_7:3; then indx(D2,D1,j)=0+(indx(D2,D1,j1)+1) by XCMPLX_1:27; hence thesis; end; A154: divset(D2,indx(D2,D1,j))=divset(D1,j) proof inf divset(D1,j)=D2.indx(D2,D1,j1) & sup divset(D1,j)=D2.indx(D2,D1,j) proof inf divset(D1,j)=D1.j1 & sup divset(D1,j)=D1.j by A45,A46,INTEGRA1:def 5; hence thesis by A20,A45,A51,INTEGRA1:def 21; end; then A155: divset(D1,j)=[. D2.indx(D2,D1,j1),D2.indx(D2,D1,j).] by INTEGRA1:5; A156: indx(D2,D1,j)-1=indx(D2,D1,j1) by A153,XCMPLX_1:26; A157: indx(D2,D1,j) in dom D2 by A20,A45,INTEGRA1:def 21; indx(D2,D1,j)<>1 by A49,A153,NAT_1:38; then inf divset(D2,indx(D2,D1,j))=D2.indx(D2,D1,j1) & sup divset(D2,indx(D2,D1,j))=D2.indx(D2,D1,j) by A156,A157,INTEGRA1:def 5; hence thesis by A155,INTEGRA1:5; end; mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)).1 =upper_volume(f,D2).(indx(D2,D1,j1)+1) by A125,A126,JORDAN3:27; then mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =<*upper_volume(f,D2).(indx(D2,D1,j1)+1)*> by A152,FINSEQ_1:57; then A158: Sum mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j )) =upper_volume(f,D2).(indx(D2,D1,j1)+1) by RVSUM_1:103 .=(sup(rng(f|divset(D2,(indx(D2,D1,j1)+1))))) *vol(divset(D2,(indx(D2,D1,j1)+1))) by A150,INTEGRA1:def 7 .=Sum mid(upper_volume(f,D1),j,j) by A149,A151,A153,A154,INTEGRA1:def 7; A159: delta(D1) >= 0 by Th8; sup rng f - inf rng f >= 0 by A17,SQUARE_1:12; then (sup rng f - inf rng f)*delta(D1) >= 0*delta(D1) by A159,AXIOMS: 25; hence Sum mid(upper_volume(f,D1),j,j) -Sum mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) <= (sup rng f-inf rng f)*delta(D1) by A158,XCMPLX_1:14; suppose A160: len mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j))=2; A161: mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)).1 =upper_volume(f,D2).(indx(D2,D1,j1)+1) by A125,A126,JORDAN3:27; mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)).2 =upper_volume(f,D2).(indx(D2,D1,j1)+2) proof A162: 2+(indx(D2,D1,j1)+1)>=0+1 by A126,REAL_1:55; mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)).2 =UVf(D2).(2+(indx(D2,D1,j1)+1)-'1) by A122,A125,A126,A160,JORDAN3: 27 .=UVf(D2).(2+(indx(D2,D1,j1)+1)-1) by A162,SCMFSA_7:3 .=UVf(D2).((indx(D2,D1,j1)+1)+(2-1)) by XCMPLX_1:29 .=UVf(D2).(indx(D2,D1,j1)+(1+1)) by XCMPLX_1:1; hence thesis; end; then mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =<*upper_volume(f,D2).(indx(D2,D1,j1)+1), upper_volume(f,D2).(indx(D2,D1,j1)+2)*> by A160,A161,FINSEQ_1:61; then A163: Sum mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =upper_volume(f,D2).(indx(D2,D1,j1)+1) +upper_volume(f,D2).(indx(D2,D1,j1)+2) by RVSUM_1:107; A164: vol(divset(D1,j)) =vol(divset(D2,indx(D2,D1,j1)+1))+vol(divset(D2,indx(D2,D1,j1)+2)) proof A165: inf divset(D1,j)=D2.indx(D2,D1,j1)&sup divset(D1,j)=D2.indx(D2,D1,j) proof inf divset(D1,j)=D1.j1 & sup divset(D1,j)=D1.j by A45,A46,INTEGRA1:def 5; hence thesis by A20,A45,A51,INTEGRA1:def 21; end; A166: indx(D2,D1,j)=indx(D2,D1,j1)+2 proof indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1=2 by A122,A125,A126,A160,JORDAN3:27; then 2=indx(D2,D1,j)-(indx(D2,D1,j1)+1)+1 by A122,SCMFSA_7:3 .=indx(D2,D1,j)-indx(D2,D1,j1)-1+1 by XCMPLX_1:36 .=indx(D2,D1,j)-indx(D2,D1,j1)-(1-1) by XCMPLX_1:37 .=indx(D2,D1,j)-indx(D2,D1,j1); hence thesis by XCMPLX_1:27; end; A167: inf divset(D2,(indx(D2,D1,j1)+2))=D2.(indx(D2,D1,j1)+1) & sup divset(D2,(indx(D2,D1,j1)+2))=D2.(indx(D2,D1,j1)+2) & inf divset(D2,(indx(D2,D1,j1)+1))=D2.indx(D2,D1,j1) & sup divset(D2,(indx(D2,D1,j1)+1))=D2.(indx(D2,D1,j1)+1) proof A168: indx(D2,D1,j1)+2 in dom D2 by A20,A45,A166,INTEGRA1:def 21; indx(D2,D1,j1)+1 in Seg len upper_volume(f,D2) by A126,FINSEQ_1:3 ; then indx(D2,D1,j1)+1 in Seg len D2 by INTEGRA1:def 7; then A169: indx(D2,D1,j1)+1 in dom D2 by FINSEQ_1:def 3; A170: indx(D2,D1,j1)+1 <> 1 by A49,NAT_1:38; indx(D2,D1,j1)+1+1 > 1 by A126,NAT_1:38; then A171: indx(D2,D1,j1)+(1+1) > 1 by XCMPLX_1:1; A172: indx(D2,D1,j1)+2-1=indx(D2,D1,j1)+1 by Lm2,XCMPLX_1:29; A173: indx(D2,D1,j1)+1-1=indx(D2,D1,j1)+(1-1) by XCMPLX_1:29 .=indx(D2,D1,j1)+0; thus inf divset(D2,(indx(D2,D1,j1)+2))=D2.(indx(D2,D1,j1)+1) & sup divset(D2,(indx(D2,D1,j1)+2))=D2.(indx(D2,D1,j1)+2) by A168,A171,A172,INTEGRA1:def 5; thus thesis by A169,A170,A173,INTEGRA1:def 5; end; vol(divset(D1,j)) =sup divset(D1,j)-inf divset(D1,j) by INTEGRA1:def 6 .=D2.indx(D2,D1,j)-(D2.(indx(D2,D1,j1)+1) -D2.(indx(D2,D1,j1)+1))-D2.indx(D2,D1,j1) by A165,XCMPLX_1:17 .=D2.(indx(D2,D1,j1)+2)-D2.(indx(D2,D1,j1)+1) +D2.(indx(D2,D1,j1)+1)-D2.indx(D2,D1,j1) by A166,XCMPLX_1:37; then vol(divset(D1,j))=vol(divset(D2,indx(D2,D1,j1)+2)) +D2.(indx(D2,D1,j1)+1)-D2.indx(D2,D1,j1) by A167,INTEGRA1:def 6 .=vol(divset(D2,indx(D2,D1,j1)+2)) +(sup divset(D2,indx(D2,D1,j1)+1)-inf divset(D2,indx(D2,D1,j1)+1)) by A167,XCMPLX_1:29; hence thesis by INTEGRA1:def 6; end; then A174: upper_volume(f,D1).j=(sup(rng(f|divset(D1,j))))* (vol(divset(D2,indx(D2,D1,j1)+1))+vol(divset(D2,indx(D2,D1,j1)+2))) by A151,INTEGRA1:def 7; A175: vol(divset(D2,indx(D2,D1,j1)+1))>=0 by INTEGRA1:11; A176: vol(divset(D2,indx(D2,D1,j1)+2))>=0 by INTEGRA1:11; A177: Sum mid(UVf(D1),j,j)-Sum mid(UVf(D2),indx(D2,D1,j1)+1,indx(D2, D1,j)) <=(sup rng f - inf rng f)*(vol(divset(D2,indx(D2,D1,j1)+2)) +vol(divset(D2,indx(D2,D1,j1)+1))) proof A178: indx(D2,D1,j) in dom D2 by A20,A45,INTEGRA1:def 21; then A179: indx(D2,D1,j) in Seg len D2 by FINSEQ_1:def 3; A180: indx(D2,D1,j)=indx(D2,D1,j1)+2 proof indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1=2 by A122,A125,A126,A160,JORDAN3:27; then 2=indx(D2,D1,j)-(indx(D2,D1,j1)+1)+1 by A122,SCMFSA_7:3 .=indx(D2,D1,j)-indx(D2,D1,j1)-1+1 by XCMPLX_1:36 .=indx(D2,D1,j)-indx(D2,D1,j1)-(1-1) by XCMPLX_1:37 .=indx(D2,D1,j)-indx(D2,D1,j1); hence thesis by XCMPLX_1:27; end; set ID1=indx(D2,D1,j1)+1; set ID2=indx(D2,D1,j1)+2; A181: Sum mid(UVf(D2),ID1,indx(D2,D1,j)) =(sup rng(f|divset(D2,ID2)))*vol(divset(D2,ID2)) +UVf(D2).ID1 by A163,A179,A180,INTEGRA1:def 7 .=(sup rng(f|divset(D2,ID2)))*vol(divset(D2,ID2)) +(sup rng(f|divset(D2,ID1))) *vol(divset(D2,ID1)) by A150,INTEGRA1:def 7; divset(D2,ID2) c= A by A178,A180,INTEGRA1:10; then sup rng(f|divset(D2,ID2)) >= inf rng f by A1,Lm7; then A182: (sup rng(f|divset(D2,ID2)))*vol(divset(D2,ID2)) >=(inf rng f)*vol(divset(D2,ID2)) by A176,AXIOMS:25; ID1 in dom D2 by A150,FINSEQ_1:def 3; then divset(D2,ID1)c=A by INTEGRA1:10; then sup rng(f|divset(D2,ID1)) >= inf rng f by A1,Lm7; then A183: (sup rng(f|divset(D2,ID1)))*vol(divset(D2,ID1)) >=(inf rng f)*vol(divset(D2,ID1)) by A175,AXIOMS:25; divset(D1,j)c=A by A45,INTEGRA1:10; then sup rng(f|divset(D1,j)) <= sup rng f by A1,Lm7; then A184: (sup rng(f|divset(D1,j)))*vol(divset(D2,ID2)) <=(sup rng f)*vol(divset(D2,ID2)) & (sup rng(f|divset(D1,j)))*vol(divset(D2,ID1)) <=(sup rng f)*vol(divset(D2,ID1)) by A175,A176,AXIOMS:25; set SR = sup rng(f|divset(D2,ID1)), VR = vol(divset(D2,ID1)); Sum mid(UVf(D2),ID1,indx(D2,D1,j)) - SR*VR >=(inf rng f)*vol(divset(D2,ID2)) by A181,A182,XCMPLX_1:26; then Sum mid(UVf(D2),ID1,indx(D2,D1,j)) >=(inf rng f)*vol(divset(D2,ID2))+SR*VR by REAL_1:84; then Sum mid(UVf(D2),ID1,indx(D2,D1,j))-(inf rng f)*vol(divset(D2, ID2)) >=SR*VR by REAL_1:84; then Sum mid(UVf(D2),ID1,indx(D2,D1,j))-(inf rng f)*vol(divset(D2, ID2)) >=(inf rng f)*VR by A183,AXIOMS:22; then A185: Sum mid(UVf(D2),ID1,indx(D2,D1,j))>=(inf rng f)*vol(divset(D2, ID2))+ (inf rng f)*VR by REAL_1:84; Sum mid(UVf(D1),j,j) =(sup rng(f|divset(D1,j)))*vol(divset(D2,ID2)) +(sup rng(f|divset(D1,j)))*vol(divset(D2,ID1)) by A149,A174,XCMPLX_1:8; then Sum mid(UVf(D1),j,j)-(sup rng(f|divset(D1,j)))*vol(divset(D2, ID1)) =(sup rng(f|divset(D1,j)))*vol(divset(D2,ID2)) by XCMPLX_1:26; then Sum mid(UVf(D1),j,j)<=(sup rng(f|divset(D1,j)))*vol(divset(D2, ID1)) +(sup rng f)*vol(divset(D2,ID2)) by A184,REAL_1:86; then Sum mid(UVf(D1),j,j)-(sup rng f)*vol(divset(D2,ID2)) <=(sup rng(f|divset(D1,j)))*vol(divset(D2,ID1)) by REAL_1:86; then Sum mid(UVf(D1),j,j)-(sup rng f)*vol(divset(D2,ID2)) <=(sup rng f)*vol(divset(D2,ID1)) by A184,AXIOMS:22; then Sum mid(UVf(D1),j,j) <=(sup rng f)*vol(divset(D2,ID2))+(sup rng f)*vol(divset(D2,ID1)) by REAL_1:86; then Sum mid(UVf(D1),j,j)-Sum mid(UVf(D2),ID1,indx(D2,D1,j)) <=(sup rng f)*vol(divset(D2,ID2))+(sup rng f)*vol(divset(D2,ID1)) -((inf rng f)*vol(divset(D2,ID2))+(inf rng f)*vol(divset(D2,ID1))) by A185,REAL_1:92; then Sum mid(UVf(D1),j,j)-Sum mid(UVf(D2),ID1,indx(D2,D1,j)) <=((sup rng f)*vol(divset(D2,ID2))-(inf rng f)*vol(divset(D2,ID2))) +((sup rng f)*vol(divset(D2,ID1))-(inf rng f)*vol(divset(D2,ID1))) by Lm5; then