reserve a,b,e,r,x,y for Real,
  i,j,k,n,m for Element of NAT,
  x1 for set,
  p,q for FinSequence of REAL,
  A for non empty closed_interval Subset of REAL,
  D,D1,D2 for Division of A,
  f,g for Function of A,REAL,
  T for DivSequence of A;

theorem Th23:
  f|A is bounded implies
  for D,D1 st delta(D1)<min rng upper_volume(chi(A,A),D) holds
  ex D2 st D<=D2 & D1<=D2 & rng D2=rng D1 \/ rng D &
  upper_sum(f,D1)-upper_sum(f,D2) <=
    (len D)*(upper_bound(rng f)-lower_bound(rng f))*delta(D1)
  proof
  assume
A1: f|A is bounded; then
A2: for D,D1 ex D2 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)
      by Th22;
    let D,D1;
    assume
A8: delta(D1)<min rng upper_volume(chi(A,A),D);
    ex D2 st D<=D2 & D1<=D2 & rng D2=rng D1 \/ rng D & upper_sum(f,D1)-
    upper_sum(f,D2)<=(len D)*(upper_bound(rng f)- lower_bound(rng f))*delta(D1)
    proof
      consider D2 be Division of A such that
A9:   D<=D2 and
A10:  D1<=D2 and
A11:  rng D2=rng D1 \/ rng D and
      0<=upper_sum(f,D)-upper_sum(f,D2) and
      0<=upper_sum(f,D1)-upper_sum(f,D2) by A2;
      upper_sum(f,D1)-upper_sum(f,D2) <= (len D)*(upper_bound(rng f)-
      lower_bound(rng f))*delta(D1)
      proof
        deffunc UVf(Division of A) = upper_volume(f,$1);
        deffunc PUf(Division of A,Nat) = (PartSums(upper_volume(f,$1))).$2;
A12:    len D2 in dom D2 by FINSEQ_5:6;
A13:    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*(upper_bound(rng f)- lower_bound(rng f)
        )*delta(D1)
        proof
          defpred P[non zero 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*(upper_bound
          rng f-lower_bound rng f) *delta(D1);
          let i;
          assume
A14:      i in dom D;
          then
A15:      i in Seg len D by FINSEQ_1:def 3;
A16:      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
A17:        i in dom D;
            assume that
A18:        j in dom D1 and
A19:        D.i in divset(D1,j);
            assume j<2;
            then j<1+1;
            then
A20:        j <= 1 by NAT_1:13;
            j in Seg len D1 by A18,FINSEQ_1:def 3;
            then j >= 1 by FINSEQ_1:1;
            then j = 1 by A20,XXREAL_0:1;
            then
A21:        lower_bound divset(D1,j)=lower_bound A by A18,INTEGRA1:def 4;
A22:        D.i<=upper_bound divset(D1,j) by A19,INTEGRA2:1;
            delta(D1) >= min rng upper_volume(chi(A,A),D)
            proof
              per cases;
              suppose
A23:            i=1;
                len D in Seg len D by FINSEQ_1:3;
                then 1 <= len D by FINSEQ_1:1;
                then
A24:            1 in dom D by FINSEQ_3:25;
                then
A25:            lower_bound divset(D,1)=lower_bound A by INTEGRA1:def 4;
                1 in Seg len D by A24,FINSEQ_1:def 3;
                then 1 in Seg len upper_volume(chi(A,A),D) by INTEGRA1:def 6;
                then
A26:            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 A24,
INTEGRA1:20;
                then vol(divset(D,1)) in rng upper_volume(chi(A,A),D) by A26,
FUNCT_1:def 3;
                then
A27:            vol
(divset(D,1))>=min rng upper_volume(chi(A,A),D) by XXREAL_2:def 7;
A28:            upper_bound divset(D,1)=D. 1 by A24,INTEGRA1:def 4;
                upper_bound divset(D1,j)-lower_bound A >= D.1-lower_bound
                A by A22,A23,XREAL_1:9;
                then
                vol(divset(D1,j)) >= upper_bound divset(D,1)- lower_bound
                divset(D,1) by A21,A25,A28,INTEGRA1:def 5;
                then
A29:            vol(divset(D1,j)) >= vol(divset(D,1)) by INTEGRA1:def 5;
                vol(divset(D1,j)) <= delta(D1) by A18,Lm5;
                then delta(D1) >= vol(divset(D,1)) by A29,XXREAL_0:2;
                hence thesis by A27,XXREAL_0:2;
              end;
              suppose
A30:            i<>1;
                then D.(i-1) in A by A17,INTEGRA1:7;
                then
A31:            lower_bound A <= D.(i-1) by INTEGRA2:1;
                lower_bound divset(D,i)=D.(i-1) by A17,A30,INTEGRA1:def 4;
                then
A32:            upper_bound divset(D,i)-lower_bound A >= upper_bound
                divset(D,i)-lower_bound divset(D,i) by A31,XREAL_1:10;
                upper_bound divset(D,i) = D.i by A17,A30,INTEGRA1:def 4;
                then upper_bound divset(D1,j)-lower_bound divset(D1,j) >=
                upper_bound divset(D,i)- lower_bound A by A22,A21,XREAL_1:9;
                then upper_bound divset(D1,j)-lower_bound divset(D1,j) >=
upper_bound divset(D,i)-lower_bound divset( D,i) by A32,XXREAL_0:2;
                then vol(divset(D1,j)) >= upper_bound divset(D,i)-lower_bound
                divset(D,i) by INTEGRA1:def 5;
                then
A33:            vol(divset(D1,j)) >= vol(divset(D,i)) by INTEGRA1:def 5;
                i in Seg len D by A17,FINSEQ_1:def 3;
                then i in Seg len upper_volume(chi(A,A),D) by INTEGRA1:def 6;
                then
A34:            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 A17,
INTEGRA1:20;
                then vol(divset(D,i)) in rng upper_volume(chi(A,A),D) by A34,
FUNCT_1:def 3;
                then
A35:            vol
(divset(D,i))>=min rng upper_volume(chi(A,A),D) by XXREAL_2:def 7;
                vol(divset(D1,j)) <= delta(D1) by A18,Lm5;
                then delta(D1) >= vol(divset(D,i)) by A33,XXREAL_0:2;
                hence thesis by A35,XXREAL_0:2;
              end;
            end;
            hence contradiction by A8;
          end;
A36:      P[1]
          proof
            len D in Seg len D by FINSEQ_1:3;
            then 1 <= len D by FINSEQ_1:1;
            then
A37:        1 in dom D by FINSEQ_3:25;
            then consider j such that
A38:        j in dom D1 and
A39:        D.1 in divset(D1,j) by Th3,INTEGRA1:6;
            PUf(D1,j)-PUf(D2,indx(D2,D1,j))<= 1*(upper_bound rng f-
            lower_bound rng f)*delta(D1)
            proof
A40:          j <> 1 by A16,A37,A38,A39;
              then reconsider j1=j-1 as Element of NAT by A38,INTEGRA1:7;
A41:          j1 in dom D1 by A38,A40,INTEGRA1:7;
              then j1 in Seg len D1 by FINSEQ_1:def 3;
              then j1 in Seg len upper_volume(f,D1) by INTEGRA1:def 6;
              then
A42:          j1 in dom upper_volume(f,D1) by FINSEQ_1:def 3;
A43:          j-1 in dom D1 by A38,A40,INTEGRA1:7;
              then
A44:          indx(D2,D1,j1) in dom D2 by A10,INTEGRA1:def 19;
              then
A45:          indx(D2,D1,j1) in Seg len D2 by FINSEQ_1:def 3;
              then
A46:          1 <= indx(D2,D1,j1) by FINSEQ_1:1;
              then mid(D2,1,indx(D2,D1,j1)) is increasing by A44,INTEGRA1:35;
              then
A47:          D2|indx(D2,D1,j1) is increasing by A46,FINSEQ_6:116;
              j < j+1 by NAT_1:13;
              then j1 < j by XREAL_1:19;
              then
A48:          indx(D2,D1,j1) < indx(D2,D1,j) by A10,A38,A41,Th8;
              then
A49:          indx(D2,D1,j1)+1 <= indx(D2,D1,j) by NAT_1:13;
A50:          Sum mid(upper_volume(f,D1),j,j) -Sum mid(upper_volume(f,D2
),(indx(D2,D1,j1)+1),indx(D2,D1,j)) <= (upper_bound rng f-lower_bound rng f)*
              delta(D1)
              proof
A51:            indx(D2,D1,j)-indx(D2,D1,j1) <= 2
                proof
                  reconsider ID1=indx(D2,D1,j1)+1 as Element of NAT;
                  reconsider ID2=ID1+1 as Element of NAT;
                  assume indx(D2,D1,j)-indx(D2,D1,j1) > 2;
                  then
A52:              indx(D2,D1,j1)+(1+1)<indx(D2,D1,j) by XREAL_1:20;
A53:              ID1 < ID2 by NAT_1:13;
                  then indx (D2,D1,j1) <= ID2 by NAT_1:13;
                  then
A54:              1 <= ID2 by A46,XXREAL_0:2;
A55:              indx(D2,D1,j) in dom D2 by A10,A38,INTEGRA1:def 19;
                  then
A56:              indx(D2,D1,j) <= len D2 by FINSEQ_3:25;
                  then ID2 <= len D2 by A52,XXREAL_0:2;
                  then
A57:              ID2 in dom D2 by A54,FINSEQ_3:25;
                  then
A58:              D2.ID2<D2.indx(D2,D1, j) by A52,A55,SEQM_3:def 1;
A59:              1 <= ID1 by A46,NAT_1:13;
A60:              D1.j = D2.indx(D2,D1,j) by A10,A38,INTEGRA1:def 19;
                  ID1 <= indx(D2,D1,j) by A52,A53,XXREAL_0:2;
                  then ID1 <= len D2 by A56,XXREAL_0:2;
                  then
A61:              ID1 in dom D2 by A59,FINSEQ_3:25;
                  then
A62:              D2.ID1<D2.ID2 by A53,A57,SEQM_3:def 1;
                  indx(D2,D1,j1) < ID1 by NAT_1:13;
                  then
A63:              D2 .indx(D2,D1,j1)<D2.ID1 by A44,A61,SEQM_3:def 1;
A64:              D1.j1 = D2.indx(D2,D1,j1) by A10,A41,INTEGRA1:def 19;
A65:              not D2.ID1 in rng D1 & not D2.ID2 in rng D1
                  proof
                    assume
A66:                D2.ID1 in rng D1 or D2.ID2 in rng D1;
                    per cases by A66;
                    suppose
                      D2.ID1 in rng D1;
                      then consider n such that
A67:                  n in dom D1 and
A68:                  D1.n=D2.ID1 by PARTFUN1:3;
                      j1<n by A41,A63,A64,A67,A68,SEQ_4:137;
                      then
A69:                  j<n+1 by XREAL_1:19;
                      D2.ID1<D2.indx(D2,D1,j) by A62,A58,XXREAL_0:2;
                      then n<j by A38,A60,A67,A68,SEQ_4:137;
                      hence contradiction by A69,NAT_1:13;
                    end;
                    suppose
                      D2.ID2 in rng D1;
                      then consider n such that
A70:                  n in dom D1 and
A71:                  D1.n=D2.ID2 by PARTFUN1:3;
                      D2.indx(D2,D1,j1)<D2.ID2 by A63,A62,XXREAL_0:2;
                      then j1<n by A41,A64,A70,A71,SEQ_4:137;
                      then
A72:                  j<n+1 by XREAL_1:19;
                      n<j by A38,A58,A60,A70,A71,SEQ_4:137;
                      hence contradiction by A72,NAT_1:13;
                    end;
                  end;
                  upper_bound divset(D1,j)=D1.j by A38,A40,INTEGRA1:def 4;
                  then
A73:              upper_bound divset(D1,j)=D2. indx(D2,D1,j) by A10,A38,
INTEGRA1:def 19;
                  lower_bound divset(D1,j)=D1.j1 by A38,A40,INTEGRA1:def 4;
                  then
A74:              lower_bound divset(D1,j)=D2.indx(D2,D1,j1) by A10,A41,
INTEGRA1:def 19;
                  D2.ID2 in rng D2 by A57,FUNCT_1:def 3;
                  then
A75:              D2.ID2 in rng D by A11,A65,XBOOLE_0:def 3;
                  D2 .ID1 in rng D2 by A61,FUNCT_1:def 3;
                  then
A76:              D2.ID1 in rng D by A11,A65,XBOOLE_0:def 3;
                  D2.indx(D2,D1,j1)<=D2.ID2 by A63,A62,XXREAL_0:2;
                  then D2.ID2 in divset(D1,j) by A58,A74,A73,INTEGRA2:1;
                  then
A77:              D2.ID2 in rng D /\ divset(D1,j) by A75,XBOOLE_0:def 4;
                  D2.ID1<=D2.indx(D2,D1,j) by A62,A58,XXREAL_0:2;
                  then D2.ID1 in divset(D1,j) by A63,A74,A73,INTEGRA2:1;
                  then D2.ID1 in rng D /\ divset(D1,j) by A76,XBOOLE_0:def 4;
                  hence contradiction by A8,A38,A53,A61,A57,A77,Th5,SEQ_4:138;
                end;
A78:            1 <= indx(D2,D1,j1)+1 by A46,NAT_1:13;
                j <= len D1 by A38,FINSEQ_3:25;
                then
A79:            j <= len upper_volume(f,D1) by INTEGRA1:def 6;
A80:            1 <= j by A38,FINSEQ_3:25;
                then
A81:            mid(upper_volume(f,D1),j,j).1 = upper_volume(f,D1).j by A79,
FINSEQ_6:118;
                reconsider uv = upper_volume(f,D1).j as Element of REAL
                        by XREAL_0:def 1;
                j-'j+1 = 1 by Lm1;
                then len mid(upper_volume(f,D1),j,j) = 1 by A80,A79,
FINSEQ_6:118;
                then mid(upper_volume(f,D1),j,j) =<*uv*> by A81,FINSEQ_1:40;
                then
A82:            Sum mid(upper_volume(f,D1),j,j)= upper_volume(f,D1).j by
FINSOP_1:11;
                indx(D2,D1,j) in dom D2 by A10,A38,INTEGRA1:def 19;
                then
A83:            indx(D2,D1,j) in Seg len D2 by FINSEQ_1:def 3;
                then
A84:            1 <= indx(D2,D1,j) by FINSEQ_1:1;
                indx(D2,D1,j) in Seg len upper_volume(f,D2) by A83,
