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 Th21:
  f|A is bounded implies
  for D,D1 st delta(D1)<min rng upper_volume(chi(A,A),D) ex D2 st D<=D2 &
  D1<=D2 & rng D2=rng D1 \/ rng D & lower_sum(f,D2)-lower_sum(f,D1) <=
  (len D)*(upper_bound(rng f)-lower_bound(rng f))*delta(D1)
  proof
    assume
A1: f|A is bounded;
    then
A5: for D,D1 ex D2 st D<=D2 & D1<=D2 & rng D2=rng D1 \/ rng D &
    0 <= lower_sum(f,D2)-lower_sum(f,D) & 0<=lower_sum(f,D2)-lower_sum(f,D1)
      by Th20;
    for D,D1 st delta(D1)<min rng upper_volume(chi(A,A),D) ex D2 st D<=D2 &
D1<=D2 & rng D2=rng D1 \/ rng D & lower_sum(f,D2)-lower_sum(f,D1) <= (len D)*(
  upper_bound(rng f)-lower_bound(rng f))*delta(D1)
  proof
    let D,D1;
    assume
A11: delta(D1)<min rng upper_volume(chi(A,A),D);
    ex D2 st D<=D2 & D1<=D2 & rng D2=rng D1 \/ rng D & lower_sum(f,D2)-
    lower_sum(f,D1)<=(len D)*(upper_bound(rng f)- lower_bound(rng f))*delta(D1)
    proof
      consider D2 such that
A12:  D<=D2 and
A13:  D1<=D2 and
A14:  rng D2=rng D1 \/ rng D and
      0<=lower_sum(f,D2)-lower_sum(f,D) and
      0<=lower_sum(f,D2)-lower_sum(f,D1) by A5;
      lower_sum(f,D2)-lower_sum(f,D1) <= (len D)*(upper_bound(rng f)-
      lower_bound(rng f))*delta(D1)
      proof
        deffunc LVf(Division of A) = lower_volume(f,$1);
        deffunc PLf(Division of A,Nat) = (PartSums(lower_volume(f,$1))).$2;
A15:    len D2 in dom D2 by FINSEQ_5:6;
A16:    for i st i in dom D holds ex j st j in dom D1 & D.i in divset(D1,
j) & PLf(D2,indx(D2,D1,j))-PLf(D1,j)<=i*(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) & PLf(D2,indx(D2,D1,j))-PLf(D1,j)<= $1*(upper_bound
          rng f-lower_bound rng f)*delta(D1);
          let i;
          assume
A17:      i in dom D;
          then
A18:      i in Seg len D by FINSEQ_1:def 3;
A19:      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
A20:        i in dom D;
            assume that
A21:        j in dom D1 and
A22:        D.i in divset(D1,j);
            assume j<2;
            then j<1+1;
            then
A23:        j <= 1 by NAT_1:13;
            j in Seg len D1 by A21,FINSEQ_1:def 3;
            then j >= 1 by FINSEQ_1:1;
            then j = 1 by A23,XXREAL_0:1;
            then
A24:        lower_bound divset(D1,j)=lower_bound A by A21,INTEGRA1:def 4;
A25:        D.i<=upper_bound divset(D1,j) by A22,INTEGRA2:1;
            delta(D1) >= min rng upper_volume(chi(A,A),D)
            proof
              per cases;
              suppose
A26:            i=1;
                len D in Seg len D by FINSEQ_1:3;
                then 1 <= len D by FINSEQ_1:1;
                then
A27:            1 in Seg len D by FINSEQ_1:1;
                then
A28:            1 in dom D by FINSEQ_1:def 3;
                then
A29:            lower_bound divset(D,1)=lower_bound A by INTEGRA1:def 4;
                1 in Seg len upper_volume(chi(A,A),D) by A27,INTEGRA1:def 6;
                then
A30:            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 A28,
INTEGRA1:20;
                then vol(divset(D,1)) in rng upper_volume(chi(A,A),D) by A30,
FUNCT_1:def 3;
                then
A31:            vol
(divset(D,1))>=min rng upper_volume(chi(A,A),D) by XXREAL_2:def 7;
A32:            upper_bound divset(D,1)=D. 1 by A28,INTEGRA1:def 4;
                upper_bound divset(D1,j)-lower_bound A >= D.1-lower_bound
                A by A25,A26,XREAL_1:9;
                then vol(divset(D1,j)) >= upper_bound divset(D,1)-lower_bound
                divset(D,1) by A24,A29,A32,INTEGRA1:def 5;
                then
A33:            vol(divset(D1,j)) >= vol(divset(D,1)) by INTEGRA1:def 5;
                vol(divset(D1,j)) <= delta(D1) by A21,Lm5;
                then delta(D1) >= vol(divset(D,1)) by A33,XXREAL_0:2;
                hence thesis by A31,XXREAL_0:2;
              end;
              suppose
A34:            i<>1;
                then D.(i-1) in A by A20,INTEGRA1:7;
                then
A35:            lower_bound A <= D.(i-1) by INTEGRA2:1;
                lower_bound divset(D,i)=D.(i-1) by A20,A34,INTEGRA1:def 4;
                then
A36:            upper_bound divset(D,i)-lower_bound A >= upper_bound
                divset(D,i)-lower_bound divset(D,i) by A35,XREAL_1:10;
                upper_bound divset(D,i)= D.i by A20,A34,INTEGRA1:def 4;
                then upper_bound divset(D1,j)-lower_bound divset(D1,j) >=
                upper_bound divset(D,i)-lower_bound A by A25,A24,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 A36,XXREAL_0:2;
                then vol(divset(D1,j)) >= upper_bound divset(D,i)-lower_bound
                divset(D,i) by INTEGRA1:def 5;
                then
A37:            vol(divset(D1,j)) >= vol(divset(D,i)) by INTEGRA1:def 5;
                i in Seg len D by A20,FINSEQ_1:def 3;
                then i in Seg len upper_volume(chi(A,A),D) by INTEGRA1:def 6;
                then
A38:            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 A20,
INTEGRA1:20;
                then vol(divset(D,i)) in rng upper_volume(chi(A,A),D) by A38,
FUNCT_1:def 3;
                then
A39:            vol
(divset(D,i))>=min rng upper_volume(chi(A,A),D) by XXREAL_2:def 7;
                vol(divset(D1,j)) <= delta(D1) by A21,Lm5;
                then delta(D1) >= vol(divset(D,i)) by A37,XXREAL_0:2;
                hence thesis by A39,XXREAL_0:2;
              end;
            end;
            hence contradiction by A11;
          end;
A40:      P[1]
          proof
            len D in Seg len D by FINSEQ_1:3;
            then 1 <= len D by FINSEQ_1:1;
            then
A41:        1 in dom D by FINSEQ_3:25;
            then consider j such that
A42:        j in dom D1 and
A43:        D.1 in divset(D1,j) by Th3,INTEGRA1:6;
            PLf(D2,indx(D2,D1,j))-PLf(D1,j) <=1*(upper_bound rng f-
            lower_bound rng f)*delta(D1)
            proof
A44:          j <> 1 by A19,A41,A42,A43;
              then reconsider j1=j-1 as Element of NAT by A42,INTEGRA1:7;
A45:          j1 in dom D1 by A42,A44,INTEGRA1:7;
              then j1 in Seg len D1 by FINSEQ_1:def 3;
              then j1 in Seg len lower_volume(f,D1) by INTEGRA1:def 7;
              then
A46:          j1 in dom lower_volume(f,D1) by FINSEQ_1:def 3;
A47:          j-1 in dom D1 by A42,A44,INTEGRA1:7;
              then
A48:          indx(D2,D1,j1) in dom D2 by A13,INTEGRA1:def 19;
              then
A49:          indx(D2,D1,j1) in Seg len D2 by FINSEQ_1:def 3;
              then
A50:          1 <= indx(D2,D1,j1) by FINSEQ_1:1;
              then mid(D2,1,indx(D2,D1,j1)) is increasing by A48,INTEGRA1:35;
              then
A51:          D2|indx(D2,D1,j1) is increasing by A50,FINSEQ_6:116;
              j < j+1 by NAT_1:13;
              then j1 < j by XREAL_1:19;
              then
A52:          indx(D2,D1,j1) < indx(D2,D1,j) by A13,A42,A45,Th8;
              then
A53:          indx(D2,D1,j1)+1 <= indx(D2,D1,j) by NAT_1:13;
A54:          Sum mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1,j
)) -Sum mid(lower_volume(f,D1),j,j) <= (upper_bound rng f-lower_bound rng f)*
              delta(D1)
              proof
A55:            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
A56:              indx(D2,D1,j1)+(1+1)<indx(D2,D1,j) by XREAL_1:20;
A57:              ID1 < ID2 by NAT_1:13;
                  then indx(D2,D1,j1) <= ID2 by NAT_1:13;
                  then
A58:              1 <= ID2 by A50,XXREAL_0:2;
A59:              indx(D2,D1,j) in dom D2 by A13,A42,INTEGRA1:def 19;
                  then
A60:              indx(D2,D1,j) <= len D2 by FINSEQ_3:25;
                  then ID2 <= len D2 by A56,XXREAL_0:2;
                  then ID2 in Seg len D2 by A58,FINSEQ_1:1;
                  then
A61:              ID2 in dom D2 by FINSEQ_1:def 3;
                  then
A62:              D2.ID2<D2.indx(D2,D1, j) by A56,A59,SEQM_3:def 1;
A63:              1 <= ID1 by A50,NAT_1:13;
A64:              D1.j = D2.indx(D2,D1,j) by A13,A42,INTEGRA1:def 19;
                  ID1 <= indx(D2,D1,j) by A56,A57,XXREAL_0:2;
                  then ID1 <= len D2 by A60,XXREAL_0:2;
                  then ID1 in Seg len D2 by A63,FINSEQ_1:1;
                  then
A65:              ID1 in dom D2 by FINSEQ_1:def 3;
                  then
A66:              D2.ID1<D2.ID2 by A57,A61,SEQM_3:def 1;
                  indx(D2,D1,j1) < ID1 by NAT_1:13;
                  then
A67:              D2 .indx(D2,D1,j1)<D2.ID1 by A48,A65,SEQM_3:def 1;
A68:              D1.j1 = D2.indx(D2,D1,j1) by A13,A45,INTEGRA1:def 19;
A69:              not D2.ID1 in rng D1 & not D2.ID2 in rng D1
                  proof
                    assume
A70:                D2.ID1 in rng D1 or D2.ID2 in rng D1;
                    per cases by A70;
                    suppose
                      D2.ID1 in rng D1;
                      then consider n such that
A71:                  n in dom D1 and
A72:                  D1.n=D2.ID1 by PARTFUN1:3;
                      j1<n by A45,A67,A68,A71,A72,SEQ_4:137;
                      then
A73:                  j<n+1 by XREAL_1:19;
                      D2.ID1<D2.indx(D2,D1,j) by A66,A62,XXREAL_0:2;
                      then n<j by A42,A64,A71,A72,SEQ_4:137;
                      hence contradiction by A73,NAT_1:13;
                    end;
                    suppose
                      D2.ID2 in rng D1;
                      then consider n such that
A74:                  n in dom D1 and
A75:                  D1.n=D2.ID2 by PARTFUN1:3;
                      D2.indx(D2,D1,j1)<D2.ID2 by A67,A66,XXREAL_0:2;
                      then j1<n by A45,A68,A74,A75,SEQ_4:137;
                      then
A76:                  j<n+1 by XREAL_1:19;
                      n<j by A42,A62,A64,A74,A75,SEQ_4:137;
                      hence contradiction by A76,NAT_1:13;
                    end;
                  end;
                  upper_bound divset(D1,j)=D1.j by A42,A44,INTEGRA1:def 4;
                  then
A77:              upper_bound divset(D1,j)=D2. indx(D2,D1,j) by A13,A42,
INTEGRA1:def 19;
                  lower_bound divset(D1,j)=D1.j1 by A42,A44,INTEGRA1:def 4;
                  then
A78:              lower_bound divset(D1,j)=D2.indx(D2,D1,j1) by A13,A45,
INTEGRA1:def 19;
                  D2.ID2 in rng D \/ rng D1 by A14,A61,FUNCT_1:def 3;
                  then
A79:              D2.ID2 in rng D by A69,XBOOLE_0:def 3;
                  D2.ID1 in rng D \/ rng D1 by A14,A65,FUNCT_1:def 3;
                  then
A80:              D2.ID1 in rng D by A69,XBOOLE_0:def 3;
                  D2.indx(D2,D1,j1)<=D2.ID2 by A67,A66,XXREAL_0:2;
                  then D2.ID2 in divset(D1,j) by A62,A78,A77,INTEGRA2:1;
                  then
A81:              D2.ID2 in rng D /\ divset(D1,j) by A79,XBOOLE_0:def 4;
                  D2.ID1<=D2.indx(D2,D1,j) by A66,A62,XXREAL_0:2;
                  then D2.ID1 in divset(D1,j) by A67,A78,A77,INTEGRA2:1;
                  then D2.ID1 in rng D /\ divset(D1,j) by A80,XBOOLE_0:def 4;