Sum mid(UVf(D1),j,j)-Sum mid(UVf(D2),ID1,indx(D2,D1,j)) <=(sup rng f - inf rng f)*vol(divset(D2,ID2)) +((sup rng f)*vol(divset(D2,ID1))-(inf rng f)*vol(divset(D2,ID1))) by XCMPLX_1:40; then Sum mid(UVf(D1),j,j)-Sum mid(UVf(D2),ID1,indx(D2,D1,j)) <=(sup rng f - inf rng f)*vol(divset(D2,ID2)) +(sup rng f - inf rng f)*vol(divset(D2,ID1)) by XCMPLX_1:40; hence thesis by XCMPLX_1:8; end; A186: vol(divset(D1,j)) <= delta(D1) by A45,Lm8; sup rng f - inf rng f >= 0 by A17,SQUARE_1:12; then (sup rng f - inf rng f)*(vol(divset(D1,j))) <=(sup rng f - inf rng f)*delta(D1) by A186,AXIOMS:25; hence thesis by A164,A177,AXIOMS:22; end; hence thesis; end; A187: PUf(D2,indx(D2,D1,j1)) +Sum mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =PUf(D2,indx(D2,D1,j)) proof indx(D2,D1,j) in dom D2 by A20,A45,INTEGRA1:def 21; then A188: indx(D2,D1,j) in Seg len D2 by FINSEQ_1:def 3; then A189: 1 <= indx(D2,D1,j) & indx(D2,D1,j) <= len D2 by FINSEQ_1:3; then A190: 1 <= indx(D2,D1,j) & indx(D2,D1,j) <= len UVf(D2) by INTEGRA1:def 7; A191: indx(D2,D1,j) in Seg len UVf(D2) by A188,INTEGRA1:def 7; A192: indx(D2,D1,j1) < indx(D2,D1,j) by A122,NAT_1:38; indx(D2,D1,j1) in Seg len D2 by A49,FINSEQ_1:def 3; then indx(D2,D1,j1) in Seg len UVf(D2) by INTEGRA1:def 7; then PUf(D2,indx(D2,D1,j1)) =Sum(UVf(D2)|indx(D2,D1,j1)) by INTEGRA1:def 22; then PUf(D2,indx(D2,D1,j1)) +Sum mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j)) =Sum(UVf(D2)|indx(D2,D1,j1) ^mid(UVf(D2),(indx(D2,D1,j1)+1),indx(D2,D1,j))) by RVSUM_1:105 .=Sum(mid(UVf(D2),1,indx(D2,D1,j1)) ^mid(UVf(D2),indx(D2,D1,j1)+1,indx(D2,D1,j))) by A49,JORDAN3:25 .=Sum(mid(UVf(D2),1,indx(D2,D1,j))) by A49,A190,A192,INTEGRA2:4 .=Sum(UVf(D2)|indx(D2,D1,j)) by A189,JORDAN3:25; hence thesis by A191,INTEGRA1:def 22; end; PUf(D1,j1)+Sum mid(upper_volume(f,D1),j,j)=PUf(D1,j) proof A193: j in Seg len D1 by A45,FINSEQ_1:def 3; then A194: 1 <= j & j <= len D1 by FINSEQ_1:3; then A195: 1 <= j & j <= len UVf(D1) by INTEGRA1:def 7; A196: j in Seg len UVf(D1) by A193,INTEGRA1:def 7; A197: j1+1=j by XCMPLX_1:27; j < j+1 by NAT_1:38; then A198: j1 < j by REAL_1:84; j1 in Seg len D1 by A51,FINSEQ_1:def 3; then j1 in Seg len UVf(D1) by INTEGRA1:def 7; then PUf(D1,j1)=Sum(UVf(D1)|j1) by INTEGRA1:def 22; then PUf(D1,j1)+Sum mid(UVf(D1),j,j) =Sum((UVf(D1)|j1)^mid(UVf(D1),j,j)) by RVSUM_1:105 .=Sum(mid(UVf(D1),1,j1)^mid(UVf(D1),j1+1,j)) by A51,A197,JORDAN3:25 .=Sum(mid(UVf(D1),1,j)) by A51,A195,A198,INTEGRA2:4 .=Sum(UVf(D1)|j) by A194,JORDAN3:25; hence thesis by A196,INTEGRA1:def 22; end; then PUf(D1,j)-PUf(D2,indx(D2,D1,j))=(PUf(D1,j1)-PUf(D2,indx(D2,D1,j1) )) +(Sum mid(upper_volume(f,D1),j,j) -Sum mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j))) by A187,Lm5 .=0+(Sum mid(upper_volume(f,D1),j,j)- Sum mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j))) by A123,XCMPLX_1:14; hence thesis by A124; end; hence thesis by A45; end; A199: for i being non empty Nat st P[i] holds P[i+1] proof let i be non empty Nat; assume A200:P[i]; A201: i>=1 by RLVECT_1:99; P[i+1] proof assume A202:i+1 in dom D; then D.(i+1) in A by INTEGRA1:8; then consider j such that A203: j in dom D1 & D.(i+1) in divset(D1,j) by Th2; i+1 in Seg len D by A202,FINSEQ_1:def 3; then A204: 1 <= i+1 & i+1 <= len D by FINSEQ_1:3; i <= i+1 by NAT_1:29; then i <= len D by A204,AXIOMS:22; then A205: i in Seg len D by A201,FINSEQ_1:3; then A206: i in dom D by FINSEQ_1:def 3; consider n1 being Nat such that A207: n1 in dom D1 & D.i in divset(D1,n1) & PUf(D1,n1)-PUf(D2,indx(D2,D1,n1))<=i*(sup rng f-inf rng f)*delta(D1) by A200,A205,FINSEQ_1:def 3; A208: indx(D2,D1,n1) in dom D2 & D2.indx(D2,D1,n1) = D1.n1 by A20,A207,INTEGRA1:def 21; then A209: 1 <= indx(D2,D1,n1) & indx(D2,D1,n1) <= len D2 by FINSEQ_3:27; A210: n1 < j proof assume A211:n1 >= j; now per cases by A211,REAL_1:def 5; suppose n1=j; then D.i in divset(D1,j) & D.i in rng D by A206,A207,FUNCT_1:def 5; then A212: D.i in rng D /\ divset(D1,j) by XBOOLE_0:def 3; D.(i+1) in rng D by A202,FUNCT_1:def 5; then D.(i+1) in rng D /\ divset(D1,j) by A203,XBOOLE_0:def 3; then A213: D.(i+1)=D.i by A19,A203,A212,Th4; i+1 > i by REAL_1:69; hence contradiction by A202,A206,A213,GOBOARD2:19; suppose n1>j; then A214: n1>=j+1 by NAT_1:38; then A215: n1-1 >= j by REAL_1:84; A216: inf divset(D1,n1) <= D.i by A207,INTEGRA2:1; A217: D.(i+1)<=sup divset(D1,j) by A203,INTEGRA2:1; 1 <= j by A203,FINSEQ_3:27; then 1+1 <= j+1 by AXIOMS:24; then A218: n1 <> 1 by A214,AXIOMS:22; then n1-1 in dom D1 by A207,INTEGRA1:9; then A219: D1.j <= D1.(n1-1) by A203,A215,GOBOARD2:18; A220: sup divset(D1,j)=D1.j proof per cases; suppose j=1; hence thesis by A203,INTEGRA1:def 5; suppose j<>1; hence thesis by A203,INTEGRA1:def 5; end; inf divset(D1,n1)=D1.(n1-1) by A207,A218,INTEGRA1:def 5; then D.i >= D1.j by A216,A219,AXIOMS:22; then A221: D.i >= D.(i+1) by A217,A220,AXIOMS:22; i < i+1 by REAL_1:69; hence contradiction by A202,A206,A221,GOBOARD1:def 1; end; hence thesis; end; then A222: n1+1 <= j by NAT_1:38; then A223: j-n1 >= 1 by REAL_1:84; A224: n1 >= 1 by A207,FINSEQ_3:27; A225: 1 <= n1 & 1 <= j & j <= len D1 by A203,A207,FINSEQ_3:27; then A226: 1 <= n1+1 & n1+1 <= len D1 by A222,AXIOMS:22,NAT_1:37; then A227: n1+1 in dom D1 by FINSEQ_3:27; then A228: indx(D2,D1,n1+1) in dom D2 & D2.indx(D2,D1,n1+1)=D1.(n1+1) by A20,INTEGRA1:def 21; A229: indx(D2,D1,j) in dom D2 & D2.indx(D2,D1,j)=D1.j by A20,A203,INTEGRA1:def 21; D1.(n1+1) <= D1.j by A203,A222,A227,GOBOARD2:18; then A230: indx(D2,D1,n1+1) <= indx(D2,D1,j) by A228,A229,GOBOARD1:def 1; A231: 1 <= indx(D2,D1,n1+1) & indx(D2,D1,n1+1) <= len D2 by A228,FINSEQ_3:27 ; A232: 1 <= indx(D2,D1,j) & indx(D2,D1,j) <= len D2 by A229,FINSEQ_3:27; n1 < n1+1 by NAT_1:38; then D1.n1 < D1.(n1+1) by A207,A227,GOBOARD1:def 1; then A233: indx(D2,D1,n1) < indx(D2,D1,n1+1) by A208,A228,GOBOARD2:18; 1+indx(D2,D1,n1+1) <= indx(D2,D1,j)+1 by A230,AXIOMS:24; then 1 <= indx(D2,D1,j)+1-indx(D2,D1,n1+1) by REAL_1:84; then A234: mid(D2,indx(D2,D1,n1+1),indx(D2,D1,j)).1 =D2.(1-1+indx(D2,D1,n1+1)) by A230,A231,A232,JORDAN3:31 .=D1.(n1+1) by A20,A227,INTEGRA1:def 21; A235: Sum mid(UVf(D1),n1+1,j)-Sum mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) <= (sup rng f-inf rng f)*delta(D1) proof per cases by A222,REAL_1:def 5; suppose A236:n1+1=j; then A237: n1=j-1 by XCMPLX_1:26; A238: 1 <= n1+1 & n1+1 <= len UVf(D1) by A226,INTEGRA1:def 7; A239: j-'(n1+1)+1=j-(n1+1)+1 by A236,SCMFSA_7:3; A240: n1+1 in Seg len D1 by A227,FINSEQ_1:def 3; A241: len mid(UVf(D1),n1+1,j)=j-'(n1+1)+1 by A236,A238,JORDAN3:27 .= 1 by A236,A239,XCMPLX_1:25; (n1+1)+1 <= j+1 by A222,AXIOMS:24; then 1 <= j+1-(n1+1) by REAL_1:84; then mid(UVf(D1),n1+1,j).1 = UVf(D1).(1+(n1+1)-1) by A236,A238,JORDAN3:31 .= UVf(D1).(n1+1) by XCMPLX_1:26 .=(sup rng(f|divset(D1,n1+1)))*vol(divset(D1,n1+1)) by A240,INTEGRA1:def 7; then mid(UVf(D1),n1+1,j) =<*(sup rng(f|divset(D1,n1+1)))*vol(divset(D1,n1+1))*> by A241,FINSEQ_1:57; then A242: Sum mid(UVf(D1),n1+1,j) =(sup rng(f|divset(D1,n1+1)))*vol(divset(D1,n1+1)) by RVSUM_1:103; divset(D1,n1+1) c= A by A227,INTEGRA1:10; then A243: sup rng(f|divset(D1,n1+1)) <= sup rng f by A1,Lm7; vol(divset(D1,n1+1)) >= 0 by INTEGRA1:11; then A244: Sum mid(UVf(D1),n1+1,j) <=(sup rng f)*vol(divset(D1,n1+1)) by A242,A243,AXIOMS:25; D1.n1 < D1.j by A203,A207,A210,GOBOARD1:def 1; then A245: indx(D2,D1,n1)<indx(D2,D1,j) by A208,A229,GOBOARD2:18; then A246: indx(D2,D1,n1)+1 <= indx(D2,D1,j) by NAT_1:38; then indx(D2,D1,n1)+1 <= len D2 by A232,AXIOMS:22; then A247: indx(D2,D1,n1)+1 <= len UVf(D2) by INTEGRA1:def 7; A248: 1<=indx(D2,D1,n1)+1 by NAT_1:37; A249: indx(D2,D1,j) <= len UVf(D2) by A232,INTEGRA1:def 7; then A250: len mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) =indx(D2,D1,j)-'(indx(D2,D1,n1)+1)+1 by A232,A246,A247,A248,JORDAN3:27 .=indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1 by A246,SCMFSA_7:3 .=indx(D2,D1,j)-indx(D2,D1,n1)-1+1 by XCMPLX_1:36 .=indx(D2,D1,j)-indx(D2,D1,n1)-(1-1) by XCMPLX_1:37 .=indx(D2,D1,j)-indx(D2,D1,n1); A251: indx(D2,D1,j)-indx(D2,D1,n1)<=2 proof assume indx(D2,D1,j) - indx(D2,D1,n1) > 2; then A252: indx(D2,D1,n1)+2 < indx(D2,D1,j) by REAL_1:86; A253: indx(D2,D1,n1) < indx(D2,D1,n1)+1 by NAT_1:38; A254: indx(D2,D1,n1)+1 < indx(D2,D1,n1)+2 by REAL_1:53; A255: indx(D2,D1,n1)+1 < indx(D2,D1,j) & indx(D2,D1,n1)+1 in dom D2 proof thus indx(D2,D1,n1)+1 < indx(D2,D1,j) by A252,A254,AXIOMS:22; then 1 <= indx(D2,D1,n1)+1 & indx(D2,D1,n1)+1 <= len D2 by A209,A232,AXIOMS:22,NAT_1:38; hence thesis by FINSEQ_3:27; end; A256: D2.