INTEGRA1:def 6;
                then
A85:            indx(D2,D1,j) <= len upper_volume(f,D2) by FINSEQ_1:1;
                then
A86:            indx(D2,D1,j1)+1 <= len upper_volume(f,D2 ) by A49,XXREAL_0:2;
                then indx(D2,D1,j1)+1 in Seg len upper_volume(f,D2) by A78,
FINSEQ_1:1;
                then
A87:            indx(D2,D1,j1)+1 in Seg len D2 by INTEGRA1:def 6;
                then
A88:            indx(D2,D1,j1)+1 in dom D2 by FINSEQ_1:def 3;
                indx(D2,D1,j)-'(indx(D2,D1,j1)+1) =indx(D2,D1,j)-(indx(
                D2,D1,j1)+1) by A49,XREAL_1:233;
                then indx(D2,D1,j)-'(indx(D2,D1,j1) +1)+1 <= 2 by A51;
                then
A89:            len mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1
                ,j))<=2 by A49,A84,A85,A78,A86,FINSEQ_6:118;
                indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 >= 0+1 by XREAL_1:6;
                then
A90:            1 <= len mid(upper_volume(f,D2),(indx(D2, D1,j1)+1),indx
                (D2,D1,j )) by A49,A84,A85,A78,A86,FINSEQ_6:118;
                now
                  per cases by A90,A89,Lm2;
                  suppose
A91:                len mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),
                    indx(D2,D1,j))=1;
                    upper_bound divset(D1,j)=D1.j by A38,A40,INTEGRA1:def 4;
                    then
A92:                upper_bound divset(D1,j)=D2. indx(D2,D1,j) by A10,A38,
INTEGRA1:def 19;
                    lower_bound divset(D1,j)=D1.j1 by A38,A40,INTEGRA1:def 4;
                    then lower_bound divset(D1,j)=D2.indx(D2,D1,j1) by A10,A41,
INTEGRA1:def 19;
                    then
A93:                divset(D1,j)=[. D2.indx(D2,D1, j1),D2.indx(D2,D1,j)
                    .] by A92,INTEGRA1:4;
A94:                delta(D1) >= 0 by Th9;
A95:                upper_bound rng f - lower_bound rng f >= 0 by A1,Lm3,
XREAL_1:48;
A96:                indx(D2,D1,j) in dom D2 by A10,A38,INTEGRA1:def 19;
                    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 A49,A84,A85,A78,A86,
FINSEQ_6:118;
                    then
A97:                indx(D2,D1,j)-(indx(D2,D1,j1)+1)=0 by A49,A91,XREAL_1:233;
                    then indx(D2,D1,j)<>1 by A45,FINSEQ_1:1;
                    then
A98:                upper_bound divset(D2,indx(D2,D1,j))=D2.indx(D2,D1,j
                    ) by A96,INTEGRA1:def 4;
                    indx(D2,D1,j)-1=indx(D2,D1,j1) by A97;
                    then lower_bound divset(D2,indx(D2,D1,j))= D2.indx(D2,D1,
                    j1) by A46,A97,A96,INTEGRA1:def 4;
                    then
A99:                divset(D2,indx(D2,D1,j))=divset(D1,j) by A93,A98,INTEGRA1:4
;
               reconsider uv = upper_volume(f,D2).(indx(D2,D1,j1)+1)
                       as Element of REAL
                  by XREAL_0:def 1;
                    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 A84,A85,A78,A86,FINSEQ_6:118;
                    then mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1
,j)) =<*uv*> by A91,FINSEQ_1:40;
                    then 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 FINSOP_1:11
                      .=(upper_bound(rng(f|divset(D2,(indx(D2,D1,j1)+1)))))
                    *vol(divset(D2,(indx(D2,D1,j1)+1))) by A88,INTEGRA1:def 6
                      .=Sum mid(upper_volume(f,D1),j,j) by A38,A82,A97,A99,
INTEGRA1:def 6;
                    hence thesis by A94,A95;
                  end;
                  suppose
A100:               len mid(upper_volume(f,D2),(indx(D2,D1,j1)+1),
                    indx(D2,D1,j))=2;
A101:               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 A84,A85,A78,A86,FINSEQ_6:118;
A102:               2+(indx(D2,D1,j1)+1)>=0+1 by XREAL_1:7;
                    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 A49,A84,A85,A78,A86,A100,
FINSEQ_6:118
                      .=UVf(D2).(2+(indx(D2,D1,j1)+1)-1) by A102,XREAL_1:233
                      .=UVf(D2).(indx(D2,D1,j1)+(1+1));
                    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 A100,A101,FINSEQ_1:44;
                    then
A103:               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_1:77;
A104:               vol(divset(D2,indx(D2,D1,j1)+1))>=0 by INTEGRA1:9;
                    upper_bound divset(D1,j)=D1.j by A38,A40,INTEGRA1:def 4;
                    then
A105:               upper_bound divset(D1,j)=D2. indx(D2,D1,j) by A10,A38,
INTEGRA1:def 19;
A106:               vol(divset(D2,indx(D2,D1,j1)+2))>=0 by INTEGRA1:9;
                    indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1=2 by A49,A84,A85,A78
,A86,A100,FINSEQ_6:118;
                    then
A107:               2=indx(D2,D1,j)-(indx(D2,D1,j1)+1)+1 by A49,XREAL_1:233
                      .=indx(D2,D1,j)-indx(D2,D1,j1);
                    then
A108:               indx (D2,D1,j1)+2 in dom D2 by A10,A38,INTEGRA1:def 19;
                    lower_bound divset(D1,j)=D1.j1 by A38,A40,INTEGRA1:def 4;
                    then lower_bound divset(D1,j)=D2.indx(D2,D1,j1) by A10,A41,
INTEGRA1:def 19;
                    then
A109:               vol(divset(D1,j)) =D2.(indx(D2,D1,j1)+2)-D2.(indx(D2
,D1,j1)+1) +D2.(indx(D2,D1,j1)+1)-D2.indx(D2,D1,j1) by A105,A107,INTEGRA1:def 5
;
                    indx(D2,D1,j1)+1 in Seg len upper_volume(f,D2) by A78,A86,
FINSEQ_1:1;
                    then indx(D2,D1,j1)+1 in Seg len D2 by INTEGRA1:def 6;
                    then
A110:               indx(D2,D1,j1)+1 in dom D2 by FINSEQ_1:def 3;
A111:               indx(D2,D1,j1)+1 <> 1 by A46,NAT_1:13;
                    then
A112:               upper_bound divset(D2,(indx( D2,D1,j1)+1))= D2.(indx
                    (D2,D1,j1)+1) by A110,INTEGRA1:def 4;
                    indx(D2,D1,j1)+1-1=indx(D2,D1,j1)+0;
                    then
A113:               lower_bound divset(D2,(indx(D2,D1,j1)+1))= D2.indx(
                    D2,D1,j1) by A110,A111,INTEGRA1:def 4;
A114:               indx(D2,D1,j1)+1+1 > 1 by A78,NAT_1:13;
                    indx(D2,D1,j1)+2-1=indx(D2,D1,j1)+1;
                    then
A115:               lower_bound divset(D2,(indx(D2,D1,j1)+2))= D2.(indx(
                    D2,D1,j1)+1) by A108,A114,INTEGRA1:def 4;
                    upper_bound divset(D2,(indx(D2,D1,j1)+2))= D2.(indx(
                    D2,D1,j1)+2) by A108,A114,INTEGRA1:def 4;
                    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 A115,A109,INTEGRA1:def 5
                      .=vol(divset(D2,indx(D2,D1,j1)+2)) +(upper_bound
divset(D2,indx(D2,D1,j1)+1)-lower_bound divset(D2, indx(D2,D1,j1)+1)) by A113
,A112;
                    then
A116:               vol(divset(D1,j)) =vol(divset(D2,indx(D2,D1,j1)+1))+
                    vol(divset(D2,indx(D2,D1,j1)+2)) by INTEGRA1:def 5;
                    then
A117:               upper_volume(f,D1).j=(upper_bound(rng(f| divset(D1,j
))))* (vol(divset(D2,indx(D2,D1,j1)+1))+vol(divset(D2,indx(D2,D1,j1)+2))) by
A38,INTEGRA1:def 6;
A118:               Sum mid(UVf(D1),j,j)-Sum mid(UVf(D2),indx(D2,D1,j1)+
1,indx(D2, D1,j)) <=(upper_bound rng f - lower_bound rng f)* (vol(divset(D2,
                    indx(D2,D1,j1)+2)) +vol(divset(D2,indx(D2,D1,j1)+1)))
                    proof
                      set ID2=indx(D2,D1,j1)+2;
                      set ID1=indx(D2,D1,j1)+1;
                      set SR = upper_bound rng(f|divset(D2,ID1)), VR = vol(
                      divset(D2,ID1));
                      set B = vol(divset(D2,ID1));
                      set C = vol(divset(D2,ID2));
                      divset(D1,j)c=A by A38,INTEGRA1:8;
                      then
A119:                 upper_bound rng(f|divset(D1,j)) <= upper_bound rng
                      f by A1,Lm4;
                      ID1 in dom D2 by A87,FINSEQ_1:def 3;
                      then divset(D2,ID1)c=A by INTEGRA1:8;
                      then upper_bound rng(f|divset(D2,ID1)) >= lower_bound
                      rng f by A1,Lm4;
                      then
A120:                 (upper_bound rng(f|divset(D2,ID1)))* vol(divset(D2
,ID1)) >=(lower_bound rng f)*vol(divset(D2,ID1)) by A104,XREAL_1:64;
                      indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1=2 by A49,A84,A85,A78
,A86,A100,FINSEQ_6:118;
                      then
A121:                 2=indx(D2,D1,j)-(indx(D2,D1,j1)+1)+1 by A49,XREAL_1:233
                        .=indx(D2,D1,j)-indx(D2,D1,j1);
A122:                 indx(D2,D1,j) in dom D2 by A10,A38,INTEGRA1:def 19;
                      then divset(D2,ID2) c= A by A121,INTEGRA1:8;
                      then
A123:                 upper_bound rng(f|divset(D2,ID2)) >= lower_bound
                      rng f by A1,Lm4;
                      reconsider A = upper_bound rng(f|divset(D1,j)) as Real;
A124:                 upper_volume(f,D1).j-A*B = A*C by A117;
                      (upper_bound rng(f|divset(D1,j)))*vol( divset(D2,
ID2)) <=(upper_bound rng f)*vol(divset(D2,ID2)) by A106,A119,XREAL_1:64;
                      then Sum mid(UVf(D1),j,j)<=(upper_bound rng(f|divset(D1
,j)))* vol(divset(D2, ID1)) +(upper_bound rng f)*vol(divset(D2,ID2)) by A82
,A124,XREAL_1:20;
                      then
A125:                 Sum mid(UVf(D1),j,j)-(upper_bound rng f)* vol(
divset(D2,ID2) ) <=(upper_bound rng(f|divset(D1,j)))*vol(divset(D2,ID1)) by
XREAL_1:20;
                      (upper_bound rng(f|divset(D1,j)))*vol(divset(D2,
ID1)) <=(upper_bound rng f)*vol(divset(D2,ID1)) by A104,A119,XREAL_1:64;
                      then Sum mid(UVf(D1),j,j)-(upper_bound rng f)* vol(
divset(D2,ID2) ) <=(upper_bound rng f)*vol(divset(D2,ID1)) by A125,XXREAL_0:2;
                      then
A126:                 Sum mid(UVf(D1),j,j) <=(upper_bound rng f)*vol(
divset(D2,ID2))+(upper_bound rng f)* vol(divset(D2,ID1)) by XREAL_1:20;
                      Sum mid(UVf(D2),ID1,indx(D2,D1,j)) =(upper_bound
rng(f|divset(D2,ID2)))*vol(divset(D2,ID2)) +UVf(D2).ID1 by A103,A122,A121,
INTEGRA1:def 6
                        .=(upper_bound rng(f|divset(D2,ID2)))* vol(divset(D2
,ID2)) +(upper_bound rng(f|divset(D2,ID1))) *vol(divset(D2,ID1)) by A88,
INTEGRA1:def 6;
                      then Sum mid(UVf(D2),ID1,indx(D2,D1,j)) - SR*VR >=(
lower_bound rng f)*vol(divset(D2,ID2)) by A106,A123,XREAL_1:64;
                      then Sum mid(UVf(D2),ID1,indx(D2,D1,j)) >=(lower_bound
                      rng f)*vol(divset(D2,ID2))+SR*VR by XREAL_1:19;
                      then Sum mid(UVf(D2),ID1,indx(D2,D1,j))-(lower_bound
                      rng f)* vol(divset(D2, ID2)) >=SR*VR by XREAL_1:19;
                      then Sum mid(UVf(D2),ID1,indx(D2,D1,j))-(lower_bound
rng f)* vol(divset(D2, ID2)) >=(lower_bound rng f)*VR by A120,XXREAL_0:2;
                      then Sum mid(UVf(D2),ID1,indx(D2,D1,j))>=(lower_bound
rng f)*vol(divset(D2, ID2))+ (lower_bound rng f)*VR by XREAL_1:19;
                      then Sum mid(UVf(D1),j,j)-Sum mid(UVf(D2),ID1,indx(D2,
D1,j)) <=(upper_bound rng f)*vol(divset(D2,ID2))+(upper_bound rng f)* vol(
divset(D2,ID1)) -((lower_bound rng f)*vol(divset(D2,ID2))+(lower_bound rng f)*
                      vol(divset(D2,ID1))) by A126,XREAL_1:13;
                      hence thesis;
                    end;
                    upper_bound rng f - lower_bound rng f >= 0 by A1,Lm3,
XREAL_1:48;
                    then (upper_bound rng f - lower_bound rng f)*( vol(divset
(D1,j))) <=(upper_bound rng f - lower_bound rng f)*delta(D1) by A38,Lm5,
XREAL_1:64;
                    hence thesis by A116,A118,XXREAL_0:2;
                  end;
                end;
                hence thesis;
              end;
              j < j+1 by NAT_1:13;
              then
A127:         j1 < j by XREAL_1:19;
              indx(D2,D1,j) in dom D2 by A10,A38,INTEGRA1:def 19;
              then
A128:         indx(D2,D1,j) in Seg len D2 by FINSEQ_1:def 3;
              then
A129:         1 <= indx(D2,D1,j) by FINSEQ_1:1;
A130:         indx(D2,D1,j1) <= len D2 by A45,FINSEQ_1:1;
              then
A131:         len (D2|indx(D2,D1,j1))=indx(D2,D1,j1) by FINSEQ_1:59;
A132:         j1 in Seg len D1 by A43,FINSEQ_1:def 3;
              then