                  hence contradiction by A11,A42,A57,A65,A61,A81,Th5,SEQ_4:138;
                end;
A82:            1 <= indx(D2,D1,j1)+1 by A50,NAT_1:13;
                j <= len D1 by A42,FINSEQ_3:25;
                then
A83:            j <= len lower_volume(f,D1) by INTEGRA1:def 7;
A84:            1 <= j by A42,FINSEQ_3:25;
                then
A85:            mid(lower_volume(f,D1),j,j).1 = lower_volume(f,D1).j by A83,
FINSEQ_6:118;
           reconsider lv = lower_volume(f,D1).j as Element of REAL
                 by XREAL_0:def 1;
                j-'j+1 = 1 by Lm1;
                then len mid(lower_volume(f,D1),j,j)= 1 by A84,A83,FINSEQ_6:118
;
                then mid(lower_volume(f,D1),j,j) =<*lv*> by A85,FINSEQ_1:40;
                then
A86:            Sum mid(lower_volume(f,D1),j,j)= lower_volume(f,D1).j by
FINSOP_1:11;
                indx(D2,D1,j) in dom D2 by A13,A42,INTEGRA1:def 19;
                then
A87:            indx(D2,D1,j) in Seg len D2 by FINSEQ_1:def 3;
                then
A88:            1 <= indx(D2,D1,j) by FINSEQ_1:1;
                indx(D2,D1,j) in Seg len lower_volume(f,D2) by A87,
INTEGRA1:def 7;
                then
A89:            indx(D2,D1,j) <= len lower_volume(f,D2) by FINSEQ_1:1;
                then
A90:            indx(D2,D1,j1)+1 <= len lower_volume(f, D2) by A53,XXREAL_0:2;
                then indx(D2,D1,j1)+1 in Seg len lower_volume(f,D2) by A82,
FINSEQ_1:1;
                then
A91:            indx(D2,D1,j1)+1 in Seg len D2 by INTEGRA1:def 7;
                then
A92:            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 A53,XREAL_1:233;
                then indx(D2,D1,j)-'(indx(D2,D1,j1) +1)+1 <= 2 by A55;
                then
A93:            len mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1
                ,j))<=2 by A53,A88,A89,A82,A90,FINSEQ_6:118;
                indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1 >= 0+1 by XREAL_1:6;
                then
A94:            1 <= len mid(lower_volume(f,D2),(indx(D2, D1,j1)+1),indx
                (D2,D1,j )) by A53,A88,A89,A82,A90,FINSEQ_6:118;
                now
                  per cases by A94,A93,Lm2;
                  suppose
A95:                len mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),
                    indx(D2,D1,j))=1;
                    upper_bound divset(D1,j)=D1.j by A42,A44,INTEGRA1:def 4;
                    then
A96:                upper_bound divset(D1,j)=D2. indx(D2,D1,j) by A13,A42,
INTEGRA1:def 19;
                    lower_bound divset(D1,j)=D1.j1 by A42,A44,INTEGRA1:def 4;
                    then lower_bound divset(D1,j)=D2.indx(D2,D1,j1) by A13,A45,
INTEGRA1:def 19;
                    then
A97:                divset(D1,j)=[. D2.indx(D2,D1, j1),D2.indx(D2,D1,j)
                    .] by A96,INTEGRA1:4;
A98:                delta(D1) >= 0 by Th9;
A99:                upper_bound rng f - lower_bound rng f >= 0 by A1,Lm3,
XREAL_1:48;
A100:               indx(D2,D1,j) in dom D2 by A13,A42,INTEGRA1:def 19;
                    indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1=1 by A53,A88,A89,A82
,A90,A95,FINSEQ_6:118;
                    then
A101:               indx(D2,D1,j)-(indx(D2,D1,j1)+1)=0 by A53,XREAL_1:233;
                    then indx(D2,D1,j)<>1 by A49,FINSEQ_1:1;
                    then
A102:               upper_bound divset(D2,indx(D2,D1,j))=D2.indx(D2,D1,j
                    ) by A100,INTEGRA1:def 4;
                    indx(D2,D1,j)-1=indx(D2,D1,j1) by A101;
                    then lower_bound divset(D2,indx(D2,D1,j)) =D2.indx(D2,D1,
                    j1) by A50,A101,A100,INTEGRA1:def 4;
                    then
A103:               divset(D2,indx(D2,D1,j))=divset(D1,j) by A97,A102,
INTEGRA1:4;
           reconsider lv = lower_volume(f,D2).(indx(D2,D1,j1)+1)
as Element of REAL
                    by XREAL_0:def 1;
                    mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1
,j)).1 =lower_volume(f,D2).(indx(D2,D1,j1)+1) by A88,A89,A82,A90,FINSEQ_6:118;
                    then mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1
,j)) =<*lv*> by A95,FINSEQ_1:40;
                    then Sum mid(lower_volume(f,D2),(indx( D2,D1,j1)+1),indx(
D2,D1,j )) =lower_volume(f,D2).(indx(D2,D1,j1)+1) by FINSOP_1:11
                      .=(lower_bound(rng(f|divset(D2,(indx(D2,D1,j1)+1)))))
                    *vol(divset(D2,(indx(D2,D1,j1)+1))) by A92,INTEGRA1:def 7
                      .=Sum mid(lower_volume(f,D1),j,j) by A42,A86,A101,A103,
INTEGRA1:def 7;
                    hence thesis by A98,A99;
                  end;
                  suppose
A104:               len mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),
                    indx(D2,D1,j))=2;
A105:               mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1
,j)).1 =lower_volume(f,D2).(indx(D2,D1,j1)+1) by A88,A89,A82,A90,FINSEQ_6:118;
A106:               2+(indx(D2,D1,j1)+1)>=0+1 by XREAL_1:7;
                    mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1
,j)).2 =LVf(D2).(2+(indx(D2,D1,j1)+1)-'1) by A53,A88,A89,A82,A90,A104,
FINSEQ_6:118
                      .=LVf(D2).(2+(indx(D2,D1,j1)+1)-1) by A106,XREAL_1:233
                      .=LVf(D2).(indx(D2,D1,j1)+(1+1));
                    then mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(D2,D1
,j)) =<*lower_volume(f,D2).(indx(D2,D1,j1)+1), lower_volume(f,D2).(indx(D2,D1,
                    j1)+2)*> by A104,A105,FINSEQ_1:44;
                    then
A107:               Sum mid(lower_volume(f,D2),(indx(D2,D1,j1)+1),indx(
D2,D1,j)) =lower_volume(f,D2).(indx(D2,D1,j1)+1) +lower_volume(f,D2).(indx(D2,
                    D1,j1)+2) by RVSUM_1:77;
A108:               vol(divset(D2,indx(D2,D1,j1)+1))>=0 by INTEGRA1:9;
                    upper_bound divset(D1,j)=D1.j by A42,A44,INTEGRA1:def 4;
                    then
A109:               upper_bound divset(D1,j)=D2. indx(D2,D1,j) by A13,A42,
INTEGRA1:def 19;
A110:               vol(divset(D2,indx(D2,D1,j1)+2))>=0 by INTEGRA1:9;
                    indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1=2 by A53,A88,A89,A82
,A90,A104,FINSEQ_6:118;
                    then
A111:               indx(D2,D1,j)-(indx(D2,D1,j1)+1)+1=2 by A53,XREAL_1:233;
                    then
A112:               indx (D2,D1,j1)+2 in dom D2 by A13,A42,INTEGRA1:def 19;
                    lower_bound divset(D1,j)=D1.j1 by A42,A44,INTEGRA1:def 4;
                    then lower_bound divset(D1,j)=D2.indx(D2,D1,j1) by A13,A45,
INTEGRA1:def 19;
                    then
A113:               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 A109,A111,INTEGRA1:def 5
;
                    indx(D2,D1,j1)+1 in Seg len lower_volume(f,D2) by A82,A90,
FINSEQ_1:1;
                    then indx(D2,D1,j1)+1 in Seg len D2 by INTEGRA1:def 7;
                    then
A114:               indx(D2,D1,j1)+1 in dom D2 by FINSEQ_1:def 3;
A115:               indx(D2,D1,j1)+1 <> 1 by A50,NAT_1:13;
                    then
A116:               upper_bound divset(D2,(indx( D2,D1,j1)+1))= D2.(indx
                    (D2,D1,j1)+1) by A114,INTEGRA1:def 4;
                    indx(D2,D1,j1)+1-1=indx(D2,D1,j1)+0;
                    then
A117:               lower_bound divset(D2,(indx(D2,D1,j1)+1))= D2.indx(
                    D2,D1,j1) by A114,A115,INTEGRA1:def 4;
A118:               indx(D2,D1,j1)+1+1 > 1 by A82,NAT_1:13;
                    indx(D2,D1,j1)+2-1=indx(D2,D1,j1)+1;
                    then
A119:               lower_bound divset(D2,(indx(D2,D1,j1)+2))= D2.(indx(
                    D2,D1,j1)+1) by A112,A118,INTEGRA1:def 4;
                    upper_bound divset(D2,(indx(D2,D1,j1)+2))= D2.(indx(
                    D2,D1,j1)+2) by A112,A118,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 A119,A113,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 A117
,A116;
                    then
A120:               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
A121:               lower_volume(f,D1).j=(lower_bound(rng(f| divset(D1,j
))))* (vol(divset(D2,indx(D2,D1,j1)+1))+vol(divset(D2,indx(D2,D1,j1)+2))) by
A42,INTEGRA1:def 7;
A122:               Sum mid(LVf(D2),indx(D2,D1,j1)+1,indx(D2,D1,j))-Sum
mid(LVf(D1) ,j,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 B = vol(divset(D2,ID1));
                      set C = vol(divset(D2,ID2));
                      divset(D1,j) c=A by A42,INTEGRA1:8;
                      then
A123:                 lower_bound rng(f|divset(D1,j)) >= lower_bound rng
                      f by A1,Lm4;
                      ID1 in dom D2 by A91,FINSEQ_1:def 3;
                      then divset(D2,ID1)c=A by INTEGRA1:8;
                      then lower_bound rng(f|divset(D2,ID1)) <= upper_bound
                      rng f by A1,Lm4;
                      then
A124:                 (lower_bound rng(f|divset(D2,ID1)))* vol(divset(D2
,ID1)) <=(upper_bound rng f)*vol(divset(D2,ID1)) by A108,XREAL_1:64;
                      indx(D2,D1,j)-'(indx(D2,D1,j1)+1)+1=2 by A53,A88,A89,A82
,A90,A104,FINSEQ_6:118;
                      then
A125:                 indx(D2,D1,j)-(indx(D2,D1,j1)+1)+1=2 by A53,XREAL_1:233;
A126:                 indx(D2,D1,j) in dom D2 by A13,A42,INTEGRA1:def 19;
                      then divset(D2,ID2)c=A by A125,INTEGRA1:8;
                      then
A127:                 lower_bound rng(f|divset(D2,ID2)) <= upper_bound
                      rng f by A1,Lm4;
                      reconsider A = lower_bound(rng(f|divset(D1,j))) as Real;
A128:                 lower_volume(f,D1).j-A*B=A*C by A121;
                      (lower_bound rng(f|divset(D1,j)))*vol( divset(D2,
ID2)) >=(lower_bound rng f)*vol(divset(D2,ID2)) by A110,A123,XREAL_1:64;
                      then Sum mid(LVf(D1),j,j)>=(lower_bound rng(f|divset(D1
,j)))* vol(divset(D2, ID1)) +(lower_bound rng f)*vol(divset(D2,ID2)) by A86
,A128,XREAL_1:19;
                      then
A129:                 Sum mid(LVf(D1),j,j)-(lower_bound rng f)* vol(
divset(D2,ID2) ) >=(lower_bound rng(f|divset(D1,j)))*vol(divset(D2,ID1)) by
XREAL_1:19;
                      (lower_bound rng(f|divset(D1,j)))*vol(divset(D2,
ID1)) >=(lower_bound rng f)*vol(divset(D2,ID1)) by A108,A123,XREAL_1:64;
                      then Sum mid(LVf(D1),j,j)-(lower_bound rng f)* vol(
divset(D2,ID2) ) >=(lower_bound rng f)*vol(divset(D2,ID1)) by A129,XXREAL_0:2;
                      then
A130:                 Sum mid(LVf(D1),j,j) >=(lower_bound rng f)*vol(
divset(D2,ID2))+(lower_bound rng f)* vol(divset(D2,ID1)) by XREAL_1:19;
                      Sum mid(LVf(D2),(indx(D2,D1,j1)+1),indx(D2,D1,j))
=(lower_bound rng(f|divset(D2,indx(D2,D1,j1)+2))) *vol(divset(D2,indx(D2,D1,j1)
+2)) +LVf(D2).(indx(D2,D1,j1)+1) by A107,A126,A125,INTEGRA1:def 7
                        .=(lower_bound rng(f|divset(D2,indx(D2,D1,j1)+2))) *
vol(divset(D2,indx(D2,D1,j1)+2)) +(lower_bound rng(f|divset(D2,indx(D2,D1,j1)+1