(indx(D2,D1,n1)+1) in rng D proof A257: D2.(indx(D2,D1,n1)+1) in rng D2 by A255,FUNCT_1:def 5; not D2.(indx(D2,D1,n1)+1) in rng D1 proof assume D2.(indx(D2,D1,n1)+1) in rng D1; then consider k1 being Nat such that A258: k1 in dom D1 & D2.(indx(D2,D1,n1)+1) = D1.k1 by PARTFUN1:26; D2.indx(D2,D1,n1) < D2.(indx(D2,D1,n1)+1) & D2.(indx(D2,D1,n1)+1) < D2.indx(D2,D1,j) by A208,A229,A253,A255,GOBOARD1:def 1; then n1 < k1 & k1 < j by A203,A207,A208,A229,A258,GOBOARD2:18; hence contradiction by A236,NAT_1:38; end; hence thesis by A20,A257,XBOOLE_0:def 2; end; A259: D2.(indx(D2,D1,n1)+1) in rng D /\ divset(D1,j) proof A260: inf divset(D1,j) = D1.(j-1) & sup divset(D1,j) = D1.j by A203,A210,A224,INTEGRA1:def 5; D2.(indx(D2,D1,n1)+1) >= D2.indx(D2,D1,n1) & D2.indx(D2,D1,j) >= D2.(indx(D2,D1,n1)+1) by A208,A229,A253,A255,GOBOARD2:18; then D2.(indx(D2,D1,n1)+1) in divset(D1,j) by A208,A229,A237,A260, INTEGRA2:1; hence thesis by A256,XBOOLE_0:def 3; end; A261: indx(D2,D1,n1) < indx(D2,D1,n1)+2 & indx(D2,D1,n1)+2 in dom D2 proof thus indx(D2,D1,n1) < indx(D2,D1,n1)+2 by A254,NAT_1:38; then 1 <= indx(D2,D1,n1)+2 & indx(D2,D1,n1)+2 <= len D2 by A209,A232,A252,AXIOMS:22; hence thesis by FINSEQ_3:27; end; A262: D2.(indx(D2,D1,n1)+2) in rng D proof A263: D2.(indx(D2,D1,n1)+2) in rng D2 by A261,FUNCT_1:def 5; not D2.(indx(D2,D1,n1)+2) in rng D1 proof assume D2.(indx(D2,D1,n1)+2) in rng D1; then consider k1 being Nat such that A264: k1 in dom D1 & D2.(indx(D2,D1,n1)+2) = D1.k1 by PARTFUN1:26; D2.indx(D2,D1,n1) < D2.(indx(D2,D1,n1)+2) & D2.(indx(D2,D1,n1)+2) < D2.indx(D2,D1,j) by A208,A229,A252,A261,GOBOARD1:def 1; then n1 < k1 & k1 < j by A203,A207,A208,A229,A264,GOBOARD2:18; hence contradiction by A236,NAT_1:38; end; hence thesis by A20,A263,XBOOLE_0:def 2; end; D2.(indx(D2,D1,n1)+2) in rng D /\ divset(D1,j) proof A265: inf divset(D1,j) = D1.(j-1) & sup divset(D1,j) = D1.j by A203,A210,A224,INTEGRA1:def 5; D2.(indx(D2,D1,n1)+2) >= D2.indx(D2,D1,n1) & D2.indx(D2,D1,j) >= D2.(indx(D2,D1,n1)+2) by A208,A229,A252,A261,GOBOARD2:18; then D2.(indx(D2,D1,n1)+2) in divset(D1,j) by A208,A229,A237,A265, INTEGRA2:1; hence thesis by A262,XBOOLE_0:def 3; end; then D2.(indx(D2,D1,n1)+1) = D2.(indx(D2,D1,n1)+2) by A19,A203,A259, Th4; hence contradiction by A254,A255,A261,GOBOARD1:def 1; end; A266: indx(D2,D1,j)-indx(D2,D1,n1)=1 or indx(D2,D1,j)-indx(D2,D1,n1)=2 proof indx(D2,D1,n1)+1 <= indx(D2,D1,j) by A245,NAT_1:38; then A267: indx(D2,D1,n1)+1 = indx(D2,D1,j) or indx(D2,D1,n1)+1 < indx(D2,D1,j) by REAL_1:def 5; indx(D2,D1,n1)+1<indx(D2,D1,j) implies indx(D2,D1,n1)+2=indx(D2,D1,j ) proof assume indx(D2,D1,n1)+1 < indx(D2,D1,j); then indx(D2,D1,n1)+1+1 <= indx(D2,D1,j) by NAT_1:38; then A268: indx(D2,D1,n1)+(1+1) <= indx(D2,D1,j) by XCMPLX_1:1; indx(D2,D1,n1)+2 >= indx(D2,D1,j) by A251,REAL_1:86; hence thesis by A268,AXIOMS:21; end; hence thesis by A267,XCMPLX_1:26; end; Sum mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) >=(inf rng f)*vol(divset(D1,n1+1)) proof now per cases by A266; suppose A269:indx(D2,D1,j)-indx(D2,D1,n1)=1; then 1=indx(D2,D1,j)-indx(D2,D1,n1)-1+1 .=indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1 by XCMPLX_1:36; then A270: mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)).1 =UVf(D2).(1+(indx(D2,D1,n1)+1)-1) by A246,A248,A249,JORDAN3:31 .=UVf(D2).((indx(D2,D1,n1)+1)-1+1) by XCMPLX_1:29 .=UVf(D2).((indx(D2,D1,n1)+1)-(1-1)) by XCMPLX_1:37 .=UVf(D2).(indx(D2,D1,n1)+1); A271: indx(D2,D1,n1)+1=indx(D2,D1,j) by A269,XCMPLX_1:27; then indx(D2,D1,n1)+1 in Seg len D2 by A229,FINSEQ_1:def 3; then A272: UVf(D2).(indx(D2,D1,n1)+1) =(sup rng(f|divset(D2,indx(D2,D1,n1)+1))) *vol(divset(D2,indx(D2,D1,n1)+1)) by INTEGRA1:def 7; A273: divset(D2,indx(D2,D1,n1)+1)=divset(D1,n1+1) proof A274: divset(D2,indx(D2,D1,n1)+1) =[.inf divset(D2,indx(D2,D1,n1)+1), sup divset(D2,indx(D2,D1,n1)+1).] by INTEGRA1:5; A275: divset(D1,n1+1)=[.inf divset(D1,n1+1), sup divset(D1,n1+1).] by INTEGRA1:5; indx(D2,D1,n1)+1 >= 1+1 by A209,AXIOMS:24; then indx(D2,D1,n1)+1 <>1; then inf divset(D2,indx(D2,D1,n1)+1)=D2.(indx(D2,D1,n1)+1-1) & sup divset(D2,indx(D2,D1,n1)+1)=D2.(indx(D2,D1,n1)+1) by A229,A271,INTEGRA1:def 5; then A276: inf divset(D2,indx(D2,D1,n1)+1)=D1.n1 & sup divset(D2,indx(D2,D1,n1)+1)=D1.j by A20,A203,A208,A271, INTEGRA1:def 21,XCMPLX_1:26; inf divset(D1,n1+1)=D1.(n1+1-1) & sup divset(D1,n1+1)=D1.(n1+1) by A210,A224,A227,A236,INTEGRA1:def 5; hence thesis by A236,A274,A275,A276,XCMPLX_1:26; end; A277: sup rng(f|divset(D2,indx(D2,D1,n1)+1))>=inf rng f by A1,A229,A271, Th18; vol(divset(D2,indx(D2,D1,n1)+1))>=0 by INTEGRA1:11; then A278: UVf(D2).(indx(D2,D1,n1)+1) >=(inf rng f)*vol(divset(D1,n1+1)) by A272,A273,A277,AXIOMS:25; mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) =<*UVf(D2).(indx(D2,D1,n1)+1)*> by A250,A269,A270,FINSEQ_1:57; hence Sum mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) >=(inf rng f)*vol(divset(D1,n1+1)) by A278,RVSUM_1:103; suppose A279:indx(D2,D1,j)-indx(D2,D1,n1)=2; then A280: indx(D2,D1,j)=indx(D2,D1,n1)+2 by XCMPLX_1:27; A281: indx(D2,D1,n1)+2-1=indx(D2,D1,n1)+(2-1) by XCMPLX_1:29 .=indx(D2,D1,n1)+1; then indx(D2,D1,j)-(indx(D2,D1,n1)+1)=1 by A280,XCMPLX_1:18; then A282: indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1>=1; A283: indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1=1+1 by A280,A281,XCMPLX_1:18; A284: 1 <= indx(D2,D1,n1)+1 & indx(D2,D1,n1)+1 <= indx(D2,D1,n1)+2 by NAT_1:37,REAL_1:53; then indx(D2,D1,n1)+1 <= len D2 by A232,A280,AXIOMS:22; then A285: indx(D2,D1,n1)+1 in dom D2 by A284,FINSEQ_3:27; A286: indx(D2,D1,j) <= len UVf(D2) by A232,INTEGRA1:def 7; then A287: mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)).1 =UVf(D2).(1+(indx(D2,D1,n1)+1)-1) by A280,A282,A284,JORDAN3:31 .=UVf(D2).((indx(D2,D1,n1)+1)-1+1) by XCMPLX_1:29 .=UVf(D2).((indx(D2,D1,n1)+1)-(1-1)) by XCMPLX_1:37 .=UVf(D2).(indx(D2,D1,n1)+1); mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)).2 =UVf(D2).(2+(indx(D2,D1,n1)+1)-1) by A280,A283,A284,A286,JORDAN3:31 .=UVf(D2).((indx(D2,D1,n1)+1)-1+2) by XCMPLX_1:29 .=UVf(D2).(indx(D2,D1,n1)+(1-1)+2) by XCMPLX_1:29 .=UVf(D2).(indx(D2,D1,n1)+0+2); then mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) =<*UVf(D2).(indx(D2,D1,n1)+1),UVf(D2).(indx(D2,D1,n1)+2)*> by A250,A279,A287,FINSEQ_1:61; then A288: Sum mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) =UVf(D2).(indx(D2,D1,n1)+1)+UVf(D2).(indx(D2,D1,n1)+2) by RVSUM_1:107; A289: UVf(D2).(indx(D2,D1,n1)+1) >= (inf rng f)*vol(divset(D2,indx(D2,D1,n1)+1)) proof indx(D2,D1,n1)+1 in Seg len D2 by A285,FINSEQ_1:def 3; then A290: UVf(D2).(indx(D2,D1,n1)+1) =(sup rng(f|divset(D2,indx(D2,D1,n1)+1))) *vol(divset(D2,indx(D2,D1,n1)+1)) by INTEGRA1:def 7; A291: sup rng(f|divset(D2,indx(D2,D1,n1)+1)) >=inf rng f by A1,A285,Th18; vol(divset(D2,indx(D2,D1,n1)+1)) >= 0 by INTEGRA1:11; hence thesis by A290,A291,AXIOMS:25; end; UVf(D2).(indx(D2,D1,n1)+2) >= (inf rng f)*vol(divset(D2,indx(D2,D1,n1)+2)) proof indx(D2,D1,n1)+2 in Seg len D2 by A229,A280,FINSEQ_1:def 3; then A292: UVf(D2).(indx(D2,D1,n1)+2) =(sup rng(f|divset(D2,indx(D2,D1,n1)+2))) *vol(divset(D2,indx(D2,D1,n1)+2)) by INTEGRA1:def 7; A293: sup rng(f|divset(D2,indx(D2,D1,n1)+2)) >=inf rng f by A1,A229,A280,Th18; vol(divset(D2,indx(D2,D1,n1)+2)) >= 0 by INTEGRA1:11; hence thesis by A292,A293,AXIOMS:25; end; then A294: Sum mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) >= (inf rng f)*vol(divset(D2,indx(D2,D1,n1)+1)) + (inf rng f)*vol(divset(D2,indx(D2,D1,n1)+2)) by A288,A289,REAL_1:55 ; vol(divset(D2,indx(D2,D1,n1)+1))+vol(divset(D2,indx(D2,D1,n1)+2)) =vol(divset(D1,n1+1)) proof indx(D2,D1,n1)+1 > 1 by A209,NAT_1:38; then A295: inf divset(D2,indx(D2,D1,n1)+1) = D2.(indx(D2,D1,n1)+1-1) & sup divset(D2,indx(D2,D1,n1)+1)=D2.(indx(D2,D1,n1)+1) by A285,INTEGRA1:def 5; then A296: inf divset(D2,indx(D2,D1,n1)+1) = D2.indx(D2,D1,n1) by XCMPLX_1:26; indx(D2,D1,n1)+2 >= 2+1 by A209,AXIOMS:24; then indx(D2,D1,n1)+2 <> 1; then A297: inf divset(D2,indx(D2,D1,n1)+2) = D2.(indx(D2,D1,n1)+1) & sup divset(D2,indx(D2,D1,n1)+2)=D2.indx(D2,D1,j) by A229,A280,A281,INTEGRA1:def 5; A298: vol(divset(D2,indx(D2,D1,n1)+1)) =D2.(indx(D2,D1,n1)+1)-D1.n1 by A208,A295,A296,INTEGRA1:def 6; A299: vol(divset(D2,indx(D2,D1,n1)+2)) =D1.j-D2.(indx(D2,D1,n1)+1) by A229,A297,INTEGRA1:def 6; vol(divset(D1,n1+1))=D1.(n1+1)-D1.n1 proof A300: inf divset(D1,n1+1)=D1.(n1+1-1) & sup divset(D1,n1+1)=D1.