A133:         j1 <= len D1 by FINSEQ_1:1;
              for x1 being object st x1 in rng(D1|j1)
                 holds x1 in rng(D2|indx(D2,D1,j1))
              proof
                let x1 be object;
                assume x1 in rng(D1|j1);
                then consider k such that
A134:           k in dom(D1|j1) and
A135:           x1=(D1|j1).k by PARTFUN1:3;
                k in Seg len(D1|j1) by A134,FINSEQ_1:def 3;
                then
A136:           k in Seg j1 by A133,FINSEQ_1:59;
                then
A137:           k in dom D1 by A41,RFINSEQ:6;
                k <= j1 by A136,FINSEQ_1:1;
                then D1.k <= D1.j1 by A43,A137,SEQ_4:137;
                then D2.indx(D2,D1,k) <= D1.j1 by A10,A137,INTEGRA1:def 19;
                then
A138:           D2.indx(D2,D1,k)<=D2.indx(D2,D1,j1) by A10,A43,INTEGRA1:def 19;
A139:           (D1|j1).k = D1.k by A41,A136,RFINSEQ:6;
                D1.k in rng D1 by A137,FUNCT_1:def 3;
                then x1 in rng D2 by A11,A135,A139,XBOOLE_0:def 3;
                then consider n such that
A140:           n in dom D2 and
A141:           x1=D2.n by PARTFUN1:3;
                D2.indx(D2,D1,k)=D2.n by A10,A135,A139,A137,A141,
INTEGRA1:def 19;
                then
A142:           n <= indx(D2,D1,j1) by A44,A140,A138,SEQM_3:def 1;
                1 <= n by A140,FINSEQ_3:25;
                then
A143:           n in Seg indx(D2,D1,j1) by A142,FINSEQ_1:1;
                then n in Seg len(D2|indx(D2,D1,j1)) by A130,FINSEQ_1:59;
                then
A144:           n in dom(D2|indx(D2,D1,j1)) by FINSEQ_1:def 3;
                D2.n = (D2|indx(D2,D1,j1)).n by A44,A143,RFINSEQ:6;
                hence thesis by A141,A144,FUNCT_1:def 3;
              end;
              then
A145:         rng(D1|j1) c= rng(D2|indx(D2,D1,j1));
A146:         1 <= j1 by A132,FINSEQ_1:1;
              lower_bound divset(D1,j) <= D.1 by A39,INTEGRA2:1;
              then
A147:         D1.j1 <= D.1 by A38,A40,INTEGRA1:def 4;
              for x1 being object st x1 in rng(D2|indx(D2,D1,j1))
                holds x1 in rng(D1|j1)
              proof
                let x1 be object;
                assume x1 in rng(D2|indx(D2,D1,j1));
                then consider k such that
A148:           k in dom(D2|indx(D2,D1,j1)) and
A149:           x1=(D2|indx(D2,D1,j1)).k by PARTFUN1:3;
                k in Seg len(D2|indx(D2,D1,j1)) by A148,FINSEQ_1:def 3;
                then
A150:           k in Seg indx(D2,D1,j1) by A130,FINSEQ_1:59;
                then
A151:           k in dom D2 by A44,RFINSEQ:6;
A152:           len(D1|j1) = j1 by A133,FINSEQ_1:59;
                k <= indx(D2,D1,j1) by A150,FINSEQ_1:1;
                then D2.k <= D2.indx(D2,D1,j1) by A44,A151,SEQ_4:137;
                then
A153:           D2.k <= D1.j1 by A10,A43,INTEGRA1:def 19;
A154:           D2.k in rng D1 implies D2.k in rng(D1|j1)
                proof
                  assume D2.k in rng D1;
                  then consider m such that
A155:             m in dom D1 and
A156:             D2.k = D1.m by PARTFUN1:3;
                  m in Seg len D1 by A155,FINSEQ_1:def 3;
                  then
A157:             1 <= m by FINSEQ_1:1;
A158:             m <= j1 by A41,A153,A155,A156,SEQM_3:def 1;
                  then m in Seg len(D1|j1) by A152,A157,FINSEQ_1:1;
                  then
A159:             m in dom (D1|j1) by FINSEQ_1:def 3;
                  m in Seg j1 by A157,A158,FINSEQ_1:1;
                  then D2.k = (D1|j1).m by A41,A156,RFINSEQ:6;
                  hence thesis by A159,FUNCT_1:def 3;
                end;
A160:           D2.k in rng D implies D2.k = D1.j1
                proof
                  assume D2.k in rng D;
                  then consider n such that
A161:             n in dom D and
A162:             D2.k=D.n by PARTFUN1:3;
                  1 <= n by A161,FINSEQ_3:25;
                  then D.1 <= D2.k by A37,A161,A162,SEQ_4:137;
                  then D1.j1 <= D2.k by A147,XXREAL_0:2;
                  hence thesis by A153,XXREAL_0:1;
                end;
A163:           D2.k in rng D implies D2.k in rng(D1|j1)
                proof
                  j1 in Seg len(D1|j1) by A146,A152,FINSEQ_1:1;
                  then j1 in dom(D1|j1) by FINSEQ_1:def 3;
                  then
A164:             (D1|j1).j1 in rng(D1|j1) by FUNCT_1:def 3;
                  assume
A165:             D2.k in rng D;
                  j1 in Seg j1 by A146,FINSEQ_1:1;
                  hence thesis by A41,A160,A165,A164,RFINSEQ:6;
                end;
                D2.k in rng D2 by A151,FUNCT_1:def 3;
                hence thesis by A11,A44,A149,A150,A163,A154,RFINSEQ:6
,XBOOLE_0:def 3;
              end;
              then rng(D2|indx(D2,D1,j1)) c= rng (D1|j1);
              then
A166:         rng (D2|indx(D2,D1,j1)) = rng (D1|j1) by A145,XBOOLE_0:def 10;
              mid(D1,1,j1) is increasing by A38,A40,A146,INTEGRA1:7,35;
              then
A167:         D1|j1 is increasing by A146,FINSEQ_6:116;
              then
A168:         D2|indx(D2,D1,j1)=D1|j1 by A47,A166,Th6;
A169:         for k st 1 <= k & k <= j1 holds k=indx(D2,D1,k)
              proof
                let k;
                assume that
A170:           1 <= k and
A171:           k <= j1;
                assume
A172:           k<>indx(D2,D1,k);
                now
                  per cases by A172,XXREAL_0:1;
                  suppose
A173:               k > indx(D2,D1,k);
                    k <= len D1 by A133,A171,XXREAL_0:2;
                    then
A174:               k in dom D1 by A170,FINSEQ_3:25;
                    then indx(D2,D1,k) in dom D2 by A10,INTEGRA1:def 19;
                    then indx(D2,D1,k) in Seg len D2 by FINSEQ_1:def 3;
                    then
A175:               1<=indx(D2,D1,k) by FINSEQ_1:1;
A176:               indx(D2,D1,k) < j1 by A171,A173,XXREAL_0:2;
                    then
A177:               indx(D2,D1,k) in Seg j1 by A175,FINSEQ_1:1;
                    indx(D2,D1,k)<= indx(D2,D1,j1) by A10,A41,A171,A174,Th7;
                    then indx(D2,D1,k) in Seg indx(D2,D1,j1) by A175,FINSEQ_1:1
;
                    then
A178:               (D2|indx(D2,D1,j1)).indx(D2,D1,k)= D2.indx(D2,D1,k)
                    by A44,RFINSEQ:6;
                    indx(D2,D1,k) <= len D1 by A133,A176,XXREAL_0:2;
                    then indx(D2,D1,k) in dom D1 by A175,FINSEQ_3:25;
                    then
A179:               D1.k > D1.indx(D2,D1,k) by A173,A174,SEQM_3:def 1;
                    D1.k=D2. indx(D2,D1,k) by A10,A174,INTEGRA1:def 19;
                    hence contradiction by A41,A168,A178,A179,A177,RFINSEQ:6;
                  end;
                  suppose
A180:               k < indx(D2,D1,k);
                    k <= len D1 by A133,A171,XXREAL_0:2;
                    then
A181:               k in dom D1 by A170,FINSEQ_3:25;
                    then indx(D2,D1,k) <= indx(D2,D1,j1) by A10,A41,A171,Th7;
                    then
A182:               k <= indx(D2,D1,j1) by A180,XXREAL_0:2;
                    then k <= len D2 by A130,XXREAL_0:2;
                    then
A183:               k in dom D2 by A170,FINSEQ_3:25;
                    k in Seg j1 by A170,A171,FINSEQ_1:1;
                    then
A184:               D1.k = (D1|j1).k by A43,RFINSEQ:6;
                    indx(D2,D1,k) in dom D2 by A10,A181,INTEGRA1:def 19;
                    then
A185:               D2.k < D2.indx(D2,D1,k) by A180,A183,SEQM_3:def 1;
A186:               k in Seg indx(D2,D1,j1) by A170,A182,FINSEQ_1:1;
                    D1.k=D2. indx(D2,D1,k) by A10,A181,INTEGRA1:def 19;
                    hence contradiction by A44,A168,A184,A185,A186,RFINSEQ:6;
                  end;
                end;
                hence contradiction;
              end;
A187:         for k be Nat 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
                indx(D2,D1,j1) in Seg len D2 by A44,FINSEQ_1:def 3;
                then indx(D2,D1,j1) in Seg len upper_volume(f,D2) by
INTEGRA1:def 6;
                then
A188:           indx(D2,D1,j1) in dom upper_volume(f,D2) by FINSEQ_1:def 3;
                let k be Nat;
                assume that
A189:           1 <= k and
A190:           k <= len(upper_volume(f,D1)|j1);
A191:           len(upper_volume(f,D1)) = len D1 by INTEGRA1:def 6;
                then
A192:           k <= j1 by A133,A190,FINSEQ_1:59;
                then
A193:           k in Seg j1 by A189,FINSEQ_1:1;
                then indx(D2,D1,k) in Seg j1 by A169,A189,A192;
                then
A194:           indx(D2,D1,k) in Seg indx(D2,D1,j1) by A146,A169;
                then indx(D2,D1,k)<=indx(D2,D1,j1) by FINSEQ_1:1;
                then
A195:           indx(D2,D1,k)<=len D2 by A130,XXREAL_0:2;
                k <= len D1 by A133,A192,XXREAL_0:2;
                then
A196:           k in Seg len D1 by A189,FINSEQ_1:1;
                then
A197:           k in dom D1 by FINSEQ_1:def 3;
                then
A198:           indx(D2,D1,k) in dom D2 by A10,INTEGRA1:def 19;
A199:           D1.k=D2.indx (D2,D1,k) by A10,A197,INTEGRA1:def 19;
A200:           lower_bound divset(D1,k)=lower_bound divset(D2, indx(D2,
D1,k)) & upper_bound divset(D1,k)=upper_bound divset(D2,indx(D2,D1,k))
                proof
                  per cases;
                  suppose
A201:               k=1;
                    then
A202:               upper_bound divset(D1,k)=D1.k by A197,INTEGRA1:def 4;
A203:               lower_bound divset(D1,k)=lower_bound A by A197,A201,
INTEGRA1:def 4;
                    indx(D2,D1,k)=1 by A146,A169,A201;
                    hence thesis by A198,A199,A203,A202,INTEGRA1:def 4;
                  end;
                  suppose
A204:               k<>1;
                    then reconsider k1=k-1 as Element of NAT by A197,INTEGRA1:7
;
                    k <= k+1 by NAT_1:11;
                    then k1 <= k by XREAL_1:20;
                    then
A205:               k1 <= j1 by A192,XXREAL_0:2;
A206:               k-1 in dom D1 by A197,A204,INTEGRA1:7;
                    then 1 <= k1 by FINSEQ_3:25;
                    then k1=indx(D2,D1,k1) by A169,A205;
                    then
A207:               D2.(indx(D2,D1,k)-1)= D2.indx(D2,D1,k1) by A169,A189,A192
,A193;
A208:               indx(D2,D1,k)<>1 by A169,A189,A192,A193,A204;
                    then
A209:               lower_bound divset(D2,indx(D2,D1,k))= D2.(indx(D2,D1
                    ,k)-1) by A198,INTEGRA1:def 4;
A210:               upper_bound divset(D2,indx(D2,D1,k))=D2.indx(D2,D1,k
                    ) by A198,A208,INTEGRA1:def 4;
A211:               upper_bound divset(D1,k)=D1.k by A197,A204,INTEGRA1:def 4;
                    lower_bound divset(D1,k)=D1.(k-1) by A197,A204,
INTEGRA1:def 4;
                    hence thesis by A10,A197,A211,A206,A209,A210,A207,
INTEGRA1:def 19;
                  end;
                end;
                divset(D2,indx(D2,D1,k))= [. lower_bound divset(D2, indx
(D2,D1,k)), upper_bound divset(D2,indx(D2,D1,k)).] by INTEGRA1:4;
                then
A212:           divset(D1,k)=divset(D2,indx(D2,D1,k)) by A200,INTEGRA1:4;
A213:           k in dom D1 by A196,FINSEQ_1:def 3;
                j1 in Seg len(upper_volume(f,D1)) by A41,A191,FINSEQ_1:def 3;
                then j1 in dom(upper_volume(f,D1)) by FINSEQ_1:def 3;
                then
A214:           (upper_volume(f,D1)|j1).k = upper_volume(f,D1).k by A193,
RFINSEQ:6
                  .=(upper_bound(rng(f|divset(D2,indx(D2,D1,k)))))*vol(
                divset(D2,indx(D2,D1,k))) by A213,A212,INTEGRA1:def 6;
                1<=indx(D2,D1,k) by A169,A189,A192,A193;
                then indx(D2,D1,k) in Seg len D2 by A195,FINSEQ_1:1;
                then
A215:           indx(D2,D1,k) in dom 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 A169,A189,A192,A193
                  .=upper_volume(f,D2).indx(D2,D1,k) by A194,A188,RFINSEQ:6
                  .=(upper_bound(rng(f|divset(D2,indx(D2,D1,k)))))*vol(
                divset(D2,indx(D2,D1,k))) by A215,INTEGRA1:def 6;
                hence thesis by A214;
              end;
              indx(D2,D1,j1) in dom D2 by A10,A43,INTEGRA1:def 19;
              then indx(D2,D1,j1) <= len D2 by FINSEQ_3:25;