))) *vol(divset(D2,indx(D2,D1,j1)+1)) by A92,INTEGRA1:def 7;
                      then Sum mid(LVf(D2),ID1,indx(D2,D1,j)) -(lower_bound
rng(f|divset(D2,ID1)))*vol(divset(D2,ID1)) <=(upper_bound rng f)*vol(divset(D2,
                      ID2)) by A110,A127,XREAL_1:64;
                      then Sum mid(LVf(D2),ID1,indx(D2,D1,j)) <=(upper_bound
rng f)*vol(divset(D2,ID2)) +(lower_bound rng(f|divset(D2,ID1)))*vol(divset(D2,
                      ID1)) by XREAL_1:20;
                      then Sum mid(LVf(D2),ID1,indx(D2,D1,j))-(upper_bound
rng f)* vol(divset(D2, ID2)) <=(lower_bound rng(f|divset(D2,ID1)))*vol(divset(
                      D2,ID1)) by XREAL_1:20;
                      then Sum mid(LVf(D2),ID1,indx(D2,D1,j))-(upper_bound
rng f)* vol(divset(D2, ID2)) <=(upper_bound rng f)*vol(divset(D2,ID1)) by A124,
XXREAL_0:2;
                      then Sum mid(LVf(D2),ID1,indx(D2,D1,j))<=(upper_bound
rng f)*vol(divset(D2,ID2 ))+ (upper_bound rng f)*vol(divset(D2,ID1)) by
XREAL_1:20;
                      then Sum mid(LVf(D2),ID1,indx(D2,D1,j))-Sum mid(LVf(D1)
,j,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 A130,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 A42,Lm5,
XREAL_1:64;
                    hence thesis by A120,A122,XXREAL_0:2;
                  end;
                end;
                hence thesis;
              end;
              j < j+1 by NAT_1:13;
              then
A131:         j1 < j by XREAL_1:19;
              indx(D2,D1,j) in dom D2 by A13,A42,INTEGRA1:def 19;
              then
A132:         indx(D2,D1,j) in Seg len D2 by FINSEQ_1:def 3;
              then
A133:         1 <= indx(D2,D1,j) by FINSEQ_1:1;
A134:         indx(D2,D1,j1) <= len D2 by A49,FINSEQ_1:1;
              then
A135:         len (D2|indx(D2,D1,j1))=indx(D2,D1,j1) by FINSEQ_1:59;
A136:         j1 in Seg len D1 by A47,FINSEQ_1:def 3;
              then
A137:         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
A138:           k in dom(D1|j1) and
A139:           x1=(D1|j1).k by PARTFUN1:3;
                k in Seg len(D1|j1) by A138,FINSEQ_1:def 3;
                then
A140:           k in Seg j1 by A137,FINSEQ_1:59;
                then
A141:           k in dom D1 by A45,RFINSEQ:6;
                k <= j1 by A140,FINSEQ_1:1;
                then D1.k <= D1.j1 by A47,A141,SEQ_4:137;
                then D2.indx(D2,D1,k) <= D1.j1 by A13,A141,INTEGRA1:def 19;
                then
A142:           D2.indx(D2,D1,k)<=D2.indx(D2,D1,j1) by A13,A47,INTEGRA1:def 19;
A143:           (D1|j1).k = D1.k by A45,A140,RFINSEQ:6;
                D1.k in rng D1 by A141,FUNCT_1:def 3;
                then x1 in rng D2 by A14,A139,A143,XBOOLE_0:def 3;
                then consider n such that
A144:           n in dom D2 and
A145:           x1=D2.n by PARTFUN1:3;
                D2.indx(D2,D1,k)=D2.n by A13,A139,A143,A141,A145,
INTEGRA1:def 19;
                then
A146:           n <= indx(D2,D1,j1) by A48,A144,A142,SEQM_3:def 1;
                1 <= n by A144,FINSEQ_3:25;
                then
A147:           n in Seg indx(D2,D1,j1) by A146,FINSEQ_1:1;
                then n in Seg len(D2|indx(D2,D1,j1)) by A134,FINSEQ_1:59;
                then
A148:           n in dom(D2|indx(D2,D1,j1)) by FINSEQ_1:def 3;
                D2.n = (D2|indx(D2,D1,j1)).n by A48,A147,RFINSEQ:6;
                hence thesis by A145,A148,FUNCT_1:def 3;
              end;
              then
A149:         rng(D1|j1) c= rng(D2|indx(D2,D1,j1));
A150:         1 <= j1 by A136,FINSEQ_1:1;
              lower_bound divset(D1,j) <= D.1 by A43,INTEGRA2:1;
              then
A151:         D1.j1 <= D.1 by A42,A44,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
A152:           k in dom(D2|indx(D2,D1,j1)) and
A153:           x1=(D2|indx(D2,D1,j1)).k by PARTFUN1:3;
                k in Seg len(D2|indx(D2,D1,j1)) by A152,FINSEQ_1:def 3;
                then
A154:           k in Seg indx(D2,D1,j1) by A134,FINSEQ_1:59;
                then
A155:           k in dom D2 by A48,RFINSEQ:6;
A156:           len(D1|j1) = j1 by A137,FINSEQ_1:59;
                k <= indx(D2,D1,j1) by A154,FINSEQ_1:1;
                then D2.k <= D2.indx(D2,D1,j1) by A48,A155,SEQ_4:137;
                then
A157:           D2.k <= D1.j1 by A13,A47,INTEGRA1:def 19;
A158:           D2.k in rng D1 implies D2.k in rng(D1|j1)
                proof
                  assume D2.k in rng D1;
                  then consider m such that
A159:             m in dom D1 and
A160:             D2.k = D1.m by PARTFUN1:3;
                  m in Seg len D1 by A159,FINSEQ_1:def 3;
                  then
A161:             1 <= m by FINSEQ_1:1;
A162:             m <= j1 by A45,A157,A159,A160,SEQM_3:def 1;
                  then m in Seg j1 by A161,FINSEQ_1:1;
                  then
A163:             D2.k = (D1|j1).m by A45,A160,RFINSEQ:6;
                  m in dom (D1|j1) by A156,A161,A162,FINSEQ_3:25;
                  hence thesis by A163,FUNCT_1:def 3;
                end;
A164:           D2.k in rng D implies D2.k = D1.j1
                proof
                  assume D2.k in rng D;
                  then consider n such that
A165:             n in dom D and
A166:             D2.k=D.n by PARTFUN1:3;
                  1 <= n by A165,FINSEQ_3:25;
                  then D.1 <= D2.k by A41,A165,A166,SEQ_4:137;
                  then D1.j1 <= D2.k by A151,XXREAL_0:2;
                  hence thesis by A157,XXREAL_0:1;
                end;
A167:           D2.k in rng D implies D2.k in rng(D1|j1)
                proof
                  j1 in Seg len(D1|j1) by A150,A156,FINSEQ_1:1;
                  then j1 in dom(D1|j1) by FINSEQ_1:def 3;
                  then
A168:             (D1|j1).j1 in rng(D1|j1) by FUNCT_1:def 3;
                  assume
A169:             D2.k in rng D;
                  j1 in Seg j1 by A150,FINSEQ_1:1;
                  hence thesis by A45,A164,A169,A168,RFINSEQ:6;
                end;
                D2.k in rng D2 by A155,FUNCT_1:def 3;
                hence thesis by A14,A48,A153,A154,A167,A158,RFINSEQ:6
,XBOOLE_0:def 3;
              end;
              then rng(D2|indx(D2,D1,j1)) c= rng (D1|j1);
              then
A170:         rng (D2|indx(D2,D1,j1)) = rng (D1|j1) by A149,XBOOLE_0:def 10;
              mid(D1,1,j1) is increasing by A42,A44,A150,INTEGRA1:7,35;
              then
A171:         D1|j1 is increasing by A150,FINSEQ_6:116;
              then
A172:         D2|indx(D2,D1,j1)=D1|j1 by A51,A170,Th6;
A173:         for k st 1 <= k & k <= j1 holds k=indx(D2,D1,k)
              proof
                let k;
                assume that
A174:           1 <= k and
A175:           k <= j1;
                assume
A176:           k<>indx(D2,D1,k);
                now
                  per cases by A176,XXREAL_0:1;
                  suppose
A177:               k > indx(D2,D1,k);
                    k <= len D1 by A137,A175,XXREAL_0:2;
                    then
A178:               k in dom D1 by A174,FINSEQ_3:25;
                    then indx(D2,D1,k) in dom D2 by A13,INTEGRA1:def 19;
                    then indx(D2,D1,k) in Seg len D2 by FINSEQ_1:def 3;
                    then
A179:               1<=indx(D2,D1,k) by FINSEQ_1:1;
A180:               indx(D2,D1,k) < j1 by A175,A177,XXREAL_0:2;
                    then
A181:               indx(D2,D1,k) in Seg j1 by A179,FINSEQ_1:1;
                    indx(D2,D1,k)<= indx(D2,D1,j1) by A13,A45,A175,A178,Th7;
                    then indx(D2,D1,k) in Seg indx(D2,D1,j1) by A179,FINSEQ_1:1
;
                    then
A182:               (D2|indx(D2,D1,j1)).indx(D2,D1,k)= D2.indx(D2,D1,k)
                    by A48,RFINSEQ:6;
                    indx(D2,D1,k) <= len D1 by A137,A180,XXREAL_0:2;
                    then indx(D2,D1,k) in dom D1 by A179,FINSEQ_3:25;
                    then
A183:               D1.k > D1.indx(D2,D1,k) by A177,A178,SEQM_3:def 1;
                    D1.k=D2. indx(D2,D1,k) by A13,A178,INTEGRA1:def 19;
                    hence contradiction by A45,A172,A182,A183,A181,RFINSEQ:6;
                  end;
                  suppose
A184:               k < indx(D2,D1,k);
                    k <= len D1 by A137,A175,XXREAL_0:2;
                    then
A185:               k in dom D1 by A174,FINSEQ_3:25;
                    then indx(D2,D1,k) <= indx(D2,D1,j1) by A13,A45,A175,Th7;
                    then
A186:               k <= indx(D2,D1,j1) by A184,XXREAL_0:2;
                    then k <= len D2 by A134,XXREAL_0:2;
                    then
A187:               k in dom D2 by A174,FINSEQ_3:25;
                    k in Seg j1 by A174,A175,FINSEQ_1:1;
                    then
A188:               D1.k = (D1|j1).k by A47,RFINSEQ:6;
                    indx(D2,D1,k) in dom D2 by A13,A185,INTEGRA1:def 19;
                    then
A189:               D2.k < D2.indx(D2,D1,k) by A184,A187,SEQM_3:def 1;
A190:               k in Seg indx(D2,D1,j1) by A174,A186,FINSEQ_1:1;
                    D1.k=D2. indx(D2,D1,k) by A13,A185,INTEGRA1:def 19;
                    hence contradiction by A48,A172,A188,A189,A190,RFINSEQ:6;
                  end;
                end;
                hence contradiction;
              end;
A191:         for k be Nat st 1 <= k & k <= len(lower_volume(f,D1)|j1)
holds (lower_volume(f,D1)|j1).k = (lower_volume(f,D2)|indx(D2,D1,j1)).k
              proof
                indx(D2,D1,j1) in Seg len D2 by A48,FINSEQ_1:def 3;
                then indx(D2,D1,j1) in Seg len lower_volume(f,D2) by
INTEGRA1:def 7;
                then
A192:           indx(D2,D1,j1) in dom lower_volume(f,D2) by FINSEQ_1:def 3;
                let k be Nat;
                assume that
A193:           1 <= k and
A194:           k <= len(lower_volume(f,D1)|j1);
                reconsider k as Element of NAT by ORDINAL1:def 12;
A195:           len(lower_volume(f,D1)) = len D1 by INTEGRA1:def 7;
                then
A196:           k <= j1 by A137,A194,FINSEQ_1:59;
                then k <= len D1 by A137,XXREAL_0:2;
                then
A197:           k in Seg len D1 by A193,FINSEQ_1:1;
                then
A198:           k in dom D1 by FINSEQ_1:def 3;
                then
A199:           indx(D2,D1,k) in dom D2 by A13,INTEGRA1:def 19;
A200:           k in Seg j1 by A193,A196,FINSEQ_1:1;
                then indx(D2,D1,k) in Seg j1 by A173,A193,A196;
                then
A201:           indx(D2,D1,k) in Seg indx(D2,D1,j1) by A150,A173;
                then indx(D2,D1,k)<=indx(D2,D1,j1) by FINSEQ_1:1;
                then
A202:           indx(D2,D1,k)<=len D2 by A134,XXREAL_0:2;
A203:           D1.k=D2.indx (D2,D1,k) by A13,A198,INTEGRA1:def 19;
A204:           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
A205:               k=1;
                    then
A206:               upper_bound divset(D1,k)=D1.k by A198,INTEGRA1:def 4;
A207:               lower_bound divset(D1,k)= lower_bound A by A198,A205,
INTEGRA1:def 4;
                    indx(D2,D1,k)=1 by A150,A173,A205;