(n1+1) by A210,A224,A227,A236,INTEGRA1:def 5; then inf divset(D1,n1+1)=D1.n1 by XCMPLX_1:26; hence thesis by A300,INTEGRA1:def 6; end; hence thesis by A236,A298,A299,XCMPLX_1:39; end; hence Sum mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) >=(inf rng f)*vol(divset(D1,n1+1)) by A294,XCMPLX_1:8; end; hence thesis; end; then Sum mid(UVf(D1),n1+1,j)-Sum mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2, D1,j)) <=(sup rng f)*vol(divset(D1,n1+1))-(inf rng f)*vol(divset(D1,n1+1)) by A244,REAL_1:92; then A301: Sum mid(UVf(D1),n1+1,j)-Sum mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1, j)) <=(sup rng f-inf rng f)*vol(divset(D1,n1+1)) by XCMPLX_1:40; sup rng f >= inf rng f by A1,Lm6; then A302: sup rng f-inf rng f >= 0 by SQUARE_1:12; vol(divset(D1,n1+1))<=delta(D1) proof A303: n1+1 in Seg len D1 by A227,FINSEQ_1:def 3; then A304: vol(divset(D1,n1+1))=upper_volume(chi(A,A),D1).(n1+1) by INTEGRA1:22; n1+1 in Seg len upper_volume(chi(A,A),D1) by A303,INTEGRA1:def 7; then n1+1 in dom upper_volume(chi(A,A),D1) by FINSEQ_1:def 3; then A305: vol(divset(D1,n1+1)) in rng upper_volume(chi(A,A),D1) by A304,FUNCT_1:def 5; delta(D1)=max rng upper_volume(chi(A,A),D1) by INTEGRA1:def 19; hence thesis by A305,PRE_CIRC:def 1; end; then (sup rng f-inf rng f)*vol(divset(D1,n1+1)) <=(sup rng f-inf rng f)*delta(D1) by A302,AXIOMS:25; hence thesis by A301,AXIOMS:22; suppose A306:n1+1 < j; A307: n1 < n1+1 by NAT_1:38; then A308: D1.n1 < D1.(n1+1) by A207,A227,GOBOARD1:def 1; then consider B being closed-interval Subset of REAL, MD1,MD2 being Element of divs B such that A309: D1.n1=inf B & sup B=MD2.(len MD2) & sup B=MD1.(len MD1) & MD1 <= MD2 & MD1=mid(D1,n1+1,j) & MD2=mid(D2,indx(D2,D1,n1+1),indx(D2,D1,j)) by A20,A203,A222,A227,A234,Th14; A310: len MD1 = j-'(n1+1)+1 by A222,A225,A226,A309,JORDAN3:27; A311: j-'(n1+1) = j-(n1+1) by A222,SCMFSA_7:3; A312: len MD1 = j-(n1+1)+1 by A222,A310,SCMFSA_7:3; A313: j-'(n1+1)+1=j-n1-1+1 by A311,XCMPLX_1:36 .= j-n1-(1-1) by XCMPLX_1:37 .=j-n1; A314: len MD1+(n1+1)-1 =j-((n1+1)-1)+(n1+1)-1 by A312,XCMPLX_1:37 .=j-((n1+1)-1)+((n1+1)-1) by XCMPLX_1:29 .=j-(((n1+1)-1)-((n1+1)-1)) by XCMPLX_1:37 .=j by XCMPLX_1:17; A315: B c= A proof let x1; assume A316:x1 in B; then reconsider x1 as Real; A317: rng D1 c= A by INTEGRA1:def 2; A318: D1.n1 <= x1 & x1 <= MD1.(len MD1) by A309,A316,INTEGRA2:1; D1.n1 in rng D1 & D1.j in rng D1 by A203,A207,FUNCT_1:def 5; then A319: inf A <= D1.n1 & D1.j <= sup A by A317,INTEGRA2:1; MD1.(len MD1)=D1.(j-n1-1+(n1+1)) by A222,A223,A225,A226,A309,A310,A313,JORDAN3:31 .=D1.(j-(n1+1)+(n1+1)) by XCMPLX_1:36 .=D1.(j-((n1+1)-(n1+1))) by XCMPLX_1:37 .=D1.j by XCMPLX_1:17; then inf A <= x1 & x1 <= sup A by A318,A319,AXIOMS:22; hence thesis by INTEGRA2:1; end; then reconsider g=f|B as Function of B,REAL by FUNCT_2:38; A320: g is_bounded_on B proof A321: f is_bounded_above_on A&f is_bounded_below_on A by A1,RFUNCT_1:def 11 ; then consider a be real number such that A322: for x being Element of A st x in A /\ dom f holds a<=f.x by RFUNCT_1:def 10; for x being Element of B st x in B /\ dom g holds a<=g.x proof let x be Element of B; assume x in B /\ dom g; then A323: x in dom g by XBOOLE_0:def 3; then A324: x in dom f /\ B by FUNCT_1:68; reconsider x as Element of A by A315,TARSKI:def 3; dom f /\ B c= dom f /\ A by A315,XBOOLE_1:26; then a <= f.x by A322,A324; hence thesis by A323,FUNCT_1:68; end; then A325: g is_bounded_below_on B by RFUNCT_1:def 10; consider a be real number such that A326: for x being Element of A st x in A /\ dom f holds f.x<=a by A321,RFUNCT_1:def 9; for x being Element of B st x in B /\ dom g holds g.x<=a proof let x be Element of B; assume x in B /\ dom g; then A327: x in dom g by XBOOLE_0:def 3; then A328: x in dom f /\ B by FUNCT_1:68; reconsider x as Element of A by A315,TARSKI:def 3; dom f /\ B c= dom f /\ A by A315,XBOOLE_1:26; then a >= f.x by A326,A328; hence thesis by A327,FUNCT_1:68; end; then g is_bounded_above_on B by RFUNCT_1:def 9; hence thesis by A325,RFUNCT_1:def 11; end; rng MD2 <> {}; then 1 in dom MD2 by FINSEQ_3:34; then A329: 1 <= len MD2 by FINSEQ_3:27; A330: len MD2=indx(D2,D1,j)-'indx(D2,D1,n1+1)+1 by A230,A231,A232,A309,JORDAN3:27; A331: 1 <= indx(D2,D1,j)-'indx(D2,D1,n1+1)+1 by A230,A231,A232,A309,A329,JORDAN3:27; A332: len MD2 = indx(D2,D1,j)-indx(D2,D1,n1+1)+1 by A230,A330,SCMFSA_7:3; then len MD2-1=indx(D2,D1,j)-indx(D2,D1,n1+1) by XCMPLX_1:26; then A333: len MD2-1+indx(D2,D1,n1+1)=indx(D2,D1,j) by XCMPLX_1:27; A334: rng MD2 = rng MD1 \/ {D.(i+1)} proof x1 in rng MD2 implies x1 in (rng MD1 \/ {D.(i+1)}) proof assume A335:x1 in rng MD2; then reconsider x1 as Real; A336: MD2.1 <= x1 & x1 <= MD2.(len MD2) by A335,Th15; A337: MD2.1=D2.(1+indx(D2,D1,n1+1)-1) by A230,A231,A232,A309,A331,JORDAN3 :31 .=D2.indx(D2,D1,n1+1) by XCMPLX_1:26; A338: MD2.(len MD2)=D2.indx(D2,D1,j) by A230,A231,A232,A309,A329,A330 ,A333,JORDAN3:31; A339: D1.(n1+1) <= x1 & x1 <= D1.j by A20,A227,A229,A230,A231,A232,A309, A329,A330,A333,A336,A337,INTEGRA1:def 21,JORDAN3:31; A340: rng MD2 c= rng D2 by A309,JORDAN3:28; now per cases by A20,A335,A340,XBOOLE_0:def 2; suppose x1 in rng D1; then consider k such that A341: k in dom D1 & D1.k = x1 by PARTFUN1:26; A342: n1+1 <= k & k <= j by A203,A227,A228,A229,A336,A337,A338,A341, GOBOARD1:def 1; then A343: 1 <= k-n1 & k-n1 <= j-n1 by REAL_1:49,84; A344: 1 <= k-n1 & k-n1 <= len MD1 by A310,A313,A342,REAL_1:49,84; A345: j-(n1+1)+1=j-n1-1+1 by XCMPLX_1:36 .=j-n1-(1-1) by XCMPLX_1:37 .=j-n1; n1 <= n1+1 by NAT_1:29; then n1 <= k by A342,AXIOMS:22; then consider n such that A346: k=n1+n by NAT_1:28; A347: n = k-n1 by A346,XCMPLX_1:26; then n in dom MD1 by A344,FINSEQ_3:27; then A348: MD1.n in rng MD1 by FUNCT_1:def 5; MD1.n = D1.(k-n1-1+(n1+1)) by A222,A225,A226,A309,A343,A345,A347,JORDAN3:31 .= D1.(k-(n1+1)+(n1+1)) by XCMPLX_1:36 .= D1.(k-((n1+1)-(n1+1))) by XCMPLX_1:37 .= D1.k by XCMPLX_1:17; hence x1 in (rng MD1 \/ {D.(i+1)}) by A341,A348,XBOOLE_0:def 2; suppose x1 in rng D; then consider n such that A349: n in dom D & D.n = x1 by PARTFUN1:26; A350: D.i <= sup divset(D1,n1) by A207,INTEGRA2:1; sup divset(D1,n1)=D1.n1 proof per cases; suppose n1=1; hence thesis by A207,INTEGRA1:def 5; suppose n1<>1; hence thesis by A207,INTEGRA1:def 5; end; then D.i < D1.(n1+1) by A308,A350,AXIOMS:22; then D.i < D.n by A228,A336,A337,A349,AXIOMS:22; then i < n by A206,A349,GOBOARD2:18; then i+1 <= n by NAT_1:38; then A351: i+1 = n or i+1 < n by REAL_1:def 5; not i+1 < n proof assume i+1 < n; then A352: D.(i+1) < D.n by A202,A349,GOBOARD1:def 1; inf divset(D1,j) <= D.(i+1) by A203,INTEGRA2:1; then A353: inf divset(D1,j) <= D.n by A352,AXIOMS:22; sup divset(D1,j)=D1.j proof per cases; suppose j=1; hence thesis by A203,INTEGRA1:def 5; suppose j<>1; hence thesis by A203,INTEGRA1:def 5; end; then D.n in rng D&D.n in divset(D1,j) by A339,A349,A353,FUNCT_1:def 5,INTEGRA2:1; then A354: x1 in rng D /\ divset(D1,j) by A349,XBOOLE_0:def 3; A355: D.(i+1) in rng D by A202,FUNCT_1:def 5; consider y1 being Real such that A356: y1=D.(i+1); y1 in rng D /\ divset(D1,j) by A203,A355,A356,XBOOLE_0:def 3; hence contradiction by A19,A203,A349,A352,A354,A356,Th4; end; then x1 in {D.(i+1)} by A349,A351,TARSKI:def 1; hence x1 in (rng MD1 \/ {D.(i+1)}) by XBOOLE_0:def 2; end; hence thesis; end; then A357: rng MD2 c= rng MD1 \/ {D.(i+1)} by TARSKI:def 3; x1 in rng MD1 \/ {D.(i+1)} implies x1 in rng MD2 proof assume A358:x1 in rng MD1 \/ {D.(i+1)}; then reconsider x1 as Real; now per cases by A358,XBOOLE_0:def 2; suppose A359:x1 in rng MD1; rng MD1 c= rng D1 by A309,JORDAN3:28; then A360: x1 in rng D1 by A359; A361: MD1.1 <= x1 & x1 <= MD1.(len MD1) by A359,Th15; rng MD1 <> {}; then 1 in dom MD1 by FINSEQ_3:34; then A362: 1 <= len MD1 by FINSEQ_3:27; A363: len MD1=j-'(n1+1)+1 by A222,A225,A226,A309,JORDAN3:27; then A364: MD1.1=D1.(1+(n1+1)-1) by A222,A225,A226,A309,A362,JORDAN3:31 .=D1.((n1+1)-1+1) by XCMPLX_1:29 .=D1.((n1+1)-(1-1)) by XCMPLX_1:37 .=D1.(n1+1); len MD1+(n1+1)-1=j-(n1+1)+1+(n1+1)-1 by A222,A363,SCMFSA_7:3 .=j-((n1+1)-1)+(n1+1)-1 by XCMPLX_1:37 .=j-((n1+1)-1)+((n1+1)-1) by XCMPLX_1:29 .=j-(((n1+1)-1)-((n1+1)-1)) by XCMPLX_1:37 .=j by XCMPLX_1:17; then A365: MD1.(len MD1)=D1.j by A222,A225,A226,A309,A362,A363,JORDAN3:31; rng D1 c= rng D2 by A20,INTEGRA1:def 20; then consider k such that A366: k in dom D2 & D2.k = x1 by A360,PARTFUN1:26; A367: indx(D2,D1,n1+1)<=k & k<=indx(D2,D1,j) by A228,A229,A361,A364,A365,A366,GOBOARD1:def 1; then indx(D2,D1,n1+1)+1<=k+1 by AXIOMS:24; then A368: 1 <= k+1-indx(D2,D1,n1+1) by REAL_1:84; then A369: 1 <= k-indx(D2,D1,n1+1)+1 by XCMPLX_1:29; k-indx(D2,D1,n1+1)<=indx(D2,D1,j)-indx(D2,D1,n1+1) by A367,REAL_1:49; then A370: k-indx(D2,D1,n1+1)+1<=indx(D2,D1,j)-indx(D2,D1,n1+1)+1 by AXIOMS: 24; indx(D2,D1,n1+1)<=k+1 by A367,NAT_1:37; then consider n such that A371: k+1 = indx(D2,D1,n1+1)+n by NAT_1:28; A372: n=k+1-indx(D2,D1,n1+1) by A371,XCMPLX_1:26 .