              then
A216:         indx(D2,D1,j1) <= len upper_volume(f,D2) by INTEGRA1:def 6;
              j1 <= len D1 by A43,FINSEQ_3:25;
              then
A217:         j1 <= len upper_volume(f,D1) by INTEGRA1:def 6;
              len (D2|indx(D2,D1,j1))=len (D1|j1) by A47,A167,A166,Th6;
              then indx(D2,D1,j1) =j1 by A133,A131,FINSEQ_1:59;
              then len(upper_volume(f,D1)|j1)=indx(D2,D1,j1) by A217,
FINSEQ_1:59;
              then len(upper_volume(f,D1)|j1)=len(upper_volume(f, D2)|indx(D2
              ,D1,j1)) by A216,FINSEQ_1:59;
              then
A218:         upper_volume(f,D2)|indx(D2,D1,j1)=upper_volume (f,D1)|j1
              by A187,FINSEQ_1:14;
A219:         j in Seg len D1 by A38,FINSEQ_1:def 3;
              then
A220:         1 <= j by FINSEQ_1:1;
              indx(D2,D1,j) in Seg len UVf(D2) by A128,INTEGRA1:def 6;
              then
A221:         indx(D2,D1,j) in dom UVf(D2) by FINSEQ_1:def 3;
              indx(D2,D1,j) <= len D2 by A128,FINSEQ_1:1;
              then
A222:         indx(D2,D1,j) <= len UVf(D2) by INTEGRA1:def 6;
              j in Seg len UVf(D1) by A219,INTEGRA1:def 6;
              then
A223:         j in dom UVf(D1) by FINSEQ_1:def 3;
              j <= len D1 by A219,FINSEQ_1:1;
              then
A224:         j <= len UVf(D1) by INTEGRA1:def 6;
              j1 in Seg len D1 by A41,FINSEQ_1:def 3;
              then j1 in Seg len UVf(D1) by INTEGRA1:def 6;
              then j1 in dom UVf(D1) by FINSEQ_1:def 3;
              then PUf(D1,j1)=Sum(UVf(D1)|j1) by INTEGRA1:def 20;
              then PUf(D1,j1)+Sum mid(UVf(D1),j,j) =Sum((UVf(D1)|j1)^mid(UVf(
              D1),j,j)) by RVSUM_1:75
                .=Sum(mid(UVf(D1),1,j1)^mid(UVf(D1),j1+1,j)) by A146,
FINSEQ_6:116
                .=Sum(mid(UVf(D1),1,j)) by A146,A224,A127,INTEGRA2:4
                .=Sum(UVf(D1)|j) by A220,FINSEQ_6:116;
              then
A225:         PUf(D1,j1)+Sum mid(upper_volume(f,D1),j,j)=PUf(D1,j) by A223,
INTEGRA1:def 20;
              indx(D2,D1,j1) in Seg len D2 by A44,FINSEQ_1:def 3;
              then indx(D2,D1,j1) in Seg len UVf(D2) by INTEGRA1:def 6;
              then indx(D2,D1,j1) in dom UVf(D2) by FINSEQ_1:def 3;
              then PUf(D2,indx(D2,D1,j1)) =Sum(UVf(D2)|indx(D2,D1,j1)) by
INTEGRA1:def 20;
              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_1:75
                .=Sum(mid(UVf(D2),1,indx(D2,D1,j1)) ^mid(UVf(D2),indx(D2,D1,
              j1)+1,indx(D2,D1,j))) by A46,FINSEQ_6:116
                .=Sum(mid(UVf(D2),1,indx(D2,D1,j))) by A46,A48,A222,INTEGRA2:4
                .=Sum(UVf(D2)|indx(D2,D1,j)) by A129,FINSEQ_6:116;
              then
A226:         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)) by A221,INTEGRA1:def 20;
              indx(D2,D1,j1) in Seg len D2 by A44,FINSEQ_1:def 3;
              then indx(D2,D1,j1) in Seg len upper_volume(f,D2) by
INTEGRA1:def 6;
              then indx(D2,D1,j1) in dom upper_volume(f,D2) by FINSEQ_1:def 3;
              then PUf(D2,indx(D2,D1,j1))=Sum(upper_volume(f,D2)|indx(D2,D1,
              j1)) by INTEGRA1:def 20
                .=PUf(D1,j1) by A218,A42,INTEGRA1:def 20;
              hence thesis by A50,A226,A225;
            end;
            hence thesis by A38,A39;
          end;
          reconsider i as non zero Element of NAT by A15,FINSEQ_1:1;
A227:     for i being non zero Nat st P[i] holds P[i+1]
          proof
            let i be non zero Nat;
A228:       i>=1 by NAT_1:14;
            assume
A229:       P[i];
            P[i+1]
            proof
A230:         i <= i+1 by NAT_1:11;
              assume
A231:         i+1 in dom D;
              then consider j such that
A232:         j in dom D1 and
A233:         D.(i+1) in divset(D1,j) by Th3,INTEGRA1:6;
A234:         D2.indx(D2,D1,j)=D1.j by A10,A232,INTEGRA1:def 19;
              i+1 in Seg len D by A231,FINSEQ_1:def 3;
              then i+1 <= len D by FINSEQ_1:1;
              then i <= len D by A230,XXREAL_0:2;
              then
A235:         i in Seg len D by A228,FINSEQ_1:1;
              then
A236:         i in dom D by FINSEQ_1:def 3;
A237:         indx(D2,D1,j) in dom D2 by A10,A232,INTEGRA1:def 19;
              then
A238:         1 <= indx(D2,D1,j) by FINSEQ_3:25;
A239:         indx(D2,D1,j) <= len D2 by A237,FINSEQ_3:25;
              then
A240:         indx(D2,D1,j) <= len UVf(D2) by INTEGRA1:def 6;
              consider n1 being Element of NAT such that
A241:         n1 in dom D1 and
A242:         D.i in divset(D1,n1) and
A243:         PUf(D1,n1)-PUf(D2,indx(D2,D1,n1))<= i*(upper_bound rng
              f- lower_bound rng f)*delta(D1) by A229,A235,FINSEQ_1:def 3;
A244:         1 <= n1+1 by NAT_1:12;
A245:         n1 < j
              proof
                assume
A246:           n1 >= j;
                now
                  per cases by A246,XXREAL_0:1;
                  suppose
A247:               n1=j;
                    D.i in rng D by A236,FUNCT_1:def 3;
                    then
A248:               D.i in rng D /\ divset(D1,j) by A242,A247,XBOOLE_0:def 4;
                    D.(i+1) in rng D by A231,FUNCT_1:def 3;
                    then
A249:               D.(i+1) in rng D /\ divset(D1,j) by A233,XBOOLE_0:def 4;
                    i+1 > i by XREAL_1:29;
                    hence contradiction by A8,A231,A232,A236,A248,A249,Th5,
SEQ_4:138;
                  end;
                  suppose
                    n1>j;
                    then
A250:               n1>=j+1 by NAT_1:13;
                    then
A251:               n1-1 >= j by XREAL_1:19;
                    1 <= j by A232,FINSEQ_3:25;
                    then 1+1 <= j+1 by XREAL_1:6;
                    then
A252:               n1 <> 1 by A250,XXREAL_0:2;
                    then n1-1 in dom D1 by A241,INTEGRA1:7;
                    then
A253:               D1.j <= D1.(n1-1) by A232,A251,SEQ_4:137;
A254:               upper_bound divset(D1,j)=D1.j
                    proof
                      per cases;
                      suppose
                        j=1;
                        hence thesis by A232,INTEGRA1:def 4;
                      end;
                      suppose
                        j<>1;
                        hence thesis by A232,INTEGRA1:def 4;
                      end;
                    end;
A255:               lower_bound divset(D1,n1) <= D.i by A242,INTEGRA2:1;
                    lower_bound divset(D1,n1)=D1.(n1-1) by A241,A252,
INTEGRA1:def 4;
                    then
A256:               D.i >= D1.j by A255,A253,XXREAL_0:2;
A257:               i < i+1 by XREAL_1:29;
                    D.(i+1)<=upper_bound divset(D1,j) by A233,INTEGRA2:1;
                    then D.i >= D.(i+1) by A254,A256,XXREAL_0:2;
                    hence contradiction by A231,A236,A257,SEQM_3:def 1;
                  end;
                end;
                hence thesis;
              end;
              then
A258:         n1+1 <= j by NAT_1:13;
A259:         1 <= n1 by A241,FINSEQ_3:25;
A260:         D2.indx(D2,D1,n1) = D1.n1 by A10,A241,INTEGRA1:def 19;
A261:         1 <= j by A232,FINSEQ_3:25;
A262:         indx(D2,D1,n1) in dom D2 by A10,A241,INTEGRA1:def 19;
              then
A263:         1 <= indx(D2,D1,n1) by FINSEQ_3:25;
A264:         j <= len D1 by A232,FINSEQ_3:25;
              then
A265:         n1+1 <= len D1 by A258,XXREAL_0:2;
              then
A266:         n1+1 in dom D1 by A244,FINSEQ_3:25;
              then
A267:         D2.indx(D2,D1,n1+ 1)=D1.(n1+1) by A10,INTEGRA1:def 19;
A268:         j <= len UVf(D1) by A264,INTEGRA1:def 6;
              then j in Seg len UVf(D1) by A261,FINSEQ_1:1;
              then
A269:         j in dom UVf(D1) by FINSEQ_1:def 3;
A270:         indx(D2,D1,n1+1) in dom D2 by A10,A266,INTEGRA1:def 19;
              then
A271:         1 <= indx(D2,D1,n1+1) by FINSEQ_3:25;
              n1 in Seg len D1 by A241,FINSEQ_1:def 3;
              then n1 in Seg len UVf(D1) by INTEGRA1:def 6;
              then n1 in dom UVf(D1) by FINSEQ_1:def 3;
              then PUf(D1,n1)=Sum(UVf(D1)|n1) by INTEGRA1:def 20
                .=Sum mid(UVf(D1),1,n1) by A259,FINSEQ_6:116;
              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_1:75
                .=Sum mid(UVf(D1),1,j) by A245,A259,A268,INTEGRA2:4
                .=Sum(UVf(D1)|j) by A261,FINSEQ_6:116;
              then
A272:         PUf(D1,j)=PUf(D1,n1)+Sum mid(UVf(D1),n1+1,j) by A269,
INTEGRA1:def 20;
              indx(D2,D1,j) in Seg len D2 by A237,FINSEQ_1:def 3;
              then indx(D2,D1,j) in Seg len UVf(D2) by INTEGRA1:def 6;
              then
A273:         indx(D2,D1,j) in dom UVf(D2) by FINSEQ_1:def 3;
A274:         indx(D2,D1,n1+1) <= len D2 by A270,FINSEQ_3:25;
              D1.(n1+1) <= D1.j by A232,A258,A266,SEQ_4:137;
              then
A275:         indx(D2,D1,n1+1) <= indx(D2,D1,j) by A270,A267,A237,A234,
SEQM_3:def 1;
              then 1+indx(D2,D1,n1+1) <= indx(D2,D1,j)+1 by XREAL_1:6;
              then 1 <= indx(D2,D1,j)+1-indx(D2,D1,n1+1) by XREAL_1:19;
              then
A276:         mid(D2,indx(D2,D1,n1+1),indx(D2,D1,j)).1 =D2.(1-1+indx(D2,
              D1,n1+1)) by A275,A271,A239,FINSEQ_6:122
                .=D1.(n1+1) by A10,A266,INTEGRA1:def 19;
A277:         n1 >= 1 by A241,FINSEQ_3:25;
A278:         j-n1 >= 1 by A258,XREAL_1:19;
              Sum mid(UVf(D1),n1+1,j)-Sum mid(UVf(D2),indx(D2,D1,n1)+1,
              indx(D2,D1,j)) <= (upper_bound rng f-lower_bound rng f)*delta(D1)
              proof
                per cases by A258,XXREAL_0:1;
                suppose
A279:             n1+1=j;
A280:             indx(D2,D1,j)-indx(D2,D1,n1)<=2
                  proof
A281:               upper_bound divset(D1,j) = D1.j by A232,A245,A277,
INTEGRA1:def 4;
A282:               lower_bound divset(D1,j) = D1.(j-1) by A232,A245,A277,
INTEGRA1:def 4;
A283:               1 <= indx(D2,D1,n1)+1 by A263,NAT_1:13;
                    assume indx(D2,D1,j) - indx(D2,D1,n1) > 2;
                    then
A284:               indx(D2,D1,n1)+2 < indx(D2,D1,j) by XREAL_1:20;
                    then
A285:               indx(D2,D1,n1)+2 <= len D2 by A239,XXREAL_0:2;
A286:               indx(D2,D1,n1)+1 < indx(D2,D1,n1)+2 by XREAL_1:6;
                    then
A287:               indx(D2,D1,n1) < indx(D2,D1,n1)+2 by NAT_1:13;
                    then 1 <= indx(D2,D1,n1)+2 by A263,XXREAL_0:2;
                    then
A288:               indx(D2,D1,n1)+2 in dom D2 by A285,FINSEQ_3:25;
                    then
A289:               D2.indx(D2,D1,j) >= D2.(indx(D2,D1,n1)+2) by A237,A284,
SEQ_4:137;
A290:               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 Element of NAT such that
A291:                 k1 in dom D1 and
A292:                 D2.(indx(D2,D1,n1)+2) = D1.k1 by PARTFUN1:3;
                      D2.(indx(D2,D1,n1)+2) < D2.indx(D2,D1,j) by A237,A284
,A288,SEQM_3:def 1;
                      then
A293:                 k1 < j by A232,A234,A291,A292,SEQ_4:137;
                      D2.indx(D2,D1,n1) < D2.(indx(D2,D1,n1)+2) by A262,A287
,A288,SEQM_3:def 1;
                      then n1 < k1 by A241,A260,A291,A292,SEQ_4:137;
                      hence contradiction by A279,A293,NAT_1:13;
                    end;
                    D2.(indx(D2,D1,n1)+2) in rng D2 by A288,FUNCT_1:def 3;
                    then
A294:               D2.(indx(D2,D1,n1)+2) in rng D by A11,A290,XBOOLE_0:def 3;
A295:               lower_bound divset(D1,j) = D1.(j-1) by A232,A245,A277,
INTEGRA1:def 4;
A296:               upper_bound divset(D1,j) = D1.j by A232,A245,A277,
INTEGRA1:def 4;
                    D2.(indx(D2,D1,n1)+2) >= D2.indx(D2,D1,n1) by A262,A287
,A288,SEQ_4:137;
                    then D2.(indx(D2,D1,n1)+2) in divset(D1,j) by A260,A234
,A279,A295,A281,A289,INTEGRA2:1;
                    then
A297:               D2.(indx(D2,D1,n1)+2) in rng D /\ divset(D1,j) by A294,
XBOOLE_0:def 4;
A298:               indx(D2,D1,n1)+1 < indx(D2,D1,j) by A284,A286,XXREAL_0:2;
                    then indx(D2,D1,n1)+1 <= len D2 by A239,XXREAL_0:2;
                    then
A299:               indx(D2,D1,n1)+1 in dom D2 by A283,FINSEQ_3:25;
                    then