                    hence thesis by A199,A203,A207,A206,INTEGRA1:def 4;
                  end;
                  suppose
A208:               k<>1;
                    then reconsider k1=k-1 as Element of NAT by A198,INTEGRA1:7
;
                    k <= k+1 by NAT_1:11;
                    then k1 <= k by XREAL_1:20;
                    then
A209:               k1 <= j1 by A196,XXREAL_0:2;
A210:               k-1 in dom D1 by A198,A208,INTEGRA1:7;
                    then k1 in Seg len D1 by FINSEQ_1:def 3;
                    then 1 <= k1 by FINSEQ_1:1;
                    then k1=indx(D2,D1,k1) by A173,A209;
                    then
A211:               D2.(indx(D2,D1,k)-1)= D2.indx(D2,D1,k1) by A173,A193,A196;
A212:               indx(D2,D1,k)<>1 by A173,A193,A196,A208;
                    then
A213:               lower_bound divset(D2,indx(D2,D1,k))= D2.(indx(D2,D1
                    ,k)-1) by A199,INTEGRA1:def 4;
A214:               upper_bound divset(D2,indx(D2,D1,k))=D2.indx(D2,D1,k
                    ) by A199,A212,INTEGRA1:def 4;
A215:               upper_bound divset(D1,k)=D1.k by A198,A208,INTEGRA1:def 4;
                    lower_bound divset(D1,k)=D1.(k-1) by A198,A208,
INTEGRA1:def 4;
                    hence thesis by A13,A198,A215,A210,A213,A214,A211,
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
A216:           divset(D1,k)=divset(D2,indx(D2,D1,k)) by A204,INTEGRA1:4;
A217:           k in dom D1 by A197,FINSEQ_1:def 3;
                j1 in Seg len(lower_volume(f,D1)) by A45,A195,FINSEQ_1:def 3;
                then j1 in dom(lower_volume(f,D1)) by FINSEQ_1:def 3;
                then
A218:           (lower_volume(f,D1)|j1).k = lower_volume(f,D1).k by A200,
RFINSEQ:6
                  .=(lower_bound(rng(f|divset(D2,indx(D2,D1,k)))))*vol(
                divset(D2,indx(D2,D1,k))) by A217,A216,INTEGRA1:def 7;
                1<=indx(D2,D1,k) by A173,A193,A196;
                then indx(D2,D1,k) in Seg len D2 by A202,FINSEQ_1:1;
                then
A219:           indx(D2,D1,k) in dom D2 by FINSEQ_1:def 3;
                (lower_volume(f,D2)|indx(D2,D1,j1)).k =(lower_volume(f,
                D2)|indx(D2,D1,j1)).indx(D2,D1,k) by A173,A193,A196
                  .=lower_volume(f,D2).indx(D2,D1,k) by A201,A192,RFINSEQ:6
                  .=(lower_bound(rng(f|divset(D2,indx(D2,D1,k)))))*vol(
                divset(D2,indx(D2,D1,k))) by A219,INTEGRA1:def 7;
                hence thesis by A218;
              end;
              indx(D2,D1,j1) in dom D2 by A13,A47,INTEGRA1:def 19;
              then indx(D2,D1,j1) <= len D2 by FINSEQ_3:25;
              then
A220:         indx(D2,D1,j1) <= len lower_volume(f,D2) by INTEGRA1:def 7;
              j1 <= len D1 by A47,FINSEQ_3:25;
              then
A221:         j1 <= len lower_volume(f,D1) by INTEGRA1:def 7;
              len (D2|indx(D2,D1,j1))=len (D1|j1) by A51,A171,A170,Th6;
              then indx(D2,D1,j1) =j1 by A137,A135,FINSEQ_1:59;
              then len(lower_volume(f,D1)|j1)=indx(D2,D1,j1) by A221,
FINSEQ_1:59;
              then len(lower_volume(f,D1)|j1)=len(lower_volume(f, D2)|indx(D2
              ,D1,j1)) by A220,FINSEQ_1:59;
              then
A222:         lower_volume(f,D2)|indx(D2,D1,j1)=lower_volume (f,D1)|j1
              by A191,FINSEQ_1:14;
A223:         j in Seg len D1 by A42,FINSEQ_1:def 3;
              then
A224:         1 <= j by FINSEQ_1:1;
              indx(D2,D1,j) in Seg len LVf(D2) by A132,INTEGRA1:def 7;
              then
A225:         indx(D2,D1,j) in dom LVf(D2) by FINSEQ_1:def 3;
              indx(D2,D1,j) <= len D2 by A132,FINSEQ_1:1;
              then
A226:         indx(D2,D1,j) <= len LVf(D2) by INTEGRA1:def 7;
              j in Seg len LVf(D1) by A223,INTEGRA1:def 7;
              then
A227:         j in dom LVf(D1) by FINSEQ_1:def 3;
              j <= len D1 by A223,FINSEQ_1:1;
              then
A228:         j <= len LVf(D1) by INTEGRA1:def 7;
              j1 in Seg len D1 by A45,FINSEQ_1:def 3;
              then j1 in Seg len LVf(D1) by INTEGRA1:def 7;
              then j1 in dom LVf(D1) by FINSEQ_1:def 3;
              then PLf(D1,j1)=Sum(LVf(D1)|j1) by INTEGRA1:def 20;
              then PLf(D1,j1)+Sum mid(LVf(D1),j,j) =Sum((LVf(D1)|j1)^mid(LVf(
              D1),j,j)) by RVSUM_1:75
                .=Sum(mid(LVf(D1),1,j1)^mid(LVf(D1),j1+1,j)) by A150,
FINSEQ_6:116
                .=Sum(mid(LVf(D1),1,j)) by A150,A228,A131,INTEGRA2:4
                .=Sum(LVf(D1)|j) by A224,FINSEQ_6:116;
              then
A229:         PLf(D1,j1)+Sum mid(lower_volume(f,D1),j,j)=PLf(D1,j) by A227,
INTEGRA1:def 20;
              indx(D2,D1,j1) in Seg len D2 by A48,FINSEQ_1:def 3;
              then indx(D2,D1,j1) in Seg len LVf(D2) by INTEGRA1:def 7;
              then indx(D2,D1,j1) in dom LVf(D2) by FINSEQ_1:def 3;
              then PLf(D2,indx(D2,D1,j1)) =Sum(LVf(D2)|indx(D2,D1,j1)) by
INTEGRA1:def 20;
              then PLf(D2,indx(D2,D1,j1)) +Sum mid(lower_volume(f,D2),(indx(
D2,D1,j1)+1),indx(D2,D1,j)) =Sum(LVf(D2)|indx(D2,D1,j1) ^mid(LVf(D2),(indx(D2,
              D1,j1)+1),indx(D2,D1,j))) by RVSUM_1:75
                .=Sum(mid(LVf(D2),1,indx(D2,D1,j1)) ^mid(LVf(D2),indx(D2,D1,
              j1)+1,indx(D2,D1,j))) by A50,FINSEQ_6:116
                .=Sum(mid(LVf(D2),1,indx(D2,D1,j))) by A50,A52,A226,INTEGRA2:4
                .=Sum(LVf(D2)|indx(D2,D1,j)) by A133,FINSEQ_6:116;
              then
A230:         PLf(D2,indx(D2,D1,j1)) +Sum mid(lower_volume(f,D2),(indx(
D2,D1,j1)+1), indx(D2,D1,j)) =PLf(D2,indx(D2,D1,j)) by A225,INTEGRA1:def 20;
              indx(D2,D1,j1) in Seg len D2 by A48,FINSEQ_1:def 3;
              then indx(D2,D1,j1) in Seg len lower_volume(f,D2) by
INTEGRA1:def 7;
              then indx(D2,D1,j1) in dom lower_volume(f,D2) by FINSEQ_1:def 3;
              then PLf(D2,indx(D2,D1,j1))=Sum(lower_volume(f,D2)|indx(D2,D1,
              j1)) by INTEGRA1:def 20
                .=PLf(D1,j1) by A222,A46,INTEGRA1:def 20;
              hence thesis by A54,A230,A229;
            end;
            hence thesis by A42,A43;
          end;
          reconsider i as non zero Element of NAT by A18,FINSEQ_1:1;
A231:     for i being non zero Nat st P[i] holds P[i+1]
          proof
            let i be non zero Nat;
A232:       i>=1 by NAT_1:14;
            assume
A233:       P[i];
            P[i+1]
            proof
A234:         i <= i+1 by NAT_1:11;
              assume
A235:         i+1 in dom D;
              then consider j such that
A236:         j in dom D1 and
A237:         D.(i+1) in divset(D1,j) by Th3,INTEGRA1:6;
A238:         D2.indx(D2,D1,j)=D1.j by A13,A236,INTEGRA1:def 19;
              i+1 <= len D by A235,FINSEQ_3:25;
              then i <= len D by A234,XXREAL_0:2;
              then
A239:         i in Seg len D by A232,FINSEQ_1:1;
              then
A240:         i in dom D by FINSEQ_1:def 3;
              consider n1 being Element of NAT such that
A241:         n1 in dom D1 and
A242:         D.i in divset(D1,n1) and
A243:         PLf(D2,indx(D2,D1,n1))-PLf(D1,n1)<=i*(upper_bound rng
              f- lower_bound rng f)*delta(D1) by A233,A239,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 A240,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 A235,FUNCT_1:def 3;
                    then
A249:               D.(i+1) in rng D /\ divset(D1,j) by A237,XBOOLE_0:def 4;
                    i+1 > i by XREAL_1:29;
                    hence contradiction by A11,A235,A236,A240,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 A236,FINSEQ_3:25;
                    then 1+1 <= j+1 by XREAL_1:6;
                    then
A252:               n1 <> 1 by A250,XXREAL_0:2;
                    lower_bound divset(D1,n1) <= D.i by A242,INTEGRA2:1;
                    then
A253:               D.i >= D1.(n1-1) by A241,A252,INTEGRA1:def 4;
                    n1-1 in dom D1 by A241,A252,INTEGRA1:7;
                    then D1.j <= D1.(n1-1) by A236,A251,SEQ_4:137;
                    then
A254:               D.i >= D1.j by A253,XXREAL_0:2;
A255:               i < i+1 by XREAL_1:29;
A256:               upper_bound divset(D1,j)=D1.j
                    proof
                      per cases;
                      suppose
                        j=1;
                        hence thesis by A236,INTEGRA1:def 4;
                      end;
                      suppose
                        j<>1;
                        hence thesis by A236,INTEGRA1:def 4;
                      end;
                    end;
                    D.(i+1)<=upper_bound divset(D1,j) by A237,INTEGRA2:1;
                    then D.i >= D.(i+1) by A256,A254,XXREAL_0:2;
                    hence contradiction by A235,A240,A255,SEQM_3:def 1;
                  end;
                end;
                hence thesis;
              end;
              then
A257:         n1+1 <= j by NAT_1:13;
A258:         1 <= n1 by A241,FINSEQ_3:25;
A259:         indx(D2,D1,n1) in dom D2 by A13,A241,INTEGRA1:def 19;
              then
A260:         1 <= indx(D2,D1,n1) by FINSEQ_3:25;
A261:         indx(D2,D1,j) in dom D2 by A13,A236,INTEGRA1:def 19;
              then
A262:         1 <= indx(D2,D1,j) by FINSEQ_3:25;
A263:         indx(D2,D1,j) <= len D2 by A261,FINSEQ_3:25;
              then
A264:         indx(D2,D1,j) <= len LVf(D2) by INTEGRA1:def 7;
A265:         1 <= j by A236,FINSEQ_3:25;
A266:         j <= len D1 by A236,FINSEQ_3:25;
              then
A267:         n1+1 <= len D1 by A257,XXREAL_0:2;
              then
A268:         n1+1 in dom D1 by A244,FINSEQ_3:25;
              then
A269:         indx(D2,D1,n1+1) in dom D2 by A13,INTEGRA1:def 19;
              then
A270:         1 <= indx(D2,D1,n1+1) by FINSEQ_3:25;
A271:         D2.indx(D2,D1,n1+ 1)=D1.(n1+1) by A13,A268,INTEGRA1:def 19;
              then D2.indx(D2,D1,n1+1) <= D2.indx(D2,D1,j) by A236,A257,A268
,A238,SEQ_4:137;
              then
A272:         indx(D2,D1,n1+1) <= indx(D2,D1,j) by A269,A261,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
A273:         mid(D2,indx(D2,D1,n1+1),indx(D2,D1,j)).1 =D2.(1-1+indx(D2,
              D1,n1+1)) by A272,A270,A263,FINSEQ_6:122
                .=D1.(n1+1) by A13,A268,INTEGRA1:def 19;
A274:         D2.indx(D2,D1,n1) = D1.n1 by A13,A241,INTEGRA1:def 19;
A275:         j <= len LVf(D1) by A266,INTEGRA1:def 7;
              then j in Seg len LVf(D1) by A265,FINSEQ_1:1;
              then
A276:         j in dom LVf(D1) by FINSEQ_1:def 3;
A277:         indx(D2,D1,n1+1) <= len D2 by A269,FINSEQ_3:25;
              n1 in Seg len D1 by A241,FINSEQ_1:def 3;
              then n1 in Seg len LVf(D1) by INTEGRA1:def 7;
              then n1 in dom LVf(D1) by FINSEQ_1:def 3;
              then PLf(D1,n1)=Sum(LVf(D1)|n1) by INTEGRA1:def 20
                .=Sum mid(LVf(D1),1,n1) by A258,FINSEQ_6:116;
              then PLf(D1,n1)+Sum mid(LVf(D1),n1+1,j) =Sum(mid(LVf(D1),1,n1)
              ^mid(LVf(D1),n1+1,j)) by RVSUM_1:75
                .=Sum mid(LVf(D1),1,j) by A245,A258,A275,INTEGRA2:4