=k-indx(D2,D1,n1+1)+1 by XCMPLX_1:29; then A373: 1 <= n & n <= len MD2 by A230,A330,A368,A370,SCMFSA_7:3, XCMPLX_1:29; n in dom MD2 by A332,A369,A370,A372,FINSEQ_3:27; then A374: MD2.n in rng MD2 by FUNCT_1:def 5; n+indx(D2,D1,n1+1)-1=k by A371,XCMPLX_1:26; hence x1 in rng MD2 by A230,A231,A232,A309,A330,A366,A373,A374, JORDAN3:31; suppose x1 in {D.(i+1)}; then A375: x1 = D.(i+1) by TARSKI:def 1; A376: D.(i+1) in rng D by A202,FUNCT_1:def 5; rng D c= rng D2 by A20,INTEGRA1:def 20; then consider k such that A377: k in dom D2 & x1 = D2.k by A375,A376,PARTFUN1:26; inf divset(D1,j)<=D.(i+1) & D.(i+1)<=sup divset(D1,j) by A203,INTEGRA2:1; then A378: D1.(j-1) <= x1 & x1 <= D1.j by A203,A210,A224,A375,INTEGRA1: def 5; A379: j-1 in dom D1 & j-1 in NAT by A203,A210,A224,INTEGRA1:9; reconsider j1 = j-1 as Nat by A203,A210,A224,INTEGRA1:9; n1<j1 by A306,REAL_1:86; then n1+1 <= j1 by NAT_1:38; then D1.(n1+1) <= D1.(j-1) by A227,A379,GOBOARD2:18; then D2.indx(D2,D1,n1+1) <= D2.k & D2.k <= D2.indx(D2,D1,j) by A20,A203,A228,A377,A378,AXIOMS:22,INTEGRA1:def 21; hence x1 in rng MD2 by A228,A229,A309,A377,Th16; end; hence thesis; end; then rng MD1 \/ {D.(i+1)} c= rng MD2 by TARSKI:def 3; hence thesis by A357,XBOOLE_0:def 10; end; A380: len MD1 in dom MD1 by SCMFSA_7:12; then A381: 1 <= len MD1 by FINSEQ_3:27; A382: inf divset(MD1,len MD1)=inf divset(D1,j) & sup divset(MD1,len MD1)=sup divset(D1,j) proof per cases; suppose A383:len MD1=1; then A384: inf divset(MD1,len MD1)=inf B & sup divset(MD1,len MD1)=MD1.(len MD1) by A380,INTEGRA1:def 5; inf divset(D1,j)=D1.(j-1) & sup divset(D1,j)=D1.j by A203,A210,A224,INTEGRA1:def 5; hence thesis by A222,A225,A226,A309,A310,A313,A314,A383,A384,JORDAN3: 31,XCMPLX_1:18; suppose A385:len MD1<>1; then A386: inf divset(MD1,len MD1)=MD1.(len MD1-1) & sup divset(MD1,len MD1)=MD1.(len MD1) by A380,INTEGRA1:def 5; A387: len MD1-1+(n1+1)-1=j-(n1+1)+(n1+1)-1 by A310,A313,XCMPLX_1:36 .=j-((n1+1)-(n1+1))-1 by XCMPLX_1:37 .=j-1 by XCMPLX_1:17; A388: len MD1-1 in dom MD1 & len MD1-1 in NAT by A380,A385,INTEGRA1:9; then A389: len MD1-1 >= 1 by FINSEQ_3:27; len MD1 <= len MD1+1 by NAT_1:29; then len MD1 -1 <= len MD1 by REAL_1:86; then A390: inf divset(MD1,len MD1) = D1.(j-1) by A222,A225,A226,A309,A310,A386,A387,A388,A389,JORDAN3:31; sup divset(MD1,len MD1)=D1.j by A222,A225,A226,A309,A310,A314,A381,A386,JORDAN3:31; hence thesis by A203,A210,A224,A390,INTEGRA1:def 5; end; inf divset(D1,j)<=D.(i+1) & D.(i+1)<=sup divset(D1,j) by A203,INTEGRA2:1; then A391: D.(i+1) in divset(MD1,len MD1) by A382,INTEGRA2:1; A392: Sum upper_volume(g,MD1)-Sum upper_volume(g,MD2)<= (sup rng g-inf rng g)*delta(MD1) proof A393: vol(B)=sup B-D1.n1 by A309,INTEGRA1:def 6; A394: len MD1 in dom MD1 by SCMFSA_7:12; vol(B)<>0 & D.(i+1) > inf B proof sup divset(MD1,len MD1)=MD1.(len MD1) proof per cases; suppose len MD1 = 1; hence thesis by A394,INTEGRA1:def 5; suppose len MD1 <> 1; hence thesis by A394,INTEGRA1:def 5; end; then A395: vol(B)=D1.j-D1.n1 by A203,A210,A224,A309,A382,A393,INTEGRA1:def 5; D1.j>D1.n1 by A203,A207,A210,GOBOARD1:def 1; hence vol(B)<>0 by A395,SQUARE_1:11; inf divset(D1,j) <= D.(i+1) by A203,INTEGRA2:1; then A396: D1.(j-1) <= D.(i+1) by A203,A210,A224,INTEGRA1:def 5; A397: n1 < j-1 by A306,REAL_1:86; j-1 in dom D1 by A203,A210,A224,INTEGRA1:9; then D1.n1 <D1.(j-1) by A207,A397,GOBOARD1:def 1; hence thesis by A309,A396,AXIOMS:22; end; hence thesis by A309,A320,A334,A391,Th13; end; dom g = B by FUNCT_2:def 1; then A398: rng g <> {} by RELAT_1:65; A399: rng g c= rng f by FUNCT_1:76; f is_bounded_above_on A & f is_bounded_below_on A by A1,RFUNCT_1:def 11; then rng f is bounded_above & rng f is bounded_below by INTEGRA1:13,15; then sup rng f >= sup rng g & inf rng f <= inf rng g by A398,A399,PSCOMP_1:12,13; then A400: sup rng f-inf rng f>=sup rng g - inf rng g by REAL_1:92; delta(MD1) >= 0 by Th8; then (sup rng f-inf rng f)*delta(MD1)>=(sup rng g-inf rng g)*delta(MD1 ) by A400,AXIOMS:25; then A401: Sum upper_volume(g,MD1)-Sum upper_volume(g,MD2)<= (sup rng f-inf rng f)*delta(MD1) by A392,AXIOMS:22; delta(MD1)=max rng upper_volume(chi(B,B),MD1) by INTEGRA1:def 19; then delta(MD1) in rng upper_volume(chi(B,B),MD1) by PRE_CIRC:def 1; then consider k such that A402: k in dom upper_volume(chi(B,B),MD1) & upper_volume(chi(B,B),MD1).k = delta(MD1) by PARTFUN1:26; k in Seg len upper_volume(chi(B,B),MD1) by A402,FINSEQ_1:def 3; then A403: k in Seg len MD1 by INTEGRA1:def 7; then A404: delta(MD1) = vol(divset(MD1,k)) by A402,INTEGRA1:22; n1+1>1 by A224,NAT_1:38; then n1>1-1 by REAL_1:84; then A405: 1 <= k & k <= len MD1 & k < k+n1 by A403,FINSEQ_1:3,REAL_1:69; then A406: 1 < k+n1 by AXIOMS:22; A407: k in dom MD1 by A403,FINSEQ_1:def 3; k+n1 <= j by A310,A313,A405,REAL_1:84; then k+n1 <= len D1 by A225,AXIOMS:22; then A408: k+n1 in dom D1 by A406,FINSEQ_3:27; A409: divset(MD1,k)=divset(D1,k+n1) proof inf divset(MD1,k)=inf divset(D1,k+n1) & sup divset(MD1,k)=sup divset(D1,k+n1) proof per cases; suppose A410:k=1; then inf divset(MD1,k)=inf B & sup divset(MD1,k)=MD1.k by A407,INTEGRA1:def 5; then A411: inf divset(MD1,k)=D1.n1 & sup divset(MD1,k)=D1.(k+(n1+1)-1) by A222,A225,A226,A309,A310,A405,JORDAN3:31; inf divset(D1,k+n1)=D1.(k+n1-1) & sup divset(D1,k+n1)=D1.(k+n1) by A405,A408,INTEGRA1:def 5; hence thesis by A410,A411,XCMPLX_1:26; suppose A412:k<>1; then A413: k-1 in dom MD1 & k-1 in NAT by A407,INTEGRA1:9; then A414: 1 <= k-1 & k-1 <= len MD1 by FINSEQ_3:27; inf divset(MD1,k)=MD1.(k-1) & sup divset(MD1,k)=MD1.k by A407,A412,INTEGRA1:def 5; then inf divset(MD1,k)=D1.(k-1+(n1+1)-1) & sup divset(MD1,k)=D1.(k+(n1+1)-1) by A222,A225,A226,A309,A310,A405,A413,A414,JORDAN3:31; then inf divset(MD1,k)=D1.(k-1+n1+1-1) & sup divset(MD1,k)=D1.(k+n1+1-1) by XCMPLX_1:1; then A415: inf divset(MD1,k)=D1.(k-1+n1) & sup divset(MD1,k)=D1.(k+n1) by XCMPLX_1:26; inf divset(D1,k+n1)=D1.(k+n1-1) & sup divset(D1,k+n1)=D1.(k+n1) by A405,A408,INTEGRA1:def 5; hence thesis by A415,XCMPLX_1:29; end; then divset(MD1,k)=[.inf divset(D1,k+n1),sup divset(D1,k+n1).] by INTEGRA1:5; hence thesis by INTEGRA1:5; end; A416: k+n1 in Seg len D1 by A408,FINSEQ_1:def 3; then A417: delta(MD1)=upper_volume(chi(A,A),D1).(k+n1) by A404,A409,INTEGRA1: 22; k+n1 in Seg len upper_volume(chi(A,A),D1) by A416,INTEGRA1:def 7; then k+n1 in dom upper_volume(chi(A,A),D1) by FINSEQ_1:def 3; then delta(MD1) in rng upper_volume(chi(A,A),D1) by A417,FUNCT_1:def 5 ; then delta(MD1) <= max rng upper_volume(chi(A,A),D1) by PRE_CIRC:def 1 ; then A418: delta(MD1) <= delta(D1) by INTEGRA1:def 19; inf rng f <= sup rng f by A1,Lm6; then sup rng f - inf rng f >= 0 by SQUARE_1:12; then A419: (sup rng f-inf rng f)*delta(MD1) <= (sup rng f-inf rng f)* delta(D1) by A418,AXIOMS:25; A420: indx(D2,D1,n1) < indx(D2,D1,n1)+1 by NAT_1:38; A421: 1 <= indx(D2,D1,n1)+1 by A209,NAT_1:38; D1.n1 < D1.(n1+1) by A207,A227,A307,GOBOARD1:def 1; then indx(D2,D1,n1) < indx(D2,D1,n1+1) by A208,A228,GOBOARD2:18; then A422: indx(D2,D1,n1)+1 <= indx(D2,D1,n1+1) by NAT_1:38; then A423: indx(D2,D1,n1)+1 <= len D2 by A231,AXIOMS:22; then A424: indx(D2,D1,n1)+1 <= len UVf(D2) by INTEGRA1:def 7; A425: indx(D2,D1,n1+1)=indx(D2,D1,n1)+1 proof assume indx(D2,D1,n1+1)<> indx(D2,D1,n1)+1; then A426: indx(D2,D1,n1+1)>indx(D2,D1,n1)+1 by A422,REAL_1:def 5; A427: indx(D2,D1,n1)+1 in dom D2 by A421,A423,FINSEQ_3:27; then A428: D2.(indx(D2,D1,n1)+1) in rng D2 by FUNCT_1:def 5; now per cases by A20,A428,XBOOLE_0:def 2; suppose D2.(indx(D2,D1,n1)+1) in rng D1; then consider n2 being Nat such that A429: n2 in dom D1 & D2.(indx(D2,D1,n1)+1) = D1.n2 by PARTFUN1:26; D2.indx(D2,D1,n1) < D2.(indx(D2,D1,n1)+1) by A208,A420,A427,GOBOARD1:def 1; then n1 < n2 by A207,A208,A429,GOBOARD2:18; then A430: n1+1 <= n2 by NAT_1:38; D1.n2 < D1.(n1+1) by A228,A426,A427,A429,GOBOARD1:def 1; hence contradiction by A227,A429,A430,GOBOARD2:18; suppose D2.(indx(D2,D1,n1)+1) in rng D; then consider n2 being Nat such that A431: n2 in dom D & D2.(indx(D2,D1,n1)+1) = D.n2 by PARTFUN1:26; A432: D1.n1 < D.n2 by A208,A420,A427,A431,GOBOARD1:def 1; A433: D.i <= sup divset(D1,n1) by A207,INTEGRA2:1; sup divset(D1,n1)=D1.n1 proof per cases; suppose n1=1; hence thesis by A207,INTEGRA1:def 5; suppose n1<>1; hence thesis by A207,INTEGRA1:def 5; end; then D.i < D.n2 by A432,A433,AXIOMS:22; then i < n2 by A206,A431,GOBOARD2:18; then A434: i+1 <= n2 by NAT_1:38; A435: D.n2 < D1.