A300:               D2.indx(D2,D1,j) >= D2.(indx(D2,D1,n1)+1) by A237,A298,
SEQ_4:137;
A301:               indx(D2,D1,n1) < indx(D2,D1,n1)+1 by NAT_1:13;
A302:               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 Element of NAT such that
A303:                 k1 in dom D1 and
A304:                 D2.(indx(D2,D1,n1)+1) = D1.k1 by PARTFUN1:3;
                      D2.(indx(D2,D1,n1)+1) < D2.indx(D2,D1,j) by A237,A298
,A299,SEQM_3:def 1;
                      then
A305:                 k1 < j by A232,A234,A303,A304,SEQ_4:137;
                      D2.indx(D2,D1,n1) < D2.(indx(D2,D1,n1)+1) by A262,A301
,A299,SEQM_3:def 1;
                      then n1 < k1 by A241,A260,A303,A304,SEQ_4:137;
                      hence contradiction by A279,A305,NAT_1:13;
                    end;
                    D2.(indx(D2,D1,n1)+1) in rng D2 by A299,FUNCT_1:def 3;
                    then
A306:               D2.(indx(D2,D1,n1)+1) in rng D by A11,A302,XBOOLE_0:def 3;
                    D2.(indx(D2,D1,n1)+1) >= D2.indx(D2,D1,n1) by A262,A301
,A299,SEQ_4:137;
                    then D2.(indx(D2,D1,n1)+1) in divset(D1,j) by A260,A234
,A279,A282,A296,A300,INTEGRA2:1;
                    then D2.(indx(D2,D1,n1)+1) in rng D /\ divset(D1,j) by A306
,XBOOLE_0:def 4;
                    then D2.(indx(D2,D1,n1)+1) = D2.(indx(D2,D1,n1)+2) by A8
,A232,A297,Th5;
                    hence contradiction by A286,A299,A288,SEQM_3:def 1;
                  end;
A307:             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
A308:               indx(D2,D1,n1)+1+1 <= indx(D2, D1,j) by NAT_1:13;
                    indx(D2,D1,n1)+2 >= indx(D2,D1,j) by A280,XREAL_1:20;
                    hence thesis by A308,XXREAL_0:1;
                  end;
A309:             1<=indx(D2,D1,n1)+1 by NAT_1:12;
A310:             indx(D2,D1,j) <= len UVf(D2) by A239,INTEGRA1:def 6;
                  D1.n1 < D1.j by A232,A241,A245,SEQM_3:def 1;
                  then
A311:             indx(D2,D1,n1)<indx(D2,D1,j) by A262,A260,A237,A234,SEQ_4:137
;
                  then
A312:             indx(D2,D1,n1)+1 <= indx(D2,D1,j) by NAT_1:13;
                  then indx(D2,D1,n1)+1 <= len D2 by A239,XXREAL_0:2;
                  then indx(D2,D1,n1)+1 <= len UVf(D2) by INTEGRA1:def 6;
                  then
A313:             len mid(UVf(D2),indx(D2,D1,n1)+1,indx( D2,D1,j)) =indx
(D2,D1,j)-'(indx(D2,D1,n1)+1)+1 by A238,A312,A309,A310,FINSEQ_6:118
                    .=indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1 by A312,XREAL_1:233
                    .=indx(D2,D1,j)-indx(D2,D1,n1);
                  indx(D2,D1,n1)+1 <= indx(D2,D1,j) by A311,NAT_1:13;
                  then
A314:             indx(D2,D1,n1)+1 = indx(D2,D1,j) or indx(D2,D1,n1)+1 <
                  indx(D2,D1,j) by XXREAL_0:1;
A315:             Sum mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) >=(
                  lower_bound rng f)*vol(divset(D1,n1+1))
                  proof
                    now
                      per cases by A314,A307;
                      suppose
A316:                   indx(D2,D1,j)-indx(D2,D1,n1)=1;
                        indx(D2,D1,n1)+1 >= 1+1 by A263,XREAL_1:6;
                        then
A317:                   indx(D2,D1,n1)+1 <>1;
                        then upper_bound divset(D2,indx(D2,D1,n1)+1)=D2.(indx
                        (D2,D1,n1)+1) by A237,A316,INTEGRA1:def 4;
                        then
A318:                   upper_bound divset(D2,indx(D2,D1,n1)+1)=D1.j by A10
,A232,A316,INTEGRA1:def 19;
                        lower_bound divset(D2,indx(D2,D1,n1)+1)=D2.(indx
                        (D2,D1,n1)+1-1) by A237,A316,A317,INTEGRA1:def 4;
                        then
A319:                   lower_bound divset(D2,indx(D2, D1,n1)+1)=D1.n1
                        by A10,A241,INTEGRA1:def 19;
                        lower_bound divset(D1,n1+1)=D1.(n1+1-1) by A245,A277
,A266,A279,INTEGRA1:def 4;
                        then
A320:                   divset(D2,indx(D2,D1,n1)+1)=divset(D1,n1+1) by A245
,A277,A266,A279,A319,A318,INTEGRA1:def 4;
A321:                   vol(divset(D2,indx(D2,D1,n1)+1))>=0 by INTEGRA1:9;
           reconsider UV = UVf(D2).(indx(D2,D1,n1)+1) as Element of REAL
                     by XREAL_0:def 1;
                        1=indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1 by A316;
                        then mid(UVf(D2),indx(D2,D1,n1) +1,indx(D2,D1,j)).1 =
UVf(D2).(1+(indx(D2,D1,n1)+1)-1) by A309,A310,FINSEQ_6:122
                          .=UVf(D2).(indx(D2,D1,n1)+1);
                        then
A322:                   mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j))
                    =<*UV*> by A313,A316,FINSEQ_1:40;
                        UVf(D2).(indx(D2,D1,n1)+1) =(upper_bound rng(f|
divset(D2,indx(D2,D1,n1)+1))) *vol(divset(D2,indx(D2,D1,n1)+1)) by A237,A316,
INTEGRA1:def 6;
                        then UVf(D2).(indx(D2,D1,n1)+1) >=(lower_bound rng f)
*vol(divset(D1,n1+1)) by A1,A237,A316,A320,A321,Th19,XREAL_1:64;
                        hence thesis by A322,FINSOP_1:11;
                      end;
                      suppose
A323:                   indx(D2,D1,j)-indx(D2,D1,n1)=2;
                        indx(D2,D1,n1)+2 >= 2+1 by A263,XREAL_1:6;
                        then
A324:                   indx(D2,D1,n1)+2 <> 1;
                        then
A325:                   upper_bound divset(D2,indx(D2,D1,n1)+2)= D2.indx
                        (D2,D1,j) by A237,A323,INTEGRA1:def 4;
                        indx(D2,D1,n1)+2-1=indx(D2,D1,n1)+1;
                        then lower_bound divset(D2,indx(D2,D1,n1)+2) = D2.(
                        indx(D2,D1,n1)+1) by A237,A323,A324,INTEGRA1:def 4;
                        then
A326:                   vol(divset(D2,indx(D2,D1,n1)+2)) =D1.j-D2.(indx(
                        D2,D1,n1)+1) by A234,A325,INTEGRA1:def 5;
A327:                   upper_bound divset(D1,n1+1)=D1.(n1+1) by A245,A277,A266
,A279,INTEGRA1:def 4;
                        lower_bound divset(D1,n1+1)=D1.(n1+1-1) by A245,A277
,A266,A279,INTEGRA1:def 4;
                        then
A328:                   vol(divset(D1,n1+1))=D1.(n1+1)-D1.n1 by A327,
INTEGRA1:def 5;
A329:                   vol(divset(D2,indx(D2,D1,n1)+2)) >= 0 by INTEGRA1:9;
A330:                   indx(D2,D1,j) <= len UVf(D2) by A239,INTEGRA1:def 6;
A331:                   vol(divset(D2,indx(D2,D1,n1)+1)) >= 0 by INTEGRA1:9;
A332:                   1 <= indx(D2,D1,n1)+1 by NAT_1:12;
A333:                   indx(D2,D1,n1)+1 <= indx(D2,D1,n1)+2 by XREAL_1:6;
                        then indx(D2,D1,n1)+1 <= len D2 by A239,A323,XXREAL_0:2
;
                        then
A334:                   indx(D2,D1,n1)+1 in dom D2 by A332,FINSEQ_3:25;
                        then UVf(D2).(indx(D2,D1,n1 )+1) =(upper_bound rng(f|
divset(D2,indx(D2,D1,n1)+1))) *vol(divset(D2,indx(D2,D1,n1)+1)) by
INTEGRA1:def 6;
                        then
A335:                   UVf(D2).(indx(D2,D1,n1)+1) >= (lower_bound rng f
)*vol(divset(D2,indx(D2, D1,n1)+1)) by A1,A334,A331,Th19,XREAL_1:64;
                        indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1=1+1 by A323;
                        then
A336:                   mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)).2 =
UVf(D2).(2+(indx(D2,D1,n1)+1)-1) by A332,A333,A330,FINSEQ_6:122
                          .=UVf(D2).(indx(D2,D1,n1)+0+2);
                        indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1>=1 by A323;
                        then mid(UVf(D2),indx(D2,D1,n1) +1,indx(D2,D1,j)).1 =
UVf(D2).(1+(indx(D2,D1,n1)+1)-1) by A323,A332,A333,A330,FINSEQ_6:122
                          .=UVf(D2).(indx(D2,D1,n1)+1);
                        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 A313,A323,A336,
FINSEQ_1:44;
                        then
A337:                   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_1:77;
A338:                   indx(D2,D1,n1)+1 > 1 by A263,NAT_1:13;
                        then
A339:                   upper_bound divset(D2,indx(D2,D1,n1)+1)=D2.(indx
                        (D2,D1,n1)+1) by A334,INTEGRA1:def 4;
                        lower_bound divset(D2,indx(D2,D1,n1)+1) = D2.(
                        indx(D2,D1,n1)+1-1) by A334,A338,INTEGRA1:def 4;
                        then
A340:                   vol(divset(D2,indx(D2,D1,n1)+1)) =D2.(indx(D2,D1
                        ,n1)+1)-D1.n1 by A260,A339,INTEGRA1:def 5;
                        UVf(D2).(indx(D2,D1,n1 )+2) =(upper_bound rng(f|
divset(D2,indx(D2,D1,n1)+2))) *vol(divset(D2,indx(D2,D1,n1)+2)) by A237,A323,
INTEGRA1:def 6;
                        then UVf(D2).(indx(D2,D1,n1)+2) >= (lower_bound rng f
)*vol(divset(D2,indx(D2, D1,n1)+2)) by A1,A237,A323,A329,Th19,XREAL_1:64;
                        then Sum mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j))
>= (lower_bound rng f)*vol(divset(D2,indx(D2,D1,n1)+1)) + (lower_bound rng f)*
vol(divset(D2,indx(D2,D1,n1)+2)) by A337,A335,XREAL_1:7;
                        hence thesis by A279,A340,A326,A328;
                      end;
                    end;
                    hence thesis;
                  end;
A341:             n1+1 <= len UVf(D1) by A265,INTEGRA1:def 6;
                  j-'(n1+1)+1=j-(n1+1)+1 by A279,XREAL_1:233;
                  then
A342:             len mid(UVf(D1),n1+1,j)= 1 by A244,A279,A341,FINSEQ_6:118;
 reconsider uv =(upper_bound rng(f|divset(D1,n1
                  +1)))* vol(divset(D1,n1+1)) as Element of REAL
                      by XREAL_0:def 1;
                  (n1+1)+1 <= j+1 by A258,XREAL_1:6;
                  then 1 <= j+1-(n1+1) by XREAL_1:19;
                  then mid(UVf(D1),n1+1,j).1 = UVf(D1).(1+(n1+1)-1) by A244
,A279,A341,FINSEQ_6:122
                    .=(upper_bound rng(f|divset(D1,n1+1)))*vol(divset(D1,n1+
                  1)) by A266,INTEGRA1:def 6;
                  then mid(UVf(D1),n1+1,j) =<*uv*> by A342,FINSEQ_1:40;
                  then
A343:             Sum mid(UVf(D1),n1+1,j) =(upper_bound rng(f|divset(D1,
                  n1+1)))*vol(divset(D1,n1+1)) by FINSOP_1:11;
                  divset(D1,n1+1) c= A by A266,INTEGRA1:8;
                  then
A344:             upper_bound rng(f|divset(D1,n1+1)) <= upper_bound rng
                  f by A1,Lm4;
                  n1+1 in Seg len D1 by A266,FINSEQ_1:def 3;
                  then n1+1 in Seg len upper_volume(chi(A,A),D1) by
INTEGRA1:def 6;
                  then
A345:             n1+1 in dom upper_volume(chi(A,A),D1) by FINSEQ_1:def 3;
                  vol(divset(D1,n1+1))=upper_volume( chi(A,A),D1).(n1+1)
                  by A266,INTEGRA1:20;
                  then vol(divset(D1,n1+1)) in rng upper_volume(chi(A,A),D1)
                  by A345,FUNCT_1:def 3;
                  then
A346:             vol(divset(D1,n1+1))<=delta(D1) by XXREAL_2:def 8;
                  upper_bound rng f-lower_bound rng f >= 0 by A1,Lm3,XREAL_1:48
;
                  then
A347:             (upper_bound rng f-lower_bound rng f)* vol(divset(D1,
n1+1)) <=(upper_bound rng f-lower_bound rng f)*delta(D1) by A346,XREAL_1:64;
                  vol(divset(D1,n1+1)) >= 0 by INTEGRA1:9;
                  then Sum mid(UVf(D1),n1+1,j) <=(upper_bound rng f)*vol(
                  divset(D1,n1+1)) by A343,A344,XREAL_1:64;
                  then Sum mid(UVf(D1),n1+1,j)-Sum mid(UVf(D2 ),indx(D2,D1,n1
)+1, indx(D2, D1,j)) <=(upper_bound rng f)*vol(divset(D1,n1+1))-(lower_bound
                  rng f)* vol(divset(D1,n1+1)) by A315,XREAL_1:13;
                  hence thesis by A347,XXREAL_0:2;
                end;
                suppose
A348:             n1+1 < j;
A349:             j-'(n1+1) = j-(n1+1) by A258,XREAL_1:233;
                  then
A350:             j-'(n1+1)+1=j-n1;
A351:             n1 < n1+1 by NAT_1:13;
                  then
A352:             D1.n1 < D1.(n1+1) by A241,A266,SEQM_3:def 1;
                  then consider
                  B being non empty closed_interval Subset of REAL, MD1,MD2
                  being Division of B such that
A353:             D1 .n1=lower_bound B and
                  upper_bound B=MD2.(len MD2) and
A354:             upper_bound B=MD1.(len MD1) and