                .=Sum(LVf(D1)|j) by A265,FINSEQ_6:116;
              then
A278:         PLf(D1,j)=PLf(D1,n1)+Sum mid(LVf(D1),n1+1,j) by A276,
INTEGRA1:def 20;
              indx(D2,D1,j) in Seg len D2 by A261,FINSEQ_1:def 3;
              then indx(D2,D1,j) in Seg len LVf(D2) by INTEGRA1:def 7;
              then
A279:         indx(D2,D1,j) in dom LVf(D2) by FINSEQ_1:def 3;
A280:         n1 >= 1 by A241,FINSEQ_3:25;
A281:         j-n1 >= 1 by A257,XREAL_1:19;
              Sum mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) -Sum mid(
LVf(D1),n1+1,j) <= (upper_bound rng f-lower_bound rng f)*delta(D1)
              proof
                now
                  per cases by A257,XXREAL_0:1;
                  suppose
A282:               n1+1=j;
A283:               indx(D2,D1,j)-indx(D2,D1,n1)<=2
                    proof
A284:                 upper_bound divset(D1,j) = D1.j by A236,A245,A280,
INTEGRA1:def 4;
A285:                 lower_bound divset(D1,j) = D1.(j-1) by A236,A245,A280,
INTEGRA1:def 4;
A286:                 1 <= indx(D2,D1,n1)+1 by A260,NAT_1:13;
                      assume indx(D2,D1,j) - indx(D2,D1,n1) > 2;
                      then
A287:                 indx(D2,D1,n1)+2 < indx(D2,D1, j) by XREAL_1:20;
                      then
A288:                 indx(D2,D1,n1)+2 <= len D2 by A263,XXREAL_0:2;
A289:                 indx(D2,D1,n1)+1 < indx(D2,D1,n1)+2 by XREAL_1:6;
                      then
A290:                 indx(D2,D1,n1) < indx(D2,D1,n1)+2 by NAT_1:13;
                      then 1 <= indx(D2,D1,n1)+2 by A260,XXREAL_0:2;
                      then
A291:                 indx(D2,D1,n1)+2 in dom D2 by A288,FINSEQ_3:25;
                      then
A292:                 D2.indx(D2,D1,j) >= D2.(indx(D2,D1,n1)+2) by A261,A287,
SEQ_4:137;
A293:                 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
A294:                   k1 in dom D1 and
A295:                   D2.(indx(D2,D1,n1)+2) = D1.k1 by PARTFUN1:3;
                        D2.(indx(D2,D1,n1)+2) < D2.indx(D2,D1,j) by A261,A287
,A291,SEQM_3:def 1;
                        then
A296:                   k1 < j by A236,A238,A294,A295,SEQ_4:137;
                        D2.indx(D2,D1,n1) < D2.(indx(D2,D1,n1)+2) by A259,A290
,A291,SEQM_3:def 1;
                        then n1 < k1 by A241,A274,A294,A295,SEQ_4:137;
                        hence contradiction by A282,A296,NAT_1:13;
                      end;
                      D2.(indx(D2,D1,n1)+2) in rng D2 by A291,FUNCT_1:def 3;
                      then
A297:                 D2.(indx(D2,D1,n1)+2) in rng D by A14,A293,XBOOLE_0:def 3
;
A298:                 lower_bound divset(D1,j) = D1.(j-1) by A236,A245,A280,
INTEGRA1:def 4;
A299:                 upper_bound divset(D1,j) = D1.j by A236,A245,A280,
INTEGRA1:def 4;
                      D2.(indx(D2,D1,n1)+2) >= D2.indx(D2,D1,n1) by A259,A290
,A291,SEQ_4:137;
                      then D2.(indx(D2,D1,n1)+2) in divset(D1,j) by A274,A238
,A282,A298,A284,A292,INTEGRA2:1;
                      then
A300:                 D2.(indx(D2,D1,n1)+2) in rng D /\ divset(D1,j) by A297,
XBOOLE_0:def 4;
A301:                 indx(D2,D1,n1)+1 < indx(D2,D1,j) by A287,A289,XXREAL_0:2;
                      then indx(D2,D1,n1)+1 <= len D2 by A263,XXREAL_0:2;
                      then
A302:                 indx(D2,D1,n1)+1 in dom D2 by A286,FINSEQ_3:25;
                      then
A303:                 D2.indx(D2,D1,j) >= D2.(indx(D2,D1,n1)+1) by A261,A301,
SEQ_4:137;
A304:                 indx(D2,D1,n1) < indx(D2,D1,n1)+1 by NAT_1:13;
A305:                 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
A306:                   k1 in dom D1 and
A307:                   D2.(indx(D2,D1,n1)+1) = D1.k1 by PARTFUN1:3;
                        D2.(indx(D2,D1,n1)+1) < D2.indx(D2,D1,j) by A261,A301
,A302,SEQM_3:def 1;
                        then
A308:                   k1 < j by A236,A238,A306,A307,SEQ_4:137;
                        D2.indx(D2,D1,n1) < D2.(indx(D2,D1,n1)+1) by A259,A304
,A302,SEQM_3:def 1;
                        then n1 < k1 by A241,A274,A306,A307,SEQ_4:137;
                        hence contradiction by A282,A308,NAT_1:13;
                      end;
                      D2.(indx(D2,D1,n1)+1) in rng D2 by A302,FUNCT_1:def 3;
                      then
A309:                 D2.(indx(D2,D1,n1)+1) in rng D by A14,A305,XBOOLE_0:def 3
;
                      D2.(indx(D2,D1,n1)+1) >= D2.indx(D2,D1,n1) by A259,A304
,A302,SEQ_4:137;
                      then D2.(indx(D2,D1,n1)+1) in divset(D1,j) by A274,A238
,A282,A285,A299,A303,INTEGRA2:1;
                      then D2.(indx(D2,D1,n1)+1) in rng D /\ divset(D1,j) by
A309,XBOOLE_0:def 4;
                      then D2.(indx(D2,D1,n1)+1) = D2.(indx(D2,D1,n1)+2) by A11
,A236,A300,Th5;
                      hence contradiction by A289,A302,A291,SEQM_3:def 1;
                    end;
A310:               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
A311:                 indx(D2,D1,n1)+1+1 <= indx (D2,D1,j) by NAT_1:13;
                      indx(D2,D1,n1)+2 >= indx(D2,D1,j) by A283,XREAL_1:20;
                      hence thesis by A311,XXREAL_0:1;
                    end;
A312:               1<=indx(D2,D1,n1)+1 by NAT_1:12;
A313:               indx(D2,D1,j) <= len LVf(D2) by A263,INTEGRA1:def 7;
                    D1.n1 < D1.j by A236,A241,A245,SEQM_3:def 1;
                    then
A314:               indx(D2,D1,n1)<indx(D2,D1,j) by A259,A274,A261,A238,
SEQ_4:137;
                    then
A315:               indx(D2,D1,n1)+1 <= indx(D2,D1,j) by NAT_1:13;
                    then indx(D2,D1,n1)+1 <= len D2 by A263,XXREAL_0:2;
                    then indx(D2,D1,n1)+1 <= len LVf(D2) by INTEGRA1:def 7;
                    then
A316:               len mid(LVf(D2),indx(D2,D1,n1)+1, indx(D2,D1,j)) =
indx(D2,D1,j)-'(indx(D2,D1,n1)+1)+1 by A262,A315,A312,A313,FINSEQ_6:118
                      .=indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1 by A315,XREAL_1:233
                      .=indx(D2,D1,j)-indx(D2,D1,n1);
                    indx(D2,D1,n1)+1 <= indx(D2,D1,j) by A314,NAT_1:13;
                    then
A317:               indx(D2,D1,n1)+1 = indx(D2,D1, j) or indx(D2,D1,n1)+
                    1 < indx(D2,D1,j) by XXREAL_0:1;
A318:               Sum mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) <=(
                    upper_bound rng f)*vol(divset(D1,n1+1))
                    proof
                      per cases by A317,A310;
                      suppose
A319:                   indx(D2,D1,j)-indx(D2,D1,n1)=1;
A320:                   indx(D2,D1,n1)+1 > 1 by A260,NAT_1:13;
                        then upper_bound divset(D2,indx(D2,D1,n1)+1)=D2.(indx
                        (D2,D1,n1)+1) by A261,A319,INTEGRA1:def 4;
                        then
A321:                   upper_bound divset(D2,indx(D2,D1,n1)+1)=D1.j by A13
,A236,A319,INTEGRA1:def 19;
                        lower_bound divset(D2,indx(D2,D1,n1)+1)=D2.(indx
                        (D2,D1,n1)+1-1) by A261,A319,A320,INTEGRA1:def 4;
                        then
A322:                   lower_bound divset(D2,indx(D2, D1,n1)+1)=D1.n1
                        by A13,A241,INTEGRA1:def 19;
                        lower_bound divset(D1,n1+1)=D1.(n1+1-1) by A245,A280
,A268,A282,INTEGRA1:def 4;
                        then
A323:                   divset(D2,indx(D2,D1,n1)+1)=divset(D1,n1+1) by A245
,A280,A268,A282,A322,A321,INTEGRA1:def 4;
A324:                   vol(divset(D2,indx(D2,D1,n1)+1))>=0 by INTEGRA1:9;
        reconsider LV = LVf(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 A319;
                        then mid(LVf(D2),indx(D2,D1,n1) +1,indx(D2,D1,j)).1 =
LVf(D2).(1+(indx(D2,D1,n1)+1)-1) by A312,A313,FINSEQ_6:122
                          .=LVf(D2).(indx(D2,D1,n1)+1);
                        then
A325:                   mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j))
            =<*LV*> by A316,A319,FINSEQ_1:40;
                        LVf(D2).(indx(D2,D1,n1)+1) =(lower_bound rng(f|
divset(D2,indx(D2,D1,n1)+1))) *vol(divset(D2,indx(D2,D1,n1)+1)) by A261,A319,
INTEGRA1:def 7;
                        then LVf(D2).(indx(D2,D1,n1)+1) <=(upper_bound rng f)
*vol(divset(D1,n1+1)) by A1,A261,A319,A323,A324,Th18,XREAL_1:64;
                        hence thesis by A325,FINSOP_1:11;
                      end;
                      suppose
A326:                   indx(D2,D1,j)-indx(D2,D1,n1)=2;
                        indx(D2,D1,n1)+2 >= 2+1 by A260,XREAL_1:6;
                        then
A327:                   indx(D2,D1,n1)+2 <> 1;
                        then
A328:                   upper_bound divset(D2,indx(D2,D1,n1)+2)= D2.indx
                        (D2,D1,j) by A261,A326,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 A261,A326,A327,INTEGRA1:def 4;
                        then
A329:                   vol(divset(D2,indx(D2,D1,n1)+2)) =D1.j-D2.(indx(
                        D2,D1,n1)+1) by A238,A328,INTEGRA1:def 5;
A330:                   upper_bound divset(D1,n1+1)=D1.(n1+1) by A245,A280,A268
,A282,INTEGRA1:def 4;
                        lower_bound divset(D1,n1+1)=D1.(n1+1-1) by A245,A280
,A268,A282,INTEGRA1:def 4;
                        then
A331:                   vol(divset(D1,n1+1))=D1.(n1+1)-D1.n1 by A330,
INTEGRA1:def 5;
A332:                   vol(divset(D2,indx(D2,D1,n1)+2)) >= 0 by INTEGRA1:9;
A333:                   indx(D2,D1,j) <= len LVf(D2) by A263,INTEGRA1:def 7;
A334:                   vol(divset(D2,indx(D2,D1,n1)+1)) >= 0 by INTEGRA1:9;
A335:                   1 <= indx(D2,D1,n1)+1 by NAT_1:12;
A336:                   indx(D2,D1,n1)+1 <= indx(D2,D1,n1)+2 by XREAL_1:6;
                        then indx(D2,D1,n1)+1 <= len D2 by A263,A326,XXREAL_0:2
;
                        then
A337:                   indx(D2,D1,n1)+1 in dom D2 by A335,FINSEQ_3:25;
                        then LVf(D2).(indx(D2,D1,n1 )+1) =(lower_bound rng(f|
divset(D2,indx(D2,D1,n1)+1))) *vol(divset(D2,indx(D2,D1,n1)+1)) by
INTEGRA1:def 7;
                        then
A338:                   LVf(D2).(indx(D2,D1,n1)+1) <= (upper_bound rng f
)*vol(divset(D2,indx(D2, D1,n1)+1)) by A1,A337,A334,Th18,XREAL_1:64;
                        indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1=1+1 by A326;
                        then
A339:                   mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)).2 =
LVf(D2).(2+(indx(D2,D1,n1)+1)-1) by A335,A336,A333,FINSEQ_6:122
                          .=LVf(D2).(indx(D2,D1,n1)+0+2);
                        indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1>=1 by A326;
                        then mid(LVf(D2),indx(D2,D1,n1) +1,indx(D2,D1,j)).1 =
LVf(D2).(1+(indx(D2,D1,n1)+1)-1) by A326,A335,A336,A333,FINSEQ_6:122
                          .=LVf(D2).(indx(D2,D1,n1)+1);
                        then mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) =<*