(n1+1) by A228,A426,A427,A431,GOBOARD1:def 1; A436: j-1 in dom D1 & j-1 in NAT by A203,A210,A224,INTEGRA1:9; n1+1 +1 <= j by A306,NAT_1:38; then n1+1 <= j-1 by REAL_1:84; then A437: D1.(n1+1) <= D1.(j-1) by A227,A436,GOBOARD2:18; A438: inf divset(D1,j) <= D.(i+1) by A203,INTEGRA2:1; inf divset(D1,j) = D1.(j-1) by A203,A210,A224,INTEGRA1:def 5; then D1.(n1+1) <= D.(i+1) by A437,A438,AXIOMS:22; then D.n2 < D.(i+1) by A435,AXIOMS:22; hence contradiction by A202,A431,A434,GOBOARD2:18; end; hence contradiction; end; A439: Sum upper_volume(g,MD2)=Sum mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) proof upper_volume(g,MD2)=mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) proof A440: len upper_volume(g,MD2)=indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1 by A332,A425,INTEGRA1:def 7; A441: indx(D2,D1,j) <= len UVf(D2) by A232,INTEGRA1:def 7; A442: indx(D2,D1,n1)+1 <= indx(D2,D1,j) by A230,A422,AXIOMS:22; then len mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) = indx(D2,D1,j)-'(indx(D2,D1,n1)+1)+1 by A232,A421,A424,A441,JORDAN3:27; then A443: len upper_volume(g,MD2) =len mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) by A440,A442,SCMFSA_7:3; for k st 1<=k & k<=len upper_volume(g,MD2) holds upper_volume(g,MD2).k=mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)).k proof let k; assume A444:1 <= k & k <= len upper_volume(g,MD2); then k in Seg len upper_volume(g,MD2) by FINSEQ_1:3; then A445: k in Seg len MD2 by INTEGRA1:def 7; then A446: upper_volume(g,MD2).k =(sup rng(g|divset(MD2,k)))*vol(divset(MD2,k)) by INTEGRA1:def 7; A447: divset(MD2,k)=[.inf divset(MD2,k),sup divset(MD2,k).] by INTEGRA1:5; A448: mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)).k =UVf(D2).(k+(indx(D2,D1,n1)+1)-1) by A421,A440,A441,A442,A444,JORDAN3:31; k <= indx(D2,D1,j)-((indx(D2,D1,n1)+1)-1) by A440,A444,XCMPLX_1:37 ; then k+((indx(D2,D1,n1)+1)-1) <= indx(D2,D1,j) by REAL_1:84; then k+(indx(D2,D1,n1)+1)-1 <= indx(D2,D1,j) by XCMPLX_1:29; then A449: k+(indx(D2,D1,n1)+1)-1 <= len UVf(D2) by A441,AXIOMS:22; 1<=indx(D2,D1,n1)+1 by NAT_1:37; then 1+1<=k+(indx(D2,D1,n1)+1) by A444,REAL_1:55; then A450: 1 <= k+(indx(D2,D1,n1)+1)-1 by REAL_1:84; consider k2 being Nat such that A451: indx(D2,D1,n1)+1=1+k2; A452: k2=(indx(D2,D1,n1)+1)-1 by A451,XCMPLX_1:26; then A453: k+k2=k+(indx(D2,D1,n1)+1)-1 by XCMPLX_1:29; then k+k2 in Seg len UVf(D2) by A449,A450,FINSEQ_1:3; then A454: k+k2 in Seg len D2 by INTEGRA1:def 7; then A455: mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)).k =(sup rng(f|divset(D2,k+k2)))*vol(divset(D2,k+k2)) by A448,A453,INTEGRA1:def 7; inf divset(MD2,k)=inf divset(D2,k+k2) & sup divset(MD2,k)=sup divset(D2,k+k2) proof k2=indx(D2,D1,n1) by A452,XCMPLX_1:26; then k+k2>=1+1 by A209,A444,REAL_1:55; then A456: k+k2>1 by NAT_1:38; A457: k in dom MD2 by A445,FINSEQ_1:def 3; A458: k+k2 in dom D2 by A454,FINSEQ_1:def 3; now per cases; suppose A459:k=1; then A460: inf divset(MD2,k)=inf B & sup divset(MD2,k)=MD2.k by A457,INTEGRA1:def 5; then A461: sup divset(MD2,k)=D2.(1+indx(D2,D1,n1+1)-1) by A230,A232,A309,A421,A425,A440,A444,A459,JORDAN3:31 .=D2.(indx(D2,D1,n1+1)-1+1) by XCMPLX_1:29 .=D2.(indx(D2,D1,n1+1)-(1-1)) by XCMPLX_1:37 .=D1.(n1+1) by A20,A227,INTEGRA1:def 21; inf divset(D2,k+k2)=D2.(1+k2-1) & sup divset(D2,k+k2)=D2.(1+k2) by A456,A458,A459,INTEGRA1:def 5; hence inf divset(MD2,k)=inf divset(D2,k+k2) & sup divset(MD2,k)=sup divset(D2,k+k2) by A20,A208,A227,A309,A425,A451,A460,A461,INTEGRA1:def 21,XCMPLX_1 :26; suppose A462:k<>1; then A463: k-1 in dom MD2 & k-1 in NAT by A457,INTEGRA1:9; then A464: 1 <= k-1 by FINSEQ_3:27; k-1 <= k by Lm1; then A465: k-1 <= indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1 by A440,A444,AXIOMS :22; inf divset(MD2,k)=MD2.(k-1) & sup divset(MD2,k)=MD2.k by A457,A462,INTEGRA1:def 5; then A466: inf divset(MD2,k)=D2.(k-1+(indx(D2,D1,n1)+1)-1) & sup divset(MD2,k)=D2.(k+(indx(D2,D1,n1)+1)-1) by A230,A232,A309,A421,A425,A440,A444,A463,A464,A465,JORDAN3:31; inf divset(D2,k+k2)=D2.(k+k2-1) &sup divset(D2,k+k2)=D2.(k+k2) by A456,A458,INTEGRA1:def 5; then inf divset(D2,k+k2)=D2.(k-1+k2) & sup divset(D2,k+k2)=D2.(k+((indx(D2,D1,n1)+1)-1)) by A451,XCMPLX_1:26,29; hence inf divset(MD2,k)=inf divset(D2,k+k2) & sup divset(MD2,k)=sup divset(D2,k+k2) by A452,A466,XCMPLX_1:29; end; hence thesis; end; then A467: divset(MD2,k)=divset(D2,k+k2) by A447,INTEGRA1:5; k in dom MD2 by A445,FINSEQ_1:def 3; then divset(D2,k+k2) c= B by A467,INTEGRA1:10; hence thesis by A446,A455,A467,FUNCT_1:82; end; hence thesis by A443,FINSEQ_1:18; end; hence thesis; end; Sum upper_volume(g,MD1)=Sum mid(UVf(D1),n1+1,j) proof upper_volume(g,MD1)=mid(UVf(D1),n1+1,j) proof A468: j-'(n1+1)+1=j-(n1+1)+1 by A222,SCMFSA_7:3; A469: len upper_volume(g,MD1)=len MD1 by INTEGRA1:def 7 .=j-(n1+1)+1 by A222,A225,A226,A309,A468,JORDAN3:27; A470: n1+1 <= len UVf(D1) by A226,INTEGRA1:def 7; A471: j <= len UVf(D1) by A225,INTEGRA1:def 7; then A472: len upper_volume(g,MD1)=len mid(UVf(D1),n1+1,j) by A222,A225,A226,A468,A469,A470,JORDAN3:27; for k st 1<=k & k<=len upper_volume(g,MD1) holds upper_volume(g,MD1).k=mid(UVf(D1),n1+1,j).k proof let k; assume A473:1 <= k & k <= len upper_volume(g,MD1); then k in Seg len upper_volume(g,MD1) by FINSEQ_1:3; then A474: k in Seg len MD1 by INTEGRA1:def 7; then A475: upper_volume(g,MD1).k =(sup rng(g|divset(MD1,k)))*vol(divset(MD1,k)) by INTEGRA1:def 7; k <= j-((n1+1)-1) by A469,A473,XCMPLX_1:37; then A476: k+((n1+1)-1) <= j by REAL_1:84; consider k2 being Nat such that A477: n1+1=1+k2; A478: k2=(n1+1)-1 by A477,XCMPLX_1:26; A479: 1 <= k+k2 by A473,NAT_1:37; k+k2 <= len D1 by A225,A476,A478,AXIOMS:22; then A480: k+k2 in Seg len D1 by A479,FINSEQ_1:3; A481: mid(UVf(D1),n1+1,j).k =UVf(D1).(k+(n1+1)-1) by A222,A226,A469,A471,A473,JORDAN3:31 .=UVf(D1).(k+((n1+1)-1)) by XCMPLX_1:29 .=(sup rng(f|divset(D1,k+k2)))*vol(divset(D1,k+k2)) by A478,A480,INTEGRA1:def 7; A482: divset(D1,k+k2)=divset(MD1,k) proof A483: k2=n1 by A478,XCMPLX_1:26; then 1+1 <= k+k2 by A225,A473,REAL_1:55; then A484: 1 < k+k2 by NAT_1:38; A485: divset(MD1,k)=[.inf divset(MD1,k),sup divset(MD1,k).] by INTEGRA1:5; A486: k in dom MD1 by A474,FINSEQ_1:def 3; k+k2 in dom D1 by A480,FINSEQ_1:def 3; then A487: inf divset(D1,k+k2)=D1.(k+k2-1) & sup divset(D1,k+k2)=D1.(k+k2) by A484,INTEGRA1:def 5; inf divset(D1,k+k2)=inf divset(MD1,k) & sup divset(D1,k+k2)=sup divset(MD1,k) proof per cases; suppose A488:k=1; then inf divset(MD1,k)=inf B & sup divset(MD1,k)=MD1.k by A486,INTEGRA1:def 5; then inf divset(MD1,k)=D1.n1 & sup divset(MD1,k)=D1.(k+(n1+1)-1) by A222,A225,A226,A309,A469,A473,JORDAN3:31; hence thesis by A483,A487,A488,XCMPLX_1:26; suppose A489:k<>1; then A490: k-1 in dom MD1 & k-1 in NAT by A486,INTEGRA1:9; then A491: 1 <= k-1 by FINSEQ_3:27; k-1 <= k by Lm1; then A492: k-1 <= j-(n1+1)+1 by A469,A473,AXIOMS:22; inf divset(MD1,k)=MD1.(k-1) & sup divset(MD1,k)=MD1.k by A486,A489,INTEGRA1:def 5; then inf divset(MD1,k)=D1.(k-1+(n1+1)-1) & sup divset(MD1,k)=D1.(k+(n1+1)-1) by A222,A225,A226,A309,A469,A473,A490,A491,A492,JORDAN3:31; then inf divset(MD1,k)=D1.(k-1+((n1+1)-1)) & sup divset(MD1,k)=D1.(k+((n1+1)-1)) by XCMPLX_1:29; hence thesis by A478,A487,XCMPLX_1:29; end; hence thesis by A485,INTEGRA1:5; end; k in dom MD1 by A474,FINSEQ_1:def 3; then divset(D1,k+k2) c= B by A482,INTEGRA1:10; hence thesis by A475,A481,A482,FUNCT_1:82; end; hence thesis by A472,FINSEQ_1:18; end; hence thesis; end; hence thesis by A401,A419,A439,AXIOMS:22; end; A493: PUf(D2,indx(D2,D1,j))= PUf(D2,indx(D2,D1,n1))+Sum mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) proof A494: indx(D2,D1,n1)<indx(D2,D1,j) by A230,A233,AXIOMS:22; indx(D2,D1,j) in Seg len D2 by A229,FINSEQ_1:def 3; then A495: indx(D2,D1,j) in Seg len UVf(D2) by INTEGRA1:def 7; A496: indx(D2,D1,j) <= len UVf(D2) by A232,INTEGRA1:def 7; indx(D2,D1,n1) in Seg len D2 by A208,FINSEQ_1:def 3; then indx(D2,D1,n1) in Seg len UVf(D2) by INTEGRA1:def 7; then PUf(D2,indx(D2,D1,n1))=Sum(UVf(D2)|indx(D2,D1,n1)) by INTEGRA1: def 22 .=Sum mid(UVf(D2),1,indx(D2,D1,n1)) by A209,JORDAN3:25; then PUf(D2,indx(D2,D1,n1))+Sum mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2, D1,j)) =Sum(mid(UVf(D2),1,indx(D2,D1,n1))^ mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j))) by RVSUM_1:105 .