A355:             MD1 <= MD2 and
A356:             MD1=mid(D1,n1+1,j) and
A357:             MD2=mid(D2,indx(D2,D1,n1+1),indx(D2,D1,j)) by A10,A232,A258
,A266,A276,Th15;
A358:             len MD1 = j-'(n1+1)+1 by A258,A261,A264,A244,A265,A356,
FINSEQ_6:118;
                  then
A359:             len MD1 = j-(n1+1)+1 by A258,XREAL_1:233;
                  then
A360:             len MD1+(n1+1)-1 =j;
A361:             len MD1 in dom MD1 by FINSEQ_5:6;
                  then
A362:             1 <= len MD1 by FINSEQ_3:25;
A363:             lower_bound divset(MD1,len MD1)=lower_bound divset(D1,
                  j) & upper_bound divset(MD1,len MD1)=upper_bound divset(D1,j)
                  proof
                    per cases;
                    suppose
A364:                 len MD1=1;
                      then
A365:                 upper_bound divset(MD1,len MD1)=MD1.(len MD1) by A361,
INTEGRA1:def 4;
A366:                 upper_bound divset(D1,j)=D1.j by A232,A245,A277,
INTEGRA1:def 4;
                      lower_bound divset(D1,j)=D1.(j-1) by A232,A245,A277,
INTEGRA1:def 4;
                      hence
                      thesis by A261,A264,A353,A356,A359,A361,A364,A365,A366,
FINSEQ_6:118,INTEGRA1:def 4;
                    end;
                    suppose
A367:                 len MD1<>1;
                      then len MD1-1 in dom MD1 by A361,INTEGRA1:7;
                      then
A368:                 len MD1-1 >= 1 by FINSEQ_3:25;
                      len MD1 <= len MD1+1 by NAT_1:11;
                      then
A369:                 len MD1 -1 <= len MD1 by XREAL_1:20;
                      upper_bound divset(MD1,len MD1)=MD1.(len MD1) by A361
,A367,INTEGRA1:def 4;
                      then
A370:                 upper_bound divset(MD1,len MD1)=D1.j by A258,A264,A244
,A356,A358,A360,A362,FINSEQ_6:122;
A371:                 len MD1-1+(n1+1)-1=j-1 by A358,A349;
                      lower_bound divset(MD1,len MD1)=MD1.( len MD1-1)
                      by A361,A367,INTEGRA1:def 4;
                      then lower_bound divset(MD1,len MD1) = D1.( j-1) by A258
,A264,A244,A356,A358,A371,A368,A369,FINSEQ_6:122;
                      hence thesis by A232,A245,A277,A370,INTEGRA1:def 4;
                    end;
                  end;
A372:             B c= A
                  proof
                    let x1 be object;
A373:               rng D1 c= A by INTEGRA1:def 2;
                    D1.n1 in rng D1 by A241,FUNCT_1:def 3;
                    then
A374:               lower_bound A <= D1.n1 by A373,INTEGRA2:1;
                    assume
A375:               x1 in B;
                    then reconsider x1 as Real;
A376:               x1 <= MD1.(len MD1) by A354,A375,INTEGRA2:1;
                    D1.j in rng D1 by A232,FUNCT_1:def 3;
                    then
A377:               D1.j <= upper_bound A by A373,INTEGRA2:1;
                    D1.n1 <= x1 by A353,A375,INTEGRA2:1;
                    then
A378:               lower_bound A <= x1 by A374,XXREAL_0:2;
                    MD1.(len MD1)=D1.(j-n1-1+(n1+1)) by A258,A278,A264,A244
,A356,A358,A349,FINSEQ_6:122
                      .=D1.j;
                    then x1 <= upper_bound A by A376,A377,XXREAL_0:2;
                    hence thesis by A378,INTEGRA2:1;
                  end;
                  then reconsider g=f|B as Function of B,REAL by FUNCT_2:32;
A379:             delta(MD1) >= 0 by Th9;
A380:             g|B is bounded
                  proof
                    consider a be Real such that
A381:               for x being object st x in A /\ dom f holds a<=f.x
                    by A1,RFUNCT_1:71;
                    for x being object st x in B /\ dom g holds a<=g.x
                    proof
                      let x be object;
A382:                 dom f /\ B c= dom f /\ A by A372,XBOOLE_1:26;
                      assume x in B /\ dom g;
                      then
A383:                 x in dom g by XBOOLE_0:def 4;
                      then x in dom f /\ B by RELAT_1:61;
                      then a <= f.x by A381,A382;
                      hence thesis by A383,FUNCT_1:47;
                    end;
                    then
A384:               g|B is bounded_below by RFUNCT_1:71;
                    consider a be Real such that
A385:               for x being object st x in A /\ dom f holds f.x<=a
                    by A1,RFUNCT_1:70;
                    for x being object st x in B /\ dom g holds g.x<=a
                    proof
                      let x be object;
A386:                 dom f /\ B c= dom f /\ A by A372,XBOOLE_1:26;
                      assume x in B /\ dom g;
                      then
A387:                 x in dom g by XBOOLE_0:def 4;
                      then x in dom f /\ B by RELAT_1:61;
                      then a >= f.x by A385,A386;
                      hence thesis by A387,FUNCT_1:47;
                    end;
                    then g|B is bounded_above by RFUNCT_1:70;
                    hence thesis by A384;
                  end;
                  lower_bound divset(D1,j) <= D.(i+1) by A233,INTEGRA2:1;
                  then
A388:             D1.(j-1) <= D.(i+1) by A232,A245,A277,INTEGRA1:def 4;
A389:             j-'(n1+1)+1=j-(n1+1)+1 by A258,XREAL_1:233;
A390:             len upper_volume(g,MD1)=len MD1 by INTEGRA1:def 6
                    .=j-(n1+1)+1 by A258,A261,A264,A244,A265,A356,A389,
FINSEQ_6:118;
A391:             j <= len UVf(D1) by A264,INTEGRA1:def 6;
A392:             for k be Nat 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 be Nat;
                    assume that
A393:               1 <= k and
A394:               k <= len upper_volume(g,MD1);
               k in Seg len upper_volume( g,MD1) by A393,A394,FINSEQ_1:1;
                    then
A395:               k in Seg len MD1 by INTEGRA1:def 6;
                    then
A396:               k in dom MD1 by FINSEQ_1:def 3;
                    k in dom MD1 by A395,FINSEQ_1:def 3;
                    then
A397:               upper_volume(g,MD1).k =(upper_bound rng(g|divset(MD1
                    ,k)))*vol(divset(MD1,k)) by INTEGRA1:def 6;
                    consider k2 being Element of NAT such that
A398:               n1+1=1+k2;
A399:               1 <= k+k2 by A393,NAT_1:12;
                    k <= j-((n1+1)-1) by A390,A394;
                    then k+((n1+1)-1) <= j by XREAL_1:19;
                    then k+k2 <= len D1 by A264,A398,XXREAL_0:2;
                    then
A400:               k+k2 in Seg len D1 by A399,FINSEQ_1:1;
                    then
A401:               k+k2 in dom D1 by FINSEQ_1:def 3;
                    1+1 <= k+k2 by A259,A393,A398,XREAL_1:7;
                    then
A402:               1 < k+k2 by NAT_1:13;
A403:               k2=(n1+1)-1 by A398;
A404:               lower_bound divset(D1,k+k2)=lower_bound divset(MD1,
                    k) & upper_bound divset(D1,k+k2)= upper_bound divset(MD1,k)
                    proof
                      per cases;
                      suppose
A405:                   k=1;
                        then upper_bound divset(MD1,k)=MD1.k by A396,
INTEGRA1:def 4;
                        then
A406:                   upper_bound divset(MD1,k)=D1.(k+(n1+1)-1) by A258,A264
,A244,A356,A390,A393,A394,FINSEQ_6:122;
                        lower_bound divset(MD1,k)=D1.n1 by A353,A396,A405,
INTEGRA1:def 4;
                        hence thesis by A403,A402,A401,A405,A406,INTEGRA1:def 4
;
                      end;
                      suppose
A407:                   k<>1;
                        then upper_bound divset(MD1,k)=MD1.k by A396,
INTEGRA1:def 4;
                        then
A408:                   upper_bound divset(MD1,k)=D1.(k+(n1+1)-1) by A258,A264
,A244,A356,A390,A393,A394,FINSEQ_6:122;
A409:                   k -1 <= j-(n1+1)+1 by A390,A394,XREAL_1:146,XXREAL_0:2;
A410:                   lower_bound divset(MD1,k)=MD1.(k-1) by A396,A407,
INTEGRA1:def 4;
A411:                   k-1 in dom MD1 by A396,A407,INTEGRA1:7;
                        then 1 <= k-1 by FINSEQ_3:25;
                        then lower_bound divset(MD1,k)=D1.(k-1+(n1+1)-1) by
A258,A264,A244,A356,A411,A409,A410,FINSEQ_6:122;
                        hence thesis by A398,A402,A401,A408,INTEGRA1:def 4;
                      end;
                    end;
                    divset(MD1,k)= [.lower_bound divset(MD1,k),
                    upper_bound divset(MD1,k).] by INTEGRA1:4;
                    then
A412:               divset(D1,k+k2)=divset(MD1,k) by A404,INTEGRA1:4;
A413:               k+k2 in dom D1 by A400,FINSEQ_1:def 3;
A414:               mid(UVf(D1),n1+1,j).k =UVf(D1).(k+(n1+1)-1) by A258,A244
,A390,A391,A393,A394,FINSEQ_6:122
                      .=(upper_bound rng(f|divset(D1,k+k2)))* vol(divset(D1,
                    k+k2)) by A398,A413,INTEGRA1:def 6;
                    k in dom MD1 by A395,FINSEQ_1:def 3;
                    then divset(D1,k+k2) c= B by A412,INTEGRA1:8;
                    hence thesis by A397,A414,A412,FUNCT_1:51;
                  end;
                  n1+1 <= len UVf(D1) by A265,INTEGRA1:def 6;
                  then len upper_volume(g,MD1)=len mid(UVf(D1),n1+1,j) by A258
,A261,A244,A389,A390,A391,FINSEQ_6:118;
                  then
A415:             Sum upper_volume(g,MD1)=Sum mid(UVf(D1),n1+1,j) by A392,
FINSEQ_1:14;
A416:             n1 < j-1 by A348,XREAL_1:20;
A417:             1 <= indx(D2,D1,n1)+1 by A263,NAT_1:13;
A418:             len MD1 in dom MD1 by FINSEQ_5:6;
A419:             upper_bound divset(MD1,len MD1)=MD1.(len MD1)
                  proof
                    per cases;
                    suppose
                      len MD1 = 1;
                      hence thesis by A418,INTEGRA1:def 4;
                    end;
                    suppose
                      len MD1 <> 1;
                      hence thesis by A418,INTEGRA1:def 4;
                    end;
                  end;
                  vol(B)=upper_bound B-D1.n1 by A353,INTEGRA1:def 5;
                  then vol(B)=D1.j-D1.n1 by A232,A245,A277,A354,A363,A419,
INTEGRA1:def 4;
                  then
A420:             vol(B)<>0 by A232,A241,A245,SEQM_3:def 1;
                  rng f is bounded_below by A1,INTEGRA1:11;
                  then
A421:             lower_bound rng f <= lower_bound rng g by RELAT_1:70,SEQ_4:47
;
                  rng f is bounded_above by A1,INTEGRA1:13;
                  then upper_bound rng f >= upper_bound rng g by RELAT_1:70
,SEQ_4:48;
                  then upper_bound rng f-lower_bound rng f>= upper_bound rng
                  g - lower_bound rng g by A421,XREAL_1:13;
                  then
A422:             (upper_bound rng f-lower_bound rng f)*delta(MD1)>= (
upper_bound rng g-lower_bound rng g)*delta(MD1 ) by A379,XREAL_1:64;
                  D1.n1 < D1.(n1+1) by A241,A266,A351,SEQM_3:def 1;
                  then indx(D2,D1,n1) < indx(D2,D1,n1+1) by A262,A260,A270,A267
,SEQ_4:137;
                  then
A423:             indx(D2,D1,n1)+1 <= indx(D2,D1,n1+1) by NAT_1:13;
                  then
A424:             indx(D2,D1,n1)+1 <= len D2 by A274,XXREAL_0:2;
A425:             indx(D2,D1,n1) < indx(D2,D1,n1)+1 by NAT_1:13;
A426:             indx(D2,D1,n1+1)=indx(D2,D1,n1)+1
                  proof
                    assume indx(D2,D1,n1+1)<> indx(D2,D1,n1)+1;
                    then
A427:               indx(D2,D1,n1+1)>indx(D2,D1,n1)+1 by A423,XXREAL_0:1;
A428:               indx(D2,D1,n1)+1 in dom D2 by A417,A424,FINSEQ_3:25;
                    then
A429:               D2.(indx(D2,D1,n1)+1) in rng D2 by FUNCT_1:def 3;
                    now
                      per cases by A11,A429,XBOOLE_0:def 3;
                      suppose
                        D2.(indx(D2,D1,n1)+1) in rng D1;
                        then consider n2 being Element of NAT such that
A430:                   n2 in dom D1 and
A431:                   D2.(indx(D2,D1,n1)+1) = D1.n2 by PARTFUN1:3;
                        D2.indx(D2,D1,n1) < D2.(indx(D2,D1,n1)+1) by A262,A425
,A428,SEQM_3:def 1;
                        then n1 < n2 by A241,A260,A430,A431,SEQ_4:137;
                        then