LVf(D2).(indx(D2,D1,n1)+1),LVf(D2).(indx(D2,D1,n1)+2)*> by A316,A326,A339,
FINSEQ_1:44;
                        then
A340:                   Sum mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j))
=LVf(D2).(indx(D2,D1,n1)+1)+LVf(D2).(indx(D2,D1,n1)+2) by RVSUM_1:77;
A341:                   indx(D2,D1,n1)+1 > 1 by A260,NAT_1:13;
                        then
A342:                   upper_bound divset(D2,indx(D2,D1,n1)+1)=D2.(indx
                        (D2,D1,n1)+1) by A337,INTEGRA1:def 4;
                        lower_bound divset(D2,indx(D2,D1,n1)+1) = D2.(
                        indx(D2,D1,n1)+1-1) by A337,A341,INTEGRA1:def 4;
                        then
A343:                   vol(divset(D2,indx(D2,D1,n1)+1)) =D2.(indx(D2,D1
                        ,n1)+1)-D1.n1 by A274,A342,INTEGRA1:def 5;
                        LVf(D2).(indx(D2,D1,n1 )+2) =(lower_bound rng(f|
divset(D2,indx(D2,D1,n1)+2))) *vol(divset(D2,indx(D2,D1,n1)+2)) by A261,A326,
INTEGRA1:def 7;
                        then LVf(D2).(indx(D2,D1,n1)+2) <= (upper_bound rng f
)*vol(divset(D2,indx(D2, D1,n1)+2)) by A1,A261,A326,A332,Th18,XREAL_1:64;
                        then Sum mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j))
<= (upper_bound rng f)*vol(divset(D2,indx(D2,D1,n1)+1)) + (upper_bound rng f)*
vol(divset(D2,indx(D2,D1,n1)+2)) by A340,A338,XREAL_1:7;
                        hence thesis by A282,A343,A329,A331;
                      end;
                    end;
A344:               n1+1 <= len LVf(D1) by A267,INTEGRA1:def 7;
                    then
A345:               len mid(LVf(D1),n1+1,j)=j-'(n1+1)+1 by A244,A282,
FINSEQ_6:118
                      .=j-j+1 by A282,XREAL_1:233
                      .= 1;
   reconsider lv =(lower_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 A257,XREAL_1:6;
                    then 1 <= j+1-(n1+1) by XREAL_1:19;
                    then mid(LVf(D1),n1+1,j).1 =LVf(D1).(1-1+(n1+1)) by A244
,A282,A344,FINSEQ_6:122
                      .=(lower_bound rng(f|divset(D1,n1+1)))* vol(divset(D1,
                    n1+1)) by A268,INTEGRA1:def 7;
                    then mid(LVf(D1),n1+1,j) =<*lv
*> by A345,FINSEQ_1:40;
                    then
A346:               Sum mid(LVf(D1),n1+1,j) =(lower_bound rng(f|divset(
                    D1,n1+1)))*vol(divset(D1,n1+1)) by FINSOP_1:11;
                    divset(D1,n1+1) c= A by A268,INTEGRA1:8;
                    then
A347:               lower_bound rng(f|divset(D1,n1+1)) >= lower_bound
                    rng f by A1,Lm4;
                    n1+1 in Seg len D1 by A268,FINSEQ_1:def 3;
                    then n1+1 in Seg len upper_volume(chi(A,A),D1) by
INTEGRA1:def 6;
                    then
A348:               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 A268,INTEGRA1:20;
                    then vol(divset(D1,n1+1)) in rng upper_volume(chi(A,A),D1
                    ) by A348,FUNCT_1:def 3;
                    then
A349:               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
A350:               (upper_bound rng f-lower_bound rng f)* vol(divset(D1
,n1+1)) <=(upper_bound rng f-lower_bound rng f)*delta(D1) by A349,XREAL_1:64;
                    vol(divset(D1,n1+1)) >= 0 by INTEGRA1:9;
                    then Sum mid(LVf(D1),n1+1,j) >=(lower_bound rng f)*vol(
                    divset(D1,n1+1)) by A346,A347,XREAL_1:64;
                    then Sum mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j))-Sum
mid(LVf(D1),n1+1 ,j) <=(upper_bound rng f)*vol(divset(D1,n1+1))-(lower_bound
                    rng f)* vol(divset(D1,n1+1)) by A318,XREAL_1:13;
                    hence thesis by A350,XXREAL_0:2;
                  end;
                  suppose
A351:               n1+1 < j;
A352:               n1 < n1+1 by NAT_1:13;
                    then
A353:               D1.n1 < D1.(n1+1) by A241,A268,SEQM_3:def 1;
                    then consider
                    B being non empty closed_interval Subset of REAL, MD1,MD2
                    being Division of B such that
A354:               D1.n1=lower_bound B and
                    upper_bound B=MD2.(len MD2) and
A355:               upper_bound B=MD1.(len MD1) and
A356:               MD1 <= MD2 and
A357:               MD1=mid(D1,n1+1,j) and
A358:               MD2=mid(D2,indx(D2,D1,n1+1),indx(D2,D1,j)) by A13,A236,A257
,A268,A273,Th15;
A359:               delta(MD1) >= 0 by Th9;
A360:               len MD1 = j-'(n1+1)+1 by A257,A265,A266,A244,A267,A357,
FINSEQ_6:118;
                    then
A361:               len MD1+(n1+1)-1 = j-(n1+1)+1+(n1+1)-1 by A257,XREAL_1:233
                      .=j;
                    j-'(n1+1) = j-(n1+1) by A257,XREAL_1:233;
                    then
A362:               j-'(n1+1)+1=j-n1;
                    then
A363:               len MD1 = j-n1 by A257,A265,A266,A244,A267,A357,
FINSEQ_6:118;
A364:               B c= A
                    proof
                      let x1 be object;
A365:                 rng D1 c= A by INTEGRA1:def 2;
                      D1.n1 in rng D1 by A241,FUNCT_1:def 3;
                      then
A366:                 lower_bound A <= D1.n1 by A365,INTEGRA2:1;
                      assume
A367:                 x1 in B;
                      then reconsider x1 as Real;
A368:                 x1 <= MD1.(len MD1) by A355,A367,INTEGRA2:1;
                      D1.j in rng D1 by A236,FUNCT_1:def 3;
                      then
A369:                 D1.j <= upper_bound A by A365,INTEGRA2:1;
                      D1 .n1 <= x1 by A354,A367,INTEGRA2:1;
                      then
A370:                 lower_bound A <= x1 by A366,XXREAL_0:2;
                      MD1.(len MD1)=D1.(j-n1-1+(n1+1)) by A257,A281,A266,A244
,A357,A362,A363,FINSEQ_6:122
                        .=D1.j;
                      then x1 <= upper_bound A by A368,A369,XXREAL_0:2;
                      hence thesis by A370,INTEGRA2:1;
                    end;
                    then reconsider g=f|B as Function of B,REAL by FUNCT_2:32;
A371:               len lower_volume(g,MD1)=len MD1 by INTEGRA1:def 7
                      .=j-'(n1+1)+1 by A257,A265,A266,A244,A267,A357,
FINSEQ_6:118
                      .=j-(n1+1)+1 by A257,XREAL_1:233;
A372:               len MD1 in dom MD1 by FINSEQ_5:6;
                    then
A373:               1 <= len MD1 by FINSEQ_3:25;
A374:               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
A375:                   len MD1=1;
                        then
A376:                   upper_bound divset(MD1,len MD1)=MD1.(len MD1) by A372,
INTEGRA1:def 4;
A377:                   upper_bound divset(D1,j)=D1.j by A236,A245,A280,
INTEGRA1:def 4;
                        lower_bound divset(D1,j)=D1.(j-1) by A236,A245,A280,
INTEGRA1:def 4;
                        hence
                        thesis by A265,A266,A354,A357,A361,A372,A375,A376,A377,
FINSEQ_6:118,INTEGRA1:def 4;
                      end;
                      suppose
A378:                   len MD1<>1;
                        then len MD1-1 in dom MD1 by A372,INTEGRA1:7;
                        then
A379:                   len MD1-1 >= 1 by FINSEQ_3:25;
                        len MD1 <= len MD1+1 by NAT_1:11;
                        then
A380:                   len MD1 -1 <= len MD1 by XREAL_1:20;
                        upper_bound divset(MD1,len MD1)=MD1.(len MD1) by A372
,A378,INTEGRA1:def 4;
                        then
A381:                   upper_bound divset(MD1,len MD1) =D1.j by A257,A266,A244
,A357,A360,A361,A373,FINSEQ_6:122;
A382:                   len MD1-1+(n1+1)-1=j-1 by A363;
                        lower_bound divset(MD1,len MD1)= MD1.(len MD1-1)
                        by A372,A378,INTEGRA1:def 4;
                        then lower_bound divset(MD1,len MD1)=D1.(j-1) by A257
,A266,A244,A357,A360,A382,A379,A380,FINSEQ_6:122;
                        hence thesis by A236,A245,A280,A381,INTEGRA1:def 4;
                      end;
                    end;
A383:               len MD1 in dom MD1 by FINSEQ_5:6;
A384:               upper_bound divset(MD1,len MD1)=MD1.(len MD1)
                    proof
                      per cases;
                      suppose
                        len MD1 = 1;
                        hence thesis by A383,INTEGRA1:def 4;
                      end;
                      suppose
                        len MD1 <> 1;
                        hence thesis by A383,INTEGRA1:def 4;
                      end;
                    end;
                    D1.n1 < D1.(n1+1) by A241,A268,A352,SEQM_3:def 1;
                    then indx(D2,D1,n1) < indx(D2,D1,n1+1) by A259,A274,A269
,A271,SEQ_4:137;
                    then
A385:               indx(D2,D1,n1)+1 <= indx(D2,D1,n1+ 1) by NAT_1:13;
                    then
A386:               indx(D2,D1,n1)+1 <= len D2 by A277,XXREAL_0:2;
                    vol(B)=upper_bound B-D1.n1 by A354,INTEGRA1:def 5;
                    then vol(B)=D1.j-D1.n1 by A236,A245,A280,A355,A374,A384,
INTEGRA1:def 4;
                    then
A387:               vol(B)<>0 by A236,A241,A245,SEQM_3:def 1;
A388:               1 <= indx(D2,D1,n1)+1 by A260,NAT_1:13;
A389:               indx(D2,D1,n1) < indx(D2,D1,n1)+1 by NAT_1:13;
A390:               indx(D2,D1,n1+1)=indx(D2,D1,n1)+1
                    proof
                      assume indx(D2,D1,n1+1)<> indx(D2,D1,n1)+1;
                      then
A391:                 indx(D2,D1,n1+1)>indx(D2,D1,n1 )+1 by A385,XXREAL_0:1;
A392:                 indx(D2,D1,n1)+1 in dom D2 by A388,A386,FINSEQ_3:25;
                      then
A393:                 D2.(indx(D2,D1,n1)+1) in rng D2 by FUNCT_1:def 3;
                      now
                        per cases by A14,A393,XBOOLE_0:def 3;
                        suppose
                          D2.(indx(D2,D1,n1)+1) in rng D1;
                          then consider n2 being Element of NAT such that
A394:                     n2 in dom D1 and
A395:                     D2.(indx(D2,D1,n1)+1) = D1.n2 by PARTFUN1:3;
                          D2.indx(D2,D1,n1) < D2.(indx(D2,D1,n1)+1) by A259
,A389,A392,SEQM_3:def 1;
                          then n1 < n2 by A241,A274,A394,A395,SEQ_4:137;
                          then
A396:                     n1+1 <= n2 by NAT_1:13;
                          D1.n2 < D1.(n1+1) by A269,A271,A391,A392,A395,
SEQM_3:def 1;
                          hence contradiction by A268,A394,A396,SEQ_4:137;
                        end;
                        suppose
A397:                     D2.(indx(D2,D1,n1)+1) in rng D;
A398:                     D.i <= upper_bound divset(D1,n1) by A242,INTEGRA2:1;
A399:                     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
A400:                     n2 in dom D and
A401:                     D2.(indx(D2,D1,n1)+1) = D.n2 by A397,PARTFUN1:3;
                          D1.n1 < D.n2 by A259,A274,A389,A392,A401,SEQM_3:def 1
;
                          then D.i < D.n2 by A398,A399,XXREAL_0:2;