=Sum mid(UVf(D2),1,indx(D2,D1,j)) by A209,A494,A496,INTEGRA2:4 .=Sum(UVf(D2)|indx(D2,D1,j)) by A232,JORDAN3:25; hence thesis by A495,INTEGRA1:def 22; end; A497: j <= len UVf(D1) by A225,INTEGRA1:def 7; then A498: j in Seg len UVf(D1) by A225,FINSEQ_1:3; n1 in Seg len D1 by A207,FINSEQ_1:def 3; then n1 in Seg len UVf(D1) by INTEGRA1:def 7; then PUf(D1,n1)=Sum(UVf(D1)|n1) by INTEGRA1:def 22 .=Sum mid(UVf(D1),1,n1) by A225,JORDAN3:25; then PUf(D1,n1)+Sum mid(UVf(D1),n1+1,j) =Sum(mid(UVf(D1),1,n1)^mid(UVf(D1),n1+1,j)) by RVSUM_1:105 .=Sum mid(UVf(D1),1,j) by A210,A225,A497,INTEGRA2:4 .=Sum(UVf(D1)|j) by A225,JORDAN3:25; then A499: PUf(D1,j)=PUf(D1,n1)+Sum mid(UVf(D1),n1+1,j) by A498,INTEGRA1:def 22; A500: i*(sup rng f-inf rng f)*delta(D1)+(sup rng f-inf rng f)*delta(D1) =i*((sup rng f-inf rng f)*delta(D1))+1* ((sup rng f-inf rng f)*delta(D1)) by XCMPLX_1:4 .=(i+1)*((sup rng f-inf rng f)*delta(D1)) by XCMPLX_1:8 .=(i+1)*(sup rng f-inf rng f)*delta(D1) by XCMPLX_1:4; A501: (PUf(D1,n1)-PUf(D2,indx(D2,D1,n1)))+ (Sum mid(UVf(D1),n1+1,j)-Sum mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j))) <=i*(sup rng f-inf rng f)*delta(D1)+(sup rng f-inf rng f)*delta(D1) by A207,A235,REAL_1:55; (PUf(D1,n1)-PUf(D2,indx(D2,D1,n1)))+ (Sum mid(UVf(D1),n1+1,j)-Sum mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j))) =PUf(D1,j)-PUf(D2,indx(D2,D1,j)) by A493,A499,Lm5; hence thesis by A203,A500,A501; end; hence thesis; end; A502: for k being non empty Nat holds P[k] from Ind_from_1(A43,A199); i in Seg len D by A22,FINSEQ_1:def 3; then reconsider i as non empty Nat by BINARITH:5; P[i] by A502; hence thesis by A22; end; len D in dom D by SCMFSA_7:12; then consider j such that A503: j in dom D1 & D.(len D) in divset(D1,j) & PUf(D1,j)-PUf(D2,indx(D2,D1,j)) <= (len D)*(sup(rng f)-inf(rng f))*delta(D1) by A21; A504:len D1 in dom D1 by SCMFSA_7:12; A505:j = len D1 proof assume A506:j<>len D1; A507: D.(len D) <= sup divset(D1,j) by A503,INTEGRA2:1; j in Seg len D1 by A503,FINSEQ_1:def 3; then j <= len D1 by FINSEQ_1:3; then j < len D1 by A506,REAL_1:def 5; then D1.j < D1.(len D1) by A503,A504,GOBOARD1:def 1; then A508: D1.j < sup A by INTEGRA1:def 2; sup divset(D1,j) < sup A proof per cases; suppose j=1; hence thesis by A503,A508,INTEGRA1:def 5; suppose j<>1; hence thesis by A503,A508,INTEGRA1:def 5; end; hence contradiction by A507,INTEGRA1:def 2; end; A509: len D2 in dom D2 by SCMFSA_7:12; indx(D2,D1,len D1)=len D2 proof A510: indx(D2,D1,len D1) in dom D2 & D1.(len D1) = D2.indx(D2,D1,len D1) by A20,A504,INTEGRA1:def 21; then sup A = D2.indx(D2,D1,len D1) by INTEGRA1:def 2; then D2.(len D2) = D2.indx(D2,D1,len D1) by INTEGRA1:def 2; hence thesis by A509,A510,GOBOARD2:19; end; then upper_sum(f,D1)-PUf(D2,len D2)<=(len D)*(sup(rng f)-inf(rng f))* delta(D1) by A503,A505,INTEGRA1:44; hence thesis by INTEGRA1:44; end; hence thesis by A20; end; hence thesis; end; A511:for e be real number st e>0 ex n st for m st n<=m holds abs((upper_sum(f,T)).m-upper_integral(f))<e proof let e be real number; assume A512:e>0; then A513: e/2>0 by REAL_2:127; reconsider e as Real by XREAL_0:def 1; A514: upper_integral(f) = inf rng upper_sum_set(f) by INTEGRA1:def 15; A515: rng upper_sum_set(f) is bounded_below by A1,INTEGRA2:35; dom upper_sum_set(f) is non empty by INTEGRA1:def 11; then rng upper_sum_set(f) is non empty by RELAT_1:65; then consider y be real number such that A516: y in rng upper_sum_set(f) & upper_integral(f)+e/2>y by A513,A514,A515,SEQ_4:def 5; ex D being Element of divs A st D in dom upper_sum_set(f) & y=(upper_sum_set(f)).D & D.1 > inf A proof consider D3 being Element of divs A such that A517: D3 in dom upper_sum_set(f) & y=(upper_sum_set(f)).D3 by A516,PARTFUN1: 26; len D3 <> 0 by FINSEQ_1:25; then A518: len D3 in Seg len D3 by FINSEQ_1:5; then 1 <= len D3 by FINSEQ_1:3; then A519: 1 in Seg len D3 by FINSEQ_1:3; then A520: 1 in dom D3 by FINSEQ_1:def 3; now per cases; suppose A521:D3.1 <> inf A; D3.1 in A by A520,INTEGRA1:8; then inf A <= D3.1 by INTEGRA2:1; then D3.1 > inf A by A521,REAL_1:def 5; hence ex D being Element of divs A st D in dom upper_sum_set(f) & y=(upper_sum_set(f)).D & D.1 > inf A by A517; suppose A522:D3.1 = inf A; ex D being Element of divs A st D in dom upper_sum_set(f) & y=(upper_sum_set(f)).D & D.1 > inf A proof A523: D3.(len D3) = sup A by INTEGRA1:def 2; A524: len D3 in dom D3 by A518,FINSEQ_1:def 3; vol(A) >= 0 by INTEGRA1:11; then sup A - inf A > 0 by A3,INTEGRA1:def 6; then sup A > inf A by REAL_2:106; then A525: len D3 > 1 by A520,A522,A523,A524,GOBOARD2:18; then reconsider D=D3/^1 as increasing FinSequence of REAL by INTEGRA1:36; A526: len D = len D3 - 1 & for m be Nat st m in dom D holds D.m = D3.(m+1) by A525,RFINSEQ:def 2; then A527: len D <> 0 by A525,SQUARE_1:11; then reconsider D as non empty increasing FinSequence of REAL by FINSEQ_1:25; rng D c= rng D3 & rng D3 c= A by FINSEQ_5:36,INTEGRA1:def 2; then A528: rng D c= A by XBOOLE_1:1; A529: len D+1=len D3 by A526,XCMPLX_1:27; A530: len D in Seg len D by A527,FINSEQ_1:5; len D in dom D by SCMFSA_7:12; then D.(len D)=sup A by A523,A525,A529,RFINSEQ:def 2; then D is DivisionPoint of A by A528,INTEGRA1:def 2; then reconsider D as Element of divs A by INTEGRA1:def 4; D in divs A; then A531: D in dom upper_sum_set(f) by INTEGRA1:def 11; A532: y=(upper_sum_set(f)).D proof A533: y=upper_sum(f,D3) by A517,INTEGRA1:def 11 .=Sum(upper_volume(f,D3)) by INTEGRA1:def 9 .=Sum((upper_volume(f,D3)|1)^(upper_volume(f,D3)/^1)) by RFINSEQ:21; A534: 1 <= len upper_volume(f,D3) by A525,INTEGRA1:def 7; then A535: len (upper_volume(f,D3)|1)=1 by TOPREAL1:3; A536: 1 in dom upper_volume(f,D3) by A534,FINSEQ_3:27; 1 in Seg 1 by FINSEQ_1:3; then (upper_volume(f,D3)|1).1 = upper_volume(f,D3).1 by A536,RFINSEQ: 19; then A537: upper_volume(f,D3)|1 = <*upper_volume(f,D3).1*> by A535, FINSEQ_1:57; A538: vol(divset(D3,1))=sup divset(D3,1)-inf divset(D3,1) by INTEGRA1:def 6 .=sup divset(D3,1) - inf A by A520,INTEGRA1:def 5 .=D3.1 - inf A by A520,INTEGRA1:def 5 .=0 by A522,XCMPLX_1:14; A539: upper_volume(f,D3).1=(sup rng(f|divset(D3,1)))*vol(divset(D3,1)) by A519,INTEGRA1:def 7; upper_volume(f,D3)/^1 = upper_volume(f,D) proof A540: 2-'1=2-1 by SCMFSA_7:3 .= 1; A541: len D3 >= 1+1 by A525,NAT_1:38; then len upper_volume(f,D3) >= 2 by INTEGRA1:def 7; then A542: mid(upper_volume(f,D3),2,len upper_volume(f,D3)) =upper_volume(f,D3)/^1 by A540,JORDAN3:26; A543: 2 <= len upper_volume(f,D3) & 1 <= len upper_volume(f,D3) by A525,A541,INTEGRA1:def 7; then A544: len(mid(upper_volume(f,D3),2,len upper_volume(f,D3))) =len upper_volume(f,D3)-'2+1 by JORDAN3:27 .=len D3-'2+1 by INTEGRA1:def 7 .=len D3-2+1 by A541,SCMFSA_7:3 .=len D3-(2-1) by XCMPLX_1:37 .=len D3-1; then A545: len(mid(upper_volume(f,D3),2,len upper_volume(f,D3))) =len upper_volume(f,D) by A526,INTEGRA1:def 7; for i st 1<=i&i<=len mid(upper_volume(f,D3),2,len upper_volume(f,D3 ) ) holds mid(upper_volume(f,D3),2,len upper_volume(f,D3)).i =upper_volume(f,D).i proof let i; assume A546: 1<=i&i<=len mid(upper_volume(f,D3),2,len upper_volume(f,D3)); then i <= len upper_volume(f,D3) - 1 by A544,INTEGRA1:def 7; then i <= len upper_volume(f,D3)-2+1 by Lm2,XCMPLX_1:37; then A547: mid(upper_volume(f,D3),2,len upper_volume(f,D3)).i =upper_volume(f,D3).(i+2-1) by A543,A546,JORDAN3:31 .=upper_volume(f,D3).(i+(2-1)) by XCMPLX_1:29 .=upper_volume(f,D3).(i+1); 1 <= i+1 & i+1 <= len D3 by A544,A546,NAT_1:37,REAL_1:84; then A548: i+1 in Seg len D3 by FINSEQ_1:3; then A549: mid(upper_volume(f,D3),2,len upper_volume(f,D3)).i =(sup rng(f|divset(D3,i+1)))*vol(divset(D3,i+1)) by A547,INTEGRA1:def 7; A550: divset(D3,i+1)=divset(D,i) proof A551: 1<>i+1 by A546,NAT_1:38; i+1 in dom D3 by A548,FINSEQ_1:def 3; then A552: sup divset(D3,i+1)=D3.(i+1) & inf divset(D3,i+1)=D3.(i+1 -1) by A551,INTEGRA1:def 5; then A553: sup divset(D3,i+1)=D3.(i+1) & inf divset(D3,i+1)=D3.i by XCMPLX_1:26; A554: i in dom D by A526,A544,A546,FINSEQ_3:27; then A555: D.i=D3.(i+1) by A525,RFINSEQ:def 2; now per cases; suppose A556:i=1; then A557: inf divset(D,i)=inf A & sup divset(D,i)=D.i by A554,INTEGRA1:def 5; divset(D3,i+1)=[.inf A, D.i.] by A522,A552,A555,A556,INTEGRA1:5; hence divset(D3,i+1)=divset(D,i) by A557,INTEGRA1:5; suppose A558:i<>1; then A559: inf divset(D,i)=D.(i-1) & sup divset(D,i)=D.i by A554,INTEGRA1:def 5; i-1 in dom D & i-1 in NAT by A554,A558,INTEGRA1:9; then D.(i-1)=D3.(i-1+1) by A525,RFINSEQ:def 2 .= D3.(i-(1-1)) by XCMPLX_1:37 .