A432:                   n1+1 <= n2 by NAT_1:13;
                        D1.n2 < D1.(n1+1) by A270,A267,A427,A428,A431,
SEQM_3:def 1;
                        hence contradiction by A266,A430,A432,SEQ_4:137;
                      end;
                      suppose
A433:                   D2.(indx(D2,D1,n1)+1) in rng D;
A434:                   D.i <= upper_bound divset(D1,n1) by A242,INTEGRA2:1;
A435:                   upper_bound divset(D1,n1)=D1.n1
                        proof
                          per cases;
                          suppose
                            n1=1;
                            hence thesis by A241,INTEGRA1:def 4;
                          end;
                          suppose
                            n1<>1;
                            hence thesis by A241,INTEGRA1:def 4;
                          end;
                        end;
                        consider n2 being Element of NAT such that
A436:                   n2 in dom D and
A437:                   D2.(indx(D2,D1,n1)+1) = D.n2 by A433,PARTFUN1:3;
                        D1.n1 < D.n2 by A262,A260,A425,A428,A437,SEQM_3:def 1;
                        then D.i < D.n2 by A434,A435,XXREAL_0:2;
                        then i < n2 by A236,A436,SEQ_4:137;
                        then
A438:                   i+1 <= n2 by NAT_1:13;
                        n1+1 +1 <= j by A348,NAT_1:13;
                        then
A439:                   n1+1 <= j-1 by XREAL_1:19;
                        j -1 in dom D1 by A232,A245,A277,INTEGRA1:7;
                        then
A440:                   D1.(n1+1) <= D1.(j-1) by A266,A439,SEQ_4:137;
A441:                   lower_bound divset(D1,j) <= D.(i+1) by A233,INTEGRA2:1;
                        lower_bound divset(D1,j) = D1.(j-1) by A232,A245,A277,
INTEGRA1:def 4;
                        then
A442:                   D1.(n1+1) <= D.(i+1) by A440,A441,XXREAL_0:2;
                        D.n2 < D1.(n1+1) by A270,A267,A427,A428,A437,
SEQM_3:def 1;
                        then D.n2 < D.(i+1) by A442,XXREAL_0:2;
                        hence contradiction by A231,A436,A438,SEQ_4:137;
                      end;
                    end;
                    hence contradiction;
                  end;
A443:             len MD2=indx(D2,D1,j)-'indx(D2,D1,n1+1)+1 by A275,A271,A274
,A238,A239,A357,FINSEQ_6:118;
                  then
A444:             len MD2 = indx(D2,D1,j)-indx(D2,D1,n1+1)+1 by A275,
XREAL_1:233;
                  then
A445:             len upper_volume(g,MD2)=indx(D2,D1,j)-(indx(D2,D1,n1)+
                  1)+1 by A426,INTEGRA1:def 6;
                 for x1 being object holds
                  x1 in rng MD1 \/ {D.(i+1)} implies x1 in rng MD2
                  proof let x1 be object;
                    assume
A446:               x1 in rng MD1 \/ {D.(i+1)};
                    then reconsider x1 as Real;
                    now
                      per cases by A446,XBOOLE_0:def 3;
                      suppose
A447:                   x1 in rng MD1;
                        rng MD1 <> {};
                        then 1 in dom MD1 by FINSEQ_3:32;
                        then
A448:                   1 <= len MD1 by FINSEQ_3:25;
A449:                   len MD1=j-'(n1+1)+1 by A258,A261,A264,A244,A265,A356,
FINSEQ_6:118;
                        then len MD1+(n1+1)-1=j-(n1+1)+1+(n1+1)-1 by A258,
XREAL_1:233
                          .=j;
                        then
A450:                   MD1.(len MD1)=D1.j by A258,A264,A244,A356,A448,A449,
FINSEQ_6:122;
                        rng MD1 c= rng D1 by A356,FINSEQ_6:119;
                        then
A451:                   x1 in rng D1 by A447;
                        rng D1 c= rng D2 by A10,INTEGRA1:def 18;
                        then consider k such that
A452:                   k in dom D2 and
A453:                   D2.k = x1 by A451,PARTFUN1:3;
                        x1 <= MD1.(len MD1) by A447,Th16;
                        then k<=indx(D2,D1,j) by A237,A234,A450,A452,A453,
SEQM_3:def 1;
                        then k-indx(D2,D1,n1+1)<=indx(D2,D1,j)-indx(D2,D1,n1+
                        1) by XREAL_1:9;
                        then
A454:                   k -indx(D2,D1,n1+1)+1<=indx(D2,D1,j)-indx(D2,D1,
                        n1+1)+1 by XREAL_1:6;
A455:                   MD1.1 <= x1 by A447,Th16;
                        MD1.1=D1.(n1+1) by A261,A264,A244,A265,A356,
FINSEQ_6:118;
                        then
A456:                   indx(D2,D1,n1+1)<=k by A270,A267,A455,A452,A453,
SEQM_3:def 1;
                        then consider n being Nat such that
A457:                   k+1 = indx(D2,D1,n1+1)+n by NAT_1:10,12;
A458:                   n+indx(D2,D1,n1+1)-1=k by A457;
                        indx(D2,D1,n1+1)+1<=k+1 by A456,XREAL_1:6;
                        then
A459:                   1 <= k+1-indx(D2,D1,n1 +1) by XREAL_1:19;
                        then n in dom MD2 by A444,A454,A457,FINSEQ_3:25;
                        then MD2.n in rng MD2 by FUNCT_1:def 3;
                        hence x1 in rng MD2 by A275,A271,A239,A357,A453,A459
,A454,A458,FINSEQ_6:122;
                      end;
                      suppose
                        x1 in {D.(i+1)};
                        then
A460:                   x1 = D.(i+1) by TARSKI:def 1;
                        reconsider j1 = j-1 as Element of NAT by A232,A245,A277
,INTEGRA1:7;
A461:                   rng D c= rng D2 by A9,INTEGRA1:def 18;
                        D.(i+1) in rng D by A231,FUNCT_1:def 3;
                        then consider k such that
A462:                   k in dom D2 and
A463:                   x1 = D2.k by A460,A461,PARTFUN1:3;
                        D.(i+1)<= upper_bound divset(D1,j ) by A233,INTEGRA2:1;
                        then x1 <= D1.j by A232,A245,A277,A460,INTEGRA1:def 4;
                        then
A464:                   D2.k <= D2.indx(D2,D1,j) by A10,A232,A463,
INTEGRA1:def 19;
                        n1<j1 by A348,XREAL_1:20;
                        then
A465:                   n1+1 <= j1 by NAT_1:13;
                        j -1 in dom D1 by A232,A245,A277,INTEGRA1:7;
                        then
A466:                   D1.(n1+1) <= D1.(j-1) by A266,A465,SEQ_4:137;
                        lower_bound divset(D1,j)<=D.(i+1) by A233,INTEGRA2:1;
                        then D1.(j-1) <= x1 by A232,A245,A277,A460,
INTEGRA1:def 4;
                        then D2.indx(D2,D1,n1+1) <= D2.k by A267,A463,A466,
XXREAL_0:2;
                        hence x1 in rng MD2 by A270,A237,A357,A462,A463,A464
,Th17;
                      end;
                    end;
                    hence thesis;
                  end;
                  then
A467:             rng MD1 \/ {D.(i+1)} c= rng MD2;
                  rng MD2 <> {};
                  then 1 in dom MD2 by FINSEQ_3:32;
                  then
A468:             1 <= len MD2 by FINSEQ_3:25;
A469:             len MD2-1+indx(D2,D1,n1+1)=indx(D2,D1, j) by A444;
                  for x1 being object holds
                  x1 in rng MD2 implies x1 in (rng MD1 \/ {D.(i+1)})
                  proof let x1 be object;
                    assume
A470:               x1 in rng MD2;
                    then reconsider x1 as Real;
A471:               MD2.1 <= x1 by A470,Th16;
A472:               MD2 .(len MD2)=D2.indx(D2,D1,j) by A275,A271,A239,A357,A468
,A443,A469,FINSEQ_6:122;
A473:               rng MD2 c= rng D2 by A357,FINSEQ_6:119;
A474:               MD2.1=D2.indx(D2,D1,n1+1) by A271,A274,A238,A239,A357,
FINSEQ_6:118;
A475:               x1 <= MD2.(len MD2) by A470,Th16;
                    then
A476:               x1 <= D1.j by A234,A275,A271,A239,A357,A468,A443,A469,
FINSEQ_6:122;
                    now
                      per cases by A11,A470,A473,XBOOLE_0:def 3;
                      suppose
                        x1 in rng D1;
                        then consider k such that
A477:                   k in dom D1 and
A478:                   D1.k = x1 by PARTFUN1:3;
A479:                   n1 +1 <= k by A266,A267,A471,A474,A477,A478,
SEQM_3:def 1;
                        then
A480:                   1 <= k-n1 by XREAL_1:19;
                        n1 <= n1+1 by NAT_1:11;
                        then consider n being Nat such that
A481:                   k=n1+n by A479,NAT_1:10,XXREAL_0:2;
A482:                   k <= j by A232,A234,A475,A472,A477,A478,SEQM_3:def 1;
                        then
A483:                   k-n1 <= j- n1 by XREAL_1:9;
A484:                   1 <= k-n1 by A479,XREAL_1:19;
A485:                   j-(n1+1)+1=j-n1;
                        k-n1 <= len MD1 by A358,A350,A482,XREAL_1:9;
                        then n in dom MD1 by A484,A481,FINSEQ_3:25;
                        then
A486:                   MD1.n in rng MD1 by FUNCT_1:def 3;
                        MD1.n = D1.(k-n1-1+(n1+1)) by A258,A264,A244,A356
,A480,A483,A485,A481,FINSEQ_6:122
                          .= D1.k;
                        hence x1 in (rng MD1 \/ {D.(i+1)}) by A478,A486,
XBOOLE_0:def 3;
                      end;
                      suppose
                        x1 in rng D;
                        then consider n such that
A487:                   n in dom D and
A488:                   D.n = x1 by PARTFUN1:3;
A489:                   not i+1 < n
                        proof
                         j=1 or j <> 1;
                         then
A490:                     upper_bound divset(D1,j)=D1.j by A232,INTEGRA1:def 4;
                          reconsider y1=D.(i+1)as Real;
A491:                     D.n in rng D by A487,FUNCT_1:def 3;
                          assume i+1 < n;
                          then
A492:                     D.(i+1) < D.n by A231,A487,SEQM_3:def 1;
                          lower_bound divset(D1,j) <= D.(i+1) by A233,
INTEGRA2:1;
                          then lower_bound divset(D1,j) <= D.n by A492,
XXREAL_0:2;
                          then D.n in divset(D1,j) by A476,A488,A490,INTEGRA2:1
;
                          then
A493:                     x1 in rng D /\ divset(D1,j) by A488,A491,
XBOOLE_0:def 4;
                          D.(i+1) in rng D by A231,FUNCT_1:def 3;
                          then y1 in rng D /\ divset(D1,j) by A233,
XBOOLE_0:def 4;
                          hence contradiction by A8,A232,A488,A492,A493,Th5;
                        end;
A494:                   upper_bound divset(D1,n1)=D1.n1
                        proof
                          per cases;
                          suppose
                            n1=1;
                            hence thesis by A241,INTEGRA1:def 4;
                          end;
                          suppose
                            n1<>1;
                            hence thesis by A241,INTEGRA1:def 4;
                          end;
                        end;
                        D.i <= upper_bound divset(D1,n1) by A242,INTEGRA2:1;
                        then D.i < D1.(n1+1) by A352,A494,XXREAL_0:2;
                        then D.i < D.n by A267,A471,A474,A488,XXREAL_0:2;
                        then i < n by A236,A487,SEQ_4:137;
                        then i+1 <= n by NAT_1:13;
                        then i+1 = n or i+1 < n by XXREAL_0:1;
                        then x1 in {D.(i+1)} by A488,A489,TARSKI:def 1;
                        hence x1 in (rng MD1 \/ {D.(i+1)}) by XBOOLE_0:def 3;
                      end;
                    end;
                    hence thesis;
                  end;
                  then rng MD2 c= rng MD1 \/ {D.(i+1)};
                  then
A495:             rng MD2 = rng MD1 \/ {D.(i+1)} by A467,XBOOLE_0:def 10;
                  delta(MD1) in rng upper_volume(chi(B,B),MD1) by
XXREAL_2:def 8;
                  then consider k such that
A496:             k in dom upper_volume(chi(B,B),MD1) and
A497:             upper_volume(chi(B,B),MD1).k = delta(MD1) by PARTFUN1:3;
A498:             k in Seg len upper_volume(chi(B,B),MD1) by A496,
FINSEQ_1:def 3;
                  then
A499:             k in Seg len MD1 by INTEGRA1:def 6;
                  then
A500:             k in dom MD1 by FINSEQ_1:def 3;
A501:             k <= len MD1 by A499,FINSEQ_1:1;