                          then i < n2 by A240,A400,SEQ_4:137;
                          then
A402:                     i+1 <= n2 by NAT_1:13;
                          n1+1 +1 <= j by A351,NAT_1:13;
                          then
A403:                     n1+1 <= j-1 by XREAL_1:19;
                          j -1 in dom D1 by A236,A245,A280,INTEGRA1:7;
                          then
A404:                     D1.(n1+1) <= D1.(j-1) by A268,A403,SEQ_4:137;
A405:                     lower_bound divset(D1,j) <= D.(i+1) by A237,
INTEGRA2:1;
                          lower_bound divset(D1,j) = D1.(j-1) by A236,A245,A280
,INTEGRA1:def 4;
                          then
A406:                     D1.(n1+1) <= D.(i+1) by A404,A405,XXREAL_0:2;
                          D .n2 < D1.(n1+1) by A269,A271,A391,A392,A401,
SEQM_3:def 1;
                          then D.n2 < D.(i+1) by A406,XXREAL_0:2;
                          hence contradiction by A235,A400,A402,SEQ_4:137;
                        end;
                      end;
                      hence contradiction;
                    end;
A407:               j <= len LVf(D1) by A266,INTEGRA1:def 7;
A408:               for k be Nat st 1<=k & k<=len lower_volume(g,MD1)
                    holds lower_volume(g,MD1).k=mid(LVf(D1),n1+1,j).k
                    proof
                      let k be Nat;
                      assume that
A409:                 1 <= k and
A410:                 k <= len lower_volume(g,MD1);
                 k in Seg len lower_volume(g,MD1) by A409,A410,FINSEQ_1:1;
                      then
A411:                 k in Seg len MD1 by INTEGRA1:def 7;
                      then
A412:                 k in dom MD1 by FINSEQ_1:def 3;
                      k in dom MD1 by A411,FINSEQ_1:def 3;
                      then
A413:                 lower_volume(g,MD1).k =(lower_bound rng(g|divset(
                      MD1,k)))*vol(divset(MD1,k)) by INTEGRA1:def 7;
                      consider k2 being Element of NAT such that
A414:                 n1+1=1+k2;
A415:                 1 <= k+k2 by A409,NAT_1:12;
                      k <= j-((n1+1)-1) by A371,A410;
                      then k+((n1+1)-1) <= j by XREAL_1:19;
                      then k+k2 <= len D1 by A266,A414,XXREAL_0:2;
                      then
A416:                 k+k2 in Seg len D1 by A415,FINSEQ_1:1;
                      then
A417:                 k+k2 in dom D1 by FINSEQ_1:def 3;
                      1+1 <= k+k2 by A258,A409,A414,XREAL_1:7;
                      then
A418:                 1 < k+k2 by NAT_1:13;
A419:                 k2=(n1+1)-1 by A414;
A420:                 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
A421:                     k=1;
                          then upper_bound divset(MD1,k) =MD1.k by A412,
INTEGRA1:def 4;
                          then
A422:                     upper_bound divset(MD1,k)=D1.(k+(n1+1)-1) by A257
,A266,A244,A357,A371,A409,A410,FINSEQ_6:122;
                          lower_bound divset(MD1,k)=D1.n1 by A354,A412,A421,
INTEGRA1:def 4;
                          hence thesis by A419,A418,A417,A421,A422,
INTEGRA1:def 4;
                        end;
                        suppose
A423:                     k<>1;
                          then upper_bound divset(MD1,k)=MD1.k by A412,
INTEGRA1:def 4;
                          then
A424:                     upper_bound divset(MD1,k)=D1.(k+(n1+1)-1) by A257
,A266,A244,A357,A371,A409,A410,FINSEQ_6:122;
A425:                     k -1 <= j-(n1+1)+1 by A371,A410,XREAL_1:146
,XXREAL_0:2;
A426:                     lower_bound divset(MD1,k)=MD1.(k-1) by A412,A423,
INTEGRA1:def 4;
A427:                     k-1 in dom MD1 by A412,A423,INTEGRA1:7;
                          then 1 <= k-1 by FINSEQ_3:25;
                          then lower_bound divset(MD1,k) =D1.(k-1+(n1+1)-1)
                          by A257,A266,A244,A357,A427,A425,A426,FINSEQ_6:122;
                          hence thesis by A414,A418,A417,A424,INTEGRA1:def 4;
                        end;
                      end;
                      divset(MD1,k)=[.lower_bound divset(MD1,k),
                      upper_bound divset(MD1,k).] by INTEGRA1:4;
                      then
A428:                 divset(D1,k+k2)=divset(MD1,k) by A420,INTEGRA1:4;
A429:                 k+k2 in dom D1 by A416,FINSEQ_1:def 3;
A430:                 mid(LVf(D1),n1+1,j).k =LVf(D1).(k+(n1+1)-1) by A257,A244
,A371,A407,A409,A410,FINSEQ_6:122
                        .=(lower_bound rng(f|divset(D1,k+k2)))* vol(divset(
                      D1,k+k2)) by A414,A429,INTEGRA1:def 7;
                      k in dom MD1 by A411,FINSEQ_1:def 3;
                      then divset(D1,k+k2) c= B by A428,INTEGRA1:8;
                      hence thesis by A413,A430,A428,FUNCT_1:51;
                    end;
A431:               g|B is bounded
                    proof
                      consider a be Real such that
A432:                 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;
A433:                   dom f /\ B c= dom f /\ A by A364,XBOOLE_1:26;
                        assume x in B /\ dom g;
                        then
A434:                   x in dom g by XBOOLE_0:def 4;
                        then x in dom f /\ B by RELAT_1:61;
                        then a <= f.x by A432,A433;
                        hence thesis by A434,FUNCT_1:47;
                      end;
                      then
A435:                 g|B is bounded_below by RFUNCT_1:71;
                      consider a be Real such that
A436:                 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;
A437:                   dom f /\ B c= dom f /\ A by A364,XBOOLE_1:26;
                        assume x in B /\ dom g;
                        then
A438:                   x in dom g by XBOOLE_0:def 4;
                        then x in dom f /\ B by RELAT_1:61;
                        then a >= f.x by A436,A437;
                        hence thesis by A438,FUNCT_1:47;
                      end;
                      then g|B is bounded_above by RFUNCT_1:70;
                      hence thesis by A435;
                    end;
                    rng f is bounded_below by A1,INTEGRA1:11;
                    then
A439:               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 A439,XREAL_1:13;
                    then
A440:               (upper_bound rng f-lower_bound rng f)*delta(MD1)>= (
upper_bound rng g-lower_bound rng g)*delta(MD1 ) by A359,XREAL_1:64;
A441:               n1 < j-1 by A351,XREAL_1:20;
A442:               indx(D2,D1,j) <= len LVf(D2) by A263,INTEGRA1:def 7;
A443:               len MD2=indx(D2,D1,j)-'indx(D2,D1,n1+1)+1 by A272,A270,A277
,A262,A263,A358,FINSEQ_6:118;
                    then
A444:               len MD2 = indx(D2,D1,j)-indx(D2,D1,n1+1)+1 by A272,
XREAL_1:233;
                    then
A445:               len lower_volume(g,MD2)=indx(D2,D1,j)-(indx(D2,D1,n1
                    )+1)+1 by A390,INTEGRA1:def 7;
                    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;
                          rng MD1 c= rng D1 by A357,FINSEQ_6:119;
                          then
A449:                     x1 in rng D1 by A447;
                          rng D1 c= rng D2 by A13,INTEGRA1:def 18;
                          then consider k such that
A450:                     k in dom D2 and
A451:                     D2.k = x1 by A449,PARTFUN1:3;
                          MD1.1=D1.(n1+1) by A265,A266,A244,A267,A357,
FINSEQ_6:118;
                          then D2.indx(D2,D1,n1+1 ) <= x1 by A271,A447,Th16;
                          then
A452:                     indx(D2,D1,n1+1)<=k by A269,A450,A451,SEQM_3:def 1;
                          then consider n being Nat such that
A453:                     k+1 = indx(D2,D1,n1+1)+n by NAT_1:10,12;
A454:                     len MD1=j-'(n1+1)+1 by A257,A265,A266,A244,A267,A357,
FINSEQ_6:118;
                          then len MD1+(n1+1)-1=j-(n1+1)+1+(n1+1)-1 by A257,
XREAL_1:233
                            .=j;
                          then MD1.(len MD1) =D1.j by A257,A266,A244,A357,A448
,A454,FINSEQ_6:122;
                          then x1 <= D2.indx(D2,D1,j) by A238,A447,Th16;
                          then k<=indx(D2,D1,j) by A261,A450,A451,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
A455:                     k -indx(D2,D1,n1+1)+1<=indx(D2,D1,j)-indx(D2,
                          D1,n1+1)+1 by XREAL_1:6;
                          indx(D2,D1,n1+1)+1<=k+1 by A452,XREAL_1:6;
                          then
A456:                     1 <= k+1-indx(D2, D1,n1+1) by XREAL_1:19;
                          then
A457:                     n in dom MD2 by A444,A455,A453,FINSEQ_3:25;
                          MD2.n=D2.(n+indx(D2,D1,n1+1)-1) by A272,A270
,A263,A358,A456,A455,A453,FINSEQ_6:122
                            .=D2.k by A453;
                          hence x1 in rng MD2 by A451,A457,FUNCT_1:def 3;
                        end;
                        suppose
                          x1 in {D.(i+1)};
                          then
A458:                     x1 = D.(i+1) by TARSKI:def 1;
                          reconsider j1 = j-1 as Element of NAT by A236,A245
,A280,INTEGRA1:7;
A459:                     rng D c= rng D2 by A12,INTEGRA1:def 18;
                          D.(i+1) in rng D by A235,FUNCT_1:def 3;
                          then consider k such that
A460:                     k in dom D2 and
A461:                     x1 = D2.k by A458,A459,PARTFUN1:3;
                          D.(i+1) <= upper_bound divset(D1,j) by A237,
INTEGRA2:1;
                          then x1 <= D1.j by A236,A245,A280,A458,INTEGRA1:def 4
;
                          then
A462:                     D2
.k <= D2.indx(D2,D1,j) by A13,A236,A461,INTEGRA1:def 19;
                          n1<j1 by A351,XREAL_1:20;
                          then
A463:                     n1+1 <= j1 by NAT_1:13;
                          j -1 in dom D1 by A236,A245,A280,INTEGRA1:7;
                          then
A464:                     D1.(n1+1) <= D1.(j-1) by A268,A463,SEQ_4:137;
                          lower_bound divset(D1,j) <= D.(i+1) by A237,
INTEGRA2:1;
                          then D1.(j-1) <= x1 by A236,A245,A280,A458,
INTEGRA1:def 4;
                          then D2.indx(D2,D1,n1+1) <= D2.k by A271,A461,A464,
XXREAL_0:2;
                          hence x1 in rng MD2 by A269,A261,A358,A460,A461,A462
,Th17;
                        end;
                      end;
                      hence thesis;
                    end;
                    then
A465:               rng MD1 \/ {D.(i+1)} c= rng MD2;
                    rng MD2 <> {};
                    then 1 in dom MD2 by FINSEQ_3:32;
                    then
A466:               1 <= len MD2 by FINSEQ_3:25;
A467:               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
A468:                 x1 in rng MD2;