=D3.i; then divset(D3,i+1)=[.inf divset(D,i),sup divset(D,i).] by A553,A555,A559,INTEGRA1:5; hence thesis by INTEGRA1:5; end; hence thesis; end; i in Seg len D by A526,A544,A546,FINSEQ_1:3; hence thesis by A549,A550,INTEGRA1:def 7; end; hence thesis by A542,A545,FINSEQ_1:18; end; then y=0+Sum(upper_volume(f,D)) by A533,A537,A538,A539,RVSUM_1:106 .=upper_sum(f,D) by INTEGRA1:def 9; hence thesis by A531,INTEGRA1:def 11; end; 1 <= len D by A530,FINSEQ_1:3; then 1 in dom D by FINSEQ_3:27; then A560: D.1=D3.(1+1) by A525,RFINSEQ:def 2 .=D3.2; 1+1 <= len D3 by A525,NAT_1:38; then 2 in dom D3 by FINSEQ_3:27; then D3.1 < D3.2 by A520,GOBOARD1:def 1; hence thesis by A522,A531,A532,A560; end; hence thesis; end; hence thesis; end; then consider D being Element of divs A such that A561: D in dom upper_sum_set(f) & y=(upper_sum_set(f)).D & D.1>inf A; y=upper_sum(f,D) by A561,INTEGRA1:def 11; then A562: upper_sum(f,D)-upper_integral(f)<e/2 by A516,REAL_1:84; set p=len D, H=sup rng f, h=inf rng f; deffunc F(Nat)=vol(divset(D,$1)); consider v being FinSequence of REAL such that A563: len v = len D & for j st j in Seg len D holds v.j=F(j) from SeqLambdaD; consider v1 being non-decreasing FinSequence of REAL such that A564: v,v1 are_fiberwise_equipotent by INTEGRA2:3; defpred P[Nat] means $1 in dom v1 & v1.$1 > 0; A565: ex k st P[k] proof consider H being Function such that A566: dom H = dom v & rng H = dom v1 & H is one-to-one & v=v1*H by A564,RFINSEQ:3; consider k such that A567: k in dom D & vol(divset(D,k)) > 0 by A3,Th1; dom D = Seg len D by FINSEQ_1:def 3; then k in dom v & v.k > 0 by A563,A567,FINSEQ_1:def 3; then H.k in dom v1 & v1.(H.k) > 0 by A566,FUNCT_1:21,22; hence thesis; end; consider k such that A568: P[k] & for n st P[n] holds k<=n from Min(A565); p <> 0 by FINSEQ_1:25; then A569:p > 0 by NAT_1:19; then A570: 2*p >0 by REAL_2:122; H >= h by A1,Lm6; then H-h >= 0 by SQUARE_1:12; then H-h+1 > 0+0 by REAL_1:67; then A571: 2*p*(H-h+1) > 0 by A570,REAL_2:122; min(v1.k,e/(2*p*(H-h+1))) > 0 proof per cases by SQUARE_1:38; suppose min(v1.k,e/(2*p*(H-h+1))) = v1.k; hence thesis by A568; suppose min(v1.k,e/(2*p*(H-h+1))) = e/(2*p*(H-h+1)); hence thesis by A512,A571,REAL_2:127; end; then consider n such that A572: for m st n<=m holds 0 < (delta(T)).m & (delta(T)).m < min(v1.k,e/(2*p*(H-h+1))) by A5; A573: for m st n<=m holds abs((upper_sum(f,T)).m-upper_integral(f))<e proof let m; assume A574:n<=m; then A575: 0 < (delta(T)).m & (delta(T)).m < min(v1.k,e/(2*p*(H-h+1))) by A572 ; reconsider D1 = T.m as Element of divs A; consider D2 being Element of divs A such that A576: D <= D2 & D1 <= D2 & rng D2 = rng D1 \/ rng D & 0 <= upper_sum(f,D)-upper_sum(f,D2) & 0 <= upper_sum(f,D1)-upper_sum(f,D2) by A13; f is_bounded_above_on A by A1,RFUNCT_1:def 11; then A577: upper_sum(f,D2) <= upper_sum(f,D) by A576,INTEGRA1:47; A578: delta(D1)=(delta(T)).m by INTEGRA2:def 3; min (v1.k,e/(2*p*(H-h+1))) <= v1.k by SQUARE_1:35; then A579: delta(D1)<v1.k by A575,A578,AXIOMS:22; A580: v1.1 > 0 proof A581: for n1 be Element of NAT st n1 in dom D holds vol(divset(D,n1))>0 proof let n1 be Element of NAT; assume A582:n1 in dom D; then A583: 1 <= n1 by FINSEQ_3:27; per cases by A583,REAL_1:def 5; suppose A584:n1=1; then inf divset(D,n1)=inf A & sup divset(D,n1)=D.n1 by A582,INTEGRA1:def 5; then vol(divset(D,n1))=D.n1-inf A by INTEGRA1:def 6; hence thesis by A561,A584,SQUARE_1:11; suppose A585:n1>1; then inf divset(D,n1)=D.(n1-1) & sup divset(D,n1)=D.n1 by A582,INTEGRA1:def 5; then A586: vol(divset(D,n1))=D.n1-D.(n1-1) by INTEGRA1:def 6; A587: n1-1 in dom D by A582,A585,INTEGRA1:9; n1 < n1+1 by REAL_1:69; then n1-1 < n1 by REAL_1:84; then D.(n1-1)<D.n1 by A582,A587,GOBOARD1:def 1; hence thesis by A586,SQUARE_1:11; end; k in Seg len v1 by A568,FINSEQ_1:def 3; then 1 <= k & k <= len v1 by FINSEQ_1:3; then 1 <= len v1 by AXIOMS:22; then 1 in dom v1 by FINSEQ_3:27; then A588: v1.1 in rng v1 by FUNCT_1:def 5; rng v = rng v1 by A564,RFINSEQ:1; then consider n1 being Element of NAT such that A589: n1 in dom v & v1.1 = v.n1 by A588,PARTFUN1:26; A590: n1 in Seg len D by A563,A589,FINSEQ_1:def 3; then A591: v1.1 = vol(divset(D,n1)) by A563,A589; n1 in dom D by A590,FINSEQ_1:def 3; hence thesis by A581,A591; end; v1.k = min rng upper_volume(chi(A,A),D) proof A592: k=1 proof assume A593:k <> 1; A594: len v1 = len v by A564,RFINSEQ:16; A595: k in Seg len v1 by A568,FINSEQ_1:def 3; k in Seg len v by A568,A594,FINSEQ_1:def 3; then A596: 1 <= k & k <= len D by A563,FINSEQ_1:3; then A597: k > 1 by A593,REAL_1:def 5; k <= len v1 by A595,FINSEQ_1:3; then 1 <= len v1 by A596,AXIOMS:22; then 1 in dom v1 by FINSEQ_3:27; hence contradiction by A568,A580,A597; end; min rng upper_volume(chi(A,A),D) in rng upper_volume(chi(A,A),D) & for x st x in rng upper_volume(chi(A,A),D) holds min rng upper_volume(chi(A,A),D) <= x by SFMASTR3:def 1; then consider m such that A598: m in dom upper_volume(chi(A,A),D) & min rng upper_volume(chi(A,A),D)=upper_volume(chi(A,A),D).m by PARTFUN1:26; m in Seg len upper_volume(chi(A,A),D) by A598,FINSEQ_1:def 3; then A599: m in Seg len D by INTEGRA1:def 7; then min rng upper_volume(chi(A,A),D)=vol(divset(D,m)) by A598,INTEGRA1:22; then A600: v.m=min rng upper_volume(chi(A,A),D) by A563,A599; m in dom v by A563,A599,FINSEQ_1:def 3; then A601: min rng upper_volume(chi(A,A),D) in rng v by A600,FUNCT_1:def 5; A602: rng v = rng v1 by A564,RFINSEQ:1; then consider m1 being Element of NAT such that A603: m1 in dom v1 & min rng upper_volume(chi(A,A),D)=v1.m1 by A601,PARTFUN1:26; m1 >= 1 by A603,FINSEQ_3:27; then A604: v1.1 <= min rng upper_volume(chi(A,A),D) by A568,A592,A603, INTEGRA2:2; v1.k in rng upper_volume(chi(A,A),D) proof v1.k in rng v by A568,A602,FUNCT_1:def 5; then consider k2 being Element of NAT such that A605: k2 in dom v & v1.k = v.k2 by PARTFUN1:26; A606: k2 in Seg len D by A563,A605,FINSEQ_1:def 3; then v1.k = vol(divset(D,k2)) by A563,A605; then A607: v1.k = upper_volume(chi(A,A),D).k2 by A606,INTEGRA1:22; k2 in Seg len upper_volume(chi(A,A),D) by A606,INTEGRA1:def 7; then k2 in dom upper_volume(chi(A,A),D) by FINSEQ_1:def 3; hence thesis by A607,FUNCT_1:def 5; end; then v1.k >= min rng upper_volume(chi(A,A),D) by SFMASTR3:def 1; hence thesis by A592,A604,AXIOMS:21; end; then consider D3 being Element of divs A such that A608: D<=D3 & D1<=D3 & rng D3=rng D1 \/ rng D & upper_sum(f,D1)-upper_sum(f,D3) <=(len D)*(sup rng f-inf rng f)*delta(D1) by A18,A579; A609: upper_sum(f,D1)-upper_sum(f,D2)<=(len D)*(sup rng f-inf rng f)*delta(D1) by A576,A608,Th5; H-h <= H-h+1 by REAL_1:69; then p*(H-h)<=p*(H-h+1) by A569,AXIOMS:25; then p*(H-h)*(delta(T)).m <= p*(H-h+1)*(delta(T)).m by A575,AXIOMS:25; then A610: upper_sum(f,T.m)-upper_sum(f,D2) <= p*(H-h+1)*(delta(T)).m by A578,A609,AXIOMS:22; A611: (delta(T)).m < min(v1.k,e/(2*p*(H-h+1))) by A572,A574; min(v1.k,e/(2*p*(H-h+1))) <= e/(2*p*(H-h+1)) by SQUARE_1:35; then (delta(T)).m < e/(2*p*(H-h+1)) by A611,AXIOMS:22; then (delta(T)).m*(2*p*(H-h+1))<e by A571,REAL_2:177; then (delta(T)).m*(2*(p*(H-h+1)))<e by XCMPLX_1:4; then ((delta(T)).m*(p*(H-h+1)))*2<e by XCMPLX_1:4; then A612: p*(H-h+1)*(delta(T)).m < e/2 by REAL_2:178; set s=upper_integral(f), sD=upper_sum(f,D); set sD1=upper_sum(f,T.m), sD2=upper_sum(f,D2); sD+sD1-sD1-s < e/2 by A562,XCMPLX_1:26; then sD+sD1-(sD1+s) < e/2 by XCMPLX_1:36; then sD1-(sD1+s)+sD < e/2 by XCMPLX_1:29; then sD1-s-sD1+sD < e/2 by XCMPLX_1:36; then sD1-s+sD-sD1 < e/2 by XCMPLX_1:29; then sD1-s+sD < sD1+e/2 by REAL_1:84; then sD1-s < sD1+e/2-sD by REAL_1:86; then A613: sD1-s < sD1-sD+e/2 by XCMPLX_1:29; sD1-sD <= sD1-sD2 by A577,REAL_2:106; then sD1-sD <= p*(H-h+1)*(delta(T)).m by A610,AXIOMS:22; then sD1-sD < e/2 by A612,AXIOMS:22; then sD1-sD+e/2 < e/2+e/2 by REAL_1:53; then sD1-sD+e/2 <= e by XCMPLX_1:66; then sD1-s < e by A613,AXIOMS:22; then A614: (upper_sum(f,T)).m-upper_integral(f) < e by INTEGRA2:def 4; T.m in divs A; then A615: T.m in dom upper_sum_set(f) by INTEGRA1:def 11; (upper_sum(f,T)).m = upper_sum(f,T.m) by INTEGRA2:def 4; then (upper_sum(f,T)).m = (upper_sum_set(f)).(T.m) by A615,INTEGRA1:def 11; then (upper_sum(f,T)).m in rng upper_sum_set(f) by A615,FUNCT_1:def 5; then inf rng upper_sum_set(f)<=(upper_sum(f,T)).m by A515,SEQ_4:def 5; then upper_integral(f)<=(upper_sum(f,T)).m by INTEGRA1:def 15; then (upper_sum(f,T)).m-upper_integral(f) >= 0 by SQUARE_1:12; hence thesis by A614,ABSVALUE:def 1; end; take n; thus thesis by A573; end; hence upper_sum(f,T) is convergent by SEQ_2:def 6; hence thesis by A511,SEQ_2:def 7; end;

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