                  then k+n1 <= j by A358,A350,XREAL_1:19;
                  then
A502:             k+n1 <= len D1 by A264,XXREAL_0:2;
A503:             1 <= k by A498,FINSEQ_1:1;
A504:             n1+1>1 by A277,NAT_1:13;
                  then n1>1-1 by XREAL_1:19;
                  then
A505:             k < k+n1 by XREAL_1:29;
                  then 1 < k+n1 by A503,XXREAL_0:2;
                  then
A506:             k+n1 in dom D1 by A502,FINSEQ_3:25;
                  lower_bound divset(MD1,k)=lower_bound divset(D1,k+n1)
                  & upper_bound divset(MD1,k)=upper_bound divset(D1,k+n1)
                  proof
                    per cases;
                    suppose
A507:                 k=1;
                      then upper_bound divset(MD1,k)=MD1.k by A500,
INTEGRA1:def 4;
                      then
A508:                 upper_bound divset(MD1,k)= D1.(k+(n1+1)-1) by A258,A264
,A244,A356,A358,A503,A501,FINSEQ_6:122;
                      lower_bound divset(D1,k+n1)=D1.(k+n1-1) by A503,A505,A506
,INTEGRA1:def 4;
                      hence thesis by A353,A504,A500,A506,A507,A508,
INTEGRA1:def 4;
                    end;
                    suppose
A509:                 k<>1;
                      then upper_bound divset(MD1,k)=MD1.k by A500,
INTEGRA1:def 4;
                      then
A510:                 upper_bound divset(MD1,k)=D1.(k+(n1+1)-1) by A258,A264
,A244,A356,A358,A503,A501,FINSEQ_6:122;
A511:                 lower_bound divset(MD1,k)=MD1.(k-1) by A500,A509,
INTEGRA1:def 4;
A512:                 k-1 in dom MD1 by A500,A509,INTEGRA1:7;
                      then
A513:                 k-1 <= len MD1 by FINSEQ_3:25;
                      1 <= k-1 by A512,FINSEQ_3:25;
                      then lower_bound divset(MD1,k)=D1.(k-1+(n1+1)-1) by A258
,A264,A244,A356,A358,A512,A513,A511,FINSEQ_6:122;
                      hence thesis by A503,A505,A506,A510,INTEGRA1:def 4;
                    end;
                  end;
                  then divset(MD1,k)=[.lower_bound divset(D1,k+n1),
                  upper_bound divset(D1,k+n1).] by INTEGRA1:4;
                  then
A514:             divset(MD1,k)=divset(D1,k+n1) by INTEGRA1:4;
                  k+n1 in Seg len D1 by A506,FINSEQ_1:def 3;
                  then k+n1 in Seg len upper_volume(chi(A,A),D1) by
INTEGRA1:def 6;
                  then
A515:             k+n1 in dom upper_volume(chi(A,A),D1) by FINSEQ_1:def 3;
                  k in dom MD1 by A499,FINSEQ_1:def 3;
                  then delta(MD1) = vol(divset(MD1,k)) by A497,INTEGRA1:20;
                  then delta(MD1)=upper_volume(chi(A,A),D1).( k+n1) by A506
,A514,INTEGRA1:20;
                  then delta(MD1) in rng upper_volume(chi(A,A),D1) by A515,
FUNCT_1:def 3;
                  then delta(MD1) <= max rng upper_volume(chi(A,A),D1) by
XXREAL_2:def 8;
                  then
A516:             delta(MD1) <= delta(D1);
A517:             D.(i+1)<= upper_bound divset(D1,j) by A233,INTEGRA2:1;
                  lower_bound divset(D1,j)<=D.(i+1) by A233,INTEGRA2:1;
                  then
A518:             D.(i+1) in divset(MD1,len MD1) by A363,A517,INTEGRA2:1;
                  j-1 in dom D1 by A232,A245,A277,INTEGRA1:7;
                  then D1.n1 <D1.(j-1) by A241,A416,SEQM_3:def 1;
                  then D.(i+1) > lower_bound B by A353,A388,XXREAL_0:2;
                  then Sum upper_volume(g,MD1)-Sum upper_volume(g,MD2)<= (
upper_bound rng g- lower_bound rng g)*delta(MD1) by A355,A380,A495,A518,A420
,Th14;
                  then
A519:             Sum upper_volume(g,MD1)-Sum upper_volume(g,MD2)<= (
upper_bound rng f-lower_bound rng f)*delta(MD1) by A422,XXREAL_0:2;
A520:             indx(D2,D1,j) <= len UVf(D2) by A239,INTEGRA1:def 6;
A521:             indx (D2,D1,n1)+1 <= indx(D2,D1,j) by A275,A423,XXREAL_0:2;
A522:             for k be Nat 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 be Nat;
                    assume that
A523:               1 <= k and
A524:               k <= len upper_volume(g,MD2);
A525:               k in Seg len upper_volume( g,MD2) by A523,A524,FINSEQ_1:1;
A526:               mid(UVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)).k =UVf(
D2).(k+(indx(D2,D1,n1)+1)-1) by A417,A445,A520,A521,A523,A524,FINSEQ_6:122;
A527:               k in Seg len MD2 by A525,INTEGRA1:def 6;
                    then k in dom MD2 by FINSEQ_1:def 3;
                    then
A528:               upper_volume(g,MD2).k =(upper_bound rng(g|divset(MD2
                    ,k)))*vol(divset(MD2,k)) by INTEGRA1:def 6;
                    1<=indx(D2,D1,n1)+1 by NAT_1:12;
                    then 1+1<=k+(indx(D2,D1,n1)+1) by A523,XREAL_1:7;
                    then
A529:               1 <= k+(indx(D2,D1,n1)+1)- 1 by XREAL_1:19;
                    consider k2 being Element of NAT such that
A530:               indx(D2,D1,n1)+1=1+k2;
                    k <= indx(D2,D1,j)-((indx(D2,D1,n1)+1)-1) by A444,A426,A524
,INTEGRA1:def 6;
                    then k+((indx(D2,D1,n1)+1)-1) <= indx(D2,D1,j) by
XREAL_1:19;
                    then k +(indx(D2,D1,n1)+1)-1 <= len UVf(D2) by A520,
XXREAL_0:2;
                    then k+k2 in Seg len UVf(D2) by A529,A530,FINSEQ_1:1;
                    then
A531:               k+k2 in Seg len D2 by INTEGRA1:def 6;
                    then k+k2 in dom D2 by FINSEQ_1:def 3;
                    then
A532:               mid(UVf(D2),indx(D2,D1,n1) +1,indx(D2,D1,j)).k =(
upper_bound rng(f|divset(D2,k+k2)))* vol(divset(D2,k+k2)) by A526,A530,
INTEGRA1:def 6;
A533:               lower_bound divset(MD2,k)=lower_bound divset(D2,k+k2
                    ) & upper_bound divset(MD2,k)=upper_bound divset(D2,k+k2)
                    proof
                      k+k2>=1+1 by A263,A523,A530,XREAL_1:7;
                      then
A534:                 k+k2>1 by NAT_1:13;
A535:                 k in dom MD2 by A527,FINSEQ_1:def 3;
A536:                 k+k2 in dom D2 by A531,FINSEQ_1:def 3;
                      per cases;
                      suppose
A537:                   k=1;
                        then
A538:                   upper_bound divset(D2,k+k2)=D2.(1+ k2 ) by A534,A536,
INTEGRA1:def 4;
A539:                   lower_bound divset(MD2,k)= lower_bound B by A535,A537,
INTEGRA1:def 4;
                        upper_bound divset(MD2,k)=MD2.k by A535,A537,
INTEGRA1:def 4;
                        then
A540:                   upper_bound divset(MD2,k)= D2.(1+indx(D2,D1,n1+1
)-1) by A275,A239,A357,A417,A426,A445,A524,A537,FINSEQ_6:122
                          .=D1.(n1+1) by A10,A266,INTEGRA1:def 19;
                        lower_bound divset(D2,k+k2)=D2.(1+k2-1) by A534,A536
,A537,INTEGRA1:def 4;
                        hence
                        thesis by A10,A241,A266,A353,A426,A530,A539,A540,A538,
INTEGRA1:def 19;
                      end;
                      suppose
A541:                   k<>1;
                        then upper_bound divset(MD2,k)=MD2.k by A535,
INTEGRA1:def 4;
                        then
A542:                   upper_bound divset(MD2,k)= D2.(k+(indx(D2,D1,n1)
+1)-1) by A275,A239,A357,A417,A426,A445,A523,A524,FINSEQ_6:122;
A543:                   k -1 <= indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1 by A445,A524
,XREAL_1:146,XXREAL_0:2;
A544:                   lower_bound divset(MD2,k)=MD2.(k-1) by A535,A541,
INTEGRA1:def 4;
A545:                   k-1 in dom MD2 by A535,A541,INTEGRA1:7;
                        then 1 <= k-1 by FINSEQ_3:25;
                        then lower_bound divset(MD2,k)=D2.(k-1+(indx(D2,D1,n1
)+1)-1) by A275,A239,A357,A417,A426,A545,A543,A544,FINSEQ_6:122;
                        hence thesis by A530,A534,A536,A542,INTEGRA1:def 4;
                      end;
                    end;
                    divset (MD2,k)=[.lower_bound divset(MD2,k),
                    upper_bound divset(MD2,k).] by INTEGRA1:4;
                    then
A546:               divset(MD2,k)=divset(D2,k+ k2) by A533,INTEGRA1:4;
                    k in dom MD2 by A527,FINSEQ_1:def 3;
                    then divset(D2,k+k2) c= B by A546,INTEGRA1:8;
                    hence thesis by A528,A532,A546,FUNCT_1:51;
                  end;
                  indx(D2,D1,n1)+1 <= len UVf(D2) by A424,INTEGRA1:def 6;
                  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 A238,A417,A520,A521,FINSEQ_6:118;
                  then len upper_volume(g,MD2) =len mid(UVf(D2),indx(D2,D1,n1
                  )+1,indx(D2,D1,j)) by A275,A423,A445,XREAL_1:233,XXREAL_0:2;
                  then
A547:             Sum upper_volume(g,MD2)=Sum mid(UVf(D2),indx(D2,D1,n1)
                  +1,indx(D2,D1,j)) by A522,FINSEQ_1:14;
                  upper_bound rng f - lower_bound rng f >= 0 by A1,Lm3,
XREAL_1:48;
                  then (upper_bound rng f-lower_bound rng f)*delta(MD1) <= (
upper_bound rng f-lower_bound rng f)* delta(D1) by A516,XREAL_1:64;
                  hence thesis by A519,A547,A415,XXREAL_0:2;
                end;
              end;
              then
A548:         (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*(upper_bound rng f-
lower_bound rng f)*delta(D1)+(upper_bound rng f- lower_bound rng f)*delta(D1)
              by A243,XREAL_1:7;
              n1 < n1+1 by NAT_1:13;
              then D1.n1 < D1.(n1+1) by A241,A266,SEQM_3:def 1;
              then indx(D2,D1,n1) < indx(D2,D1,n1+1) by A262,A260,A270,A267,
SEQ_4:137;
              then
A549:         indx(D2,D1,n1)<indx(D2,D1,j) by A275,XXREAL_0:2;
              indx(D2,D1,n1) in Seg len D2 by A262,FINSEQ_1:def 3;
              then indx(D2,D1,n1) in Seg len UVf(D2) by INTEGRA1:def 6;
              then indx(D2,D1,n1) in dom UVf(D2) by FINSEQ_1:def 3;
              then PUf(D2,indx(D2,D1,n1))=Sum(UVf(D2)|indx(D2,D1,n1)) by
INTEGRA1:def 20
                .=Sum mid(UVf(D2),1,indx(D2,D1,n1)) by A263,FINSEQ_6:116;
              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_1:75
                .=Sum mid(UVf(D2),1,indx(D2,D1,j)) by A263,A549,A240,INTEGRA2:4
                .=Sum(UVf(D2)|indx(D2,D1,j)) by A238,FINSEQ_6:116;
              then 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)) by A273,INTEGRA1:def 20;
              then (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 A272;
              hence thesis by A232,A233,A548;
            end;
            hence thesis;
          end;
          for k being non zero Nat holds P[k] from NAT_1:sch 10(A36,
          A227);
          then P[i];
          hence thesis by A14;
        end;
A550:   len D1 in dom D1 by FINSEQ_5:6;
        then D1.(len D1) = D2.indx(D2,D1,len D1) by A10,INTEGRA1:def 19;
        then upper_bound A = D2.indx(D2,D1,len D1) by INTEGRA1:def 2;
        then
A551:   D2.(len D2) = D2.indx(D2,D1,len D1) by INTEGRA1:def 2;
        len D in dom D by FINSEQ_5:6;
        then consider j such that
A552:   j in dom D1 and
A553:   D.(len D) in divset(D1,j) and
A554:   PUf(D1,j)-PUf(D2,indx(D2,D1,j)) <= (len D)*(upper_bound(rng
        f)- lower_bound(rng f))*delta(D1) by A13;
A555:   j = len D1
        proof
          j in Seg len D1 by A552,FINSEQ_1:def 3;
          then
A556:     j <= len D1 by FINSEQ_1:1;
          assume j<>len D1;
          then j < len D1 by A556,XXREAL_0:1;
          then D1.j < D1.(len D1) by A552,A550,SEQM_3:def 1;
          then
A557:     D1.j < upper_bound A by INTEGRA1:def 2;
A558:     upper_bound divset(D1,j) < upper_bound A
          proof
            per cases;
            suppose
              j=1;
              hence thesis by A552,A557,INTEGRA1:def 4;
            end;
            suppose
              j<>1;
              hence thesis by A552,A557,INTEGRA1:def 4;
            end;
          end;
          D.(len D) <= upper_bound divset(D1,j) by A553,INTEGRA2:1;
          hence contradiction by A558,INTEGRA1:def 2;
        end;
        indx(D2,D1,len D1) in dom D2 by A10,A550,INTEGRA1:def 19;
        then indx(D2,D1,len D1)=len D2 by A12,A551,SEQ_4:138;
        then upper_sum(f,D1)-PUf(D2,len D2)<= (len D)*(upper_bound(rng f)-
        lower_bound(rng f))* delta(D1) by A554,A555,INTEGRA1:42;
        hence thesis by INTEGRA1:42;
      end;
      hence thesis by A9,A10,A11;
    end;
    hence thesis;
  end;