                      then reconsider x1 as Real;
                      MD2.1=D2.indx(D2,D1,n1+1) by A270,A277,A262,A263,A358,
FINSEQ_6:118;
                      then
A469:                 D1 .(n1+1) <= x1 by A271,A468,Th16;
                      MD2.(len MD2)=D2.indx(D2,D1,j) by A272,A270,A263,A358
,A466,A443,A467,FINSEQ_6:122;
                      then
A470:                 x1 <= D1.j by A238,A468,Th16;
A471:                 rng MD2 c= rng D2 by A358,FINSEQ_6:119;
                      now
                        per cases by A14,A468,A471,XBOOLE_0:def 3;
                        suppose
                          x1 in rng D1;
                          then consider k such that
A472:                     k in dom D1 and
A473:                     D1.k = x1 by PARTFUN1:3;
A474:                     n1 +1 <= k by A268,A469,A472,A473,SEQM_3:def 1;
                          then
A475:                     1 <= k-n1 by XREAL_1:19;
                          n1 <= n1+1 by NAT_1:11;
                          then consider n being Nat such that
A476:                     k=n1+n by A474,NAT_1:10,XXREAL_0:2;
A477:                     k <= j by A236,A470,A472,A473,SEQM_3:def 1;
                          then k-n1 <= len MD1 by A363,XREAL_1:9;
                          then n in dom MD1 by A475,A476,FINSEQ_3:25;
                          then
A478:                     MD1.n in rng MD1 by FUNCT_1:def 3;
                          j-(n1+1)+1=j-n1;
                          then
A479:                     k-n1 <= j-(n1+1)+1 by A477,XREAL_1:9;
                          MD1.n = D1.(k-n1-1+(n1+1)) by A257,A266,A244
,A357,A475,A479,A476,FINSEQ_6:122
                            .= D1.k;
                          hence x1 in (rng MD1 \/ {D.(i+1)}) by A473,A478,
XBOOLE_0:def 3;
                        end;
                        suppose
                          x1 in rng D;
                          then consider n such that
A480:                     n in dom D and
A481:                     D.n = x1 by PARTFUN1:3;
A482:                     not i+1 < n
                          proof
A483:                       upper_bound divset(D1,j)=D1.j
                            proof
                              per cases;
                              suppose
                                j=1;
                                hence thesis by A236,INTEGRA1:def 4;
                              end;
                              suppose
                                j<>1;
                                hence thesis by A236,INTEGRA1:def 4;
                              end;
                            end;
                            reconsider y1=D.(i+1) as Real;
A484:                       D.n in rng D by A480,FUNCT_1:def 3;
                            assume i+1 < n;
                            then
A485:                       D.(i+1) < D.n by A235,A480,SEQM_3:def 1;
                            lower_bound divset(D1,j) <= D.(i+1) by A237,
INTEGRA2:1;
                            then lower_bound divset(D1, j) <= D.n by A485,
XXREAL_0:2;
                            then D.n in divset(D1,j) by A470,A481,A483,
INTEGRA2:1;
                            then
A486:                       x1 in rng D /\ divset(D1,j) by A481,A484,
XBOOLE_0:def 4;
                            D.(i+1) in rng D by A235,FUNCT_1:def 3;
                            then y1 in rng D /\ divset(D1,j) by A237,
XBOOLE_0:def 4;
                            hence
                            contradiction by A11,A236,A481,A485,A486,Th5;
                          end;
A487:                     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 A353,A487,XXREAL_0:2;
                          then D.i < D.n by A469,A481,XXREAL_0:2;
                          then i < n by A240,A480,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 A481,A482,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
A488:               rng MD2 = rng MD1 \/ {D.(i+1)} by A465,XBOOLE_0:def 10;
delta(MD1) in rng upper_volume(chi(B,B),MD1) by XXREAL_2:def 8;
                    then consider k such that
A489:               k in dom upper_volume(chi(B,B),MD1) and
A490:               upper_volume(chi(B,B),MD1).k = delta(MD1) by PARTFUN1:3;
A491:               k in Seg len upper_volume(chi(B,B),MD1) by A489,
FINSEQ_1:def 3;
                    then
A492:               k in Seg len MD1 by INTEGRA1:def 6;
                    then
A493:               k in dom MD1 by FINSEQ_1:def 3;
A494:               k <= len MD1 by A492,FINSEQ_1:1;
                    then k+n1 <= j by A363,XREAL_1:19;
                    then
A495:               k+n1 <= len D1 by A266,XXREAL_0:2;
A496:               1 <= k by A491,FINSEQ_1:1;
A497:               n1+1>1 by A280,NAT_1:13;
                    then n1>1-1 by XREAL_1:19;
                    then
A498:               k < k+n1 by XREAL_1:29;
                    then 1 < k+n1 by A496,XXREAL_0:2;
                    then
A499:               k+n1 in dom D1 by A495,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
A500:                   k=1;
                        then upper_bound divset(MD1,k)=MD1.k by A493,
INTEGRA1:def 4;
                        then
A501:                   upper_bound divset(MD1,k)=D1.(k+(n1+1)-1) by A257,A266
,A244,A357,A360,A496,A494,FINSEQ_6:122;
                        lower_bound divset(D1,k+n1)=D1.(k+n1-1) by A496,A498
,A499,INTEGRA1:def 4;
                        hence thesis by A354,A497,A493,A499,A500,A501,
INTEGRA1:def 4;
                      end;
                      suppose
A502:                   k<>1;
                        then upper_bound divset(MD1,k)=MD1. k by A493,
INTEGRA1:def 4;
                        then
A503:                   upper_bound divset(MD1,k)=D1.(k+(n1+1)-1) by A257,A266
,A244,A357,A360,A496,A494,FINSEQ_6:122;
A504:                   lower_bound divset(MD1,k)=MD1.(k-1) by A493,A502,
INTEGRA1:def 4;
A505:                   k-1 in dom MD1 by A493,A502,INTEGRA1:7;
                        then
A506:                   k-1 <= len MD1 by FINSEQ_3:25;
                        1 <= k-1 by A505,FINSEQ_3:25;
                        then lower_bound divset(MD1,k)=D1.(k-1+(n1+1)-1) by
A257,A266,A244,A357,A360,A505,A506,A504,FINSEQ_6:122;
                        hence thesis by A496,A498,A499,A503,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
A507:               divset(MD1,k)=divset(D1,k+n1) by INTEGRA1:4;
                    k+n1 in Seg len D1 by A499,FINSEQ_1:def 3;
                    then k+n1 in Seg len upper_volume(chi(A,A),D1) by
INTEGRA1:def 6;
                    then
A508:               k+n1 in dom upper_volume(chi(A,A),D1) by FINSEQ_1:def 3;
                    k in dom MD1 by A492,FINSEQ_1:def 3;
                    then delta(MD1) = vol(divset(MD1,k)) by A490,INTEGRA1:20;
                    then
delta(MD1)=upper_volume(chi(A,A), D1).(k+n1) by A499,A507,INTEGRA1:20;
                    then delta(MD1) in rng upper_volume(chi(A,A),D1) by A508,
FUNCT_1:def 3;
                    then delta(MD1) <= max rng upper_volume(chi(A,A),D1) by
XXREAL_2:def 8;
                    then
A509:               delta(MD1) <= delta(D1);
                    upper_bound rng f - lower_bound rng f >= 0 by A1,Lm3,
XREAL_1:48;
                    then
A510:               (upper_bound rng f-lower_bound rng f)*delta(MD1) <=
(upper_bound rng f-lower_bound rng f)* delta(D1) by A509,XREAL_1:64;
                    lower_bound divset(D1,j) <= D.(i+1) by A237,INTEGRA2:1;
                    then
A511:               D1.(j-1) <= D.(i+1) by A236,A245,A280,INTEGRA1:def 4;
A512:               D.(i+1)<=upper_bound divset(D1,j) by A237,INTEGRA2:1;
A513:               indx (D2,D1,n1)+1 <= indx(D2,D1,j) by A272,A385,XXREAL_0:2;
A514:               for k be Nat st 1<=k & k<=len lower_volume(g,MD2)
holds lower_volume(g,MD2).k=mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)).k
                    proof
                      let k be Nat;
                      assume that
A515:                 1 <= k and
A516:                 k <= len lower_volume(g,MD2);
A517:                 k in Seg len lower_volume(g,MD2) by A515,A516,FINSEQ_1:1;
A518:                 mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)).k =LVf
(D2).(k+(indx(D2,D1,n1)+1)-1) by A388,A445,A442,A513,A515,A516,FINSEQ_6:122;
                      1<=indx(D2,D1,n1)+1 by NAT_1:12;
                      then 1+1<=k+(indx(D2,D1,n1)+1) by A515,XREAL_1:7;
                      then
A519:                 1 <= k+(indx(D2,D1,n1) +1)-1 by XREAL_1:19;
                      consider k2 being Element of NAT such that
A520:                 indx(D2,D1,n1)+1=1+k2;
                      k <= indx(D2,D1,j)-((indx(D2,D1,n1)+1)-1) by A444,A390
,A516,INTEGRA1:def 7;
                      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 LVf(D2) by A442,
XXREAL_0:2;
                      then k+k2 in Seg len LVf(D2) by A519,A520,FINSEQ_1:1;
                      then
A521:                 k+k2 in Seg len D2 by INTEGRA1:def 7;
                      then k+k2 in dom D2 by FINSEQ_1:def 3;
                      then
A522:                 mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)).k =(
lower_bound rng(f|divset(D2,k+k2)))* vol(divset(D2,k+k2)) by A518,A520,
INTEGRA1:def 7;
A523:                 k in Seg len MD2 by A517,INTEGRA1:def 7;
A524:                 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 A260,A515,A520,XREAL_1:7;
                        then
A525:                   k+k2>1 by NAT_1:13;
A526:                   k in dom MD2 by A523,FINSEQ_1:def 3;
A527:                   k+k2 in dom D2 by A521,FINSEQ_1:def 3;
                        per cases;
                        suppose
A528:                     k=1;
                          then upper_bound divset(MD2,k)= MD2.k by A526,
INTEGRA1:def 4;
                          then
A529:                     upper_bound divset(MD2,k)=D2.(k+( indx(D2,D1,
n1)+1)-1) by A272,A263,A358,A388,A390,A445,A515,A516,FINSEQ_6:122;
A530:                     lower_bound divset(D2,k+k2)=D2.(k+k2-1) by A525,A527,
INTEGRA1:def 4;
                          lower_bound divset(MD2,k)=D1.n1 by A354,A526,A528,
INTEGRA1:def 4;
                          hence thesis by A13,A241,A520,A525,A527,A528,A529
,A530,INTEGRA1:def 4,def 19;
                        end;
                        suppose
A531:                     k<>1;
                          then upper_bound divset(MD2,k)=MD2.k by A526,
INTEGRA1:def 4;
                          then
A532:                     upper_bound divset(MD2,k)=D2.(k+( indx(D2,D1,
n1)+1)-1) by A272,A263,A358,A388,A390,A445,A515,A516,FINSEQ_6:122;
A533:                     k -1 <= indx(D2,D1,j)-(indx(D2,D1,n1)+1)+1 by A445
,A516,XREAL_1:146,XXREAL_0:2;
A534:                     lower_bound divset(MD2,k)=MD2.(k-1) by A526,A531,
INTEGRA1:def 4;
A535:                     k-1 in dom MD2 by A526,A531,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 A272,A263,A358,A388,A390,A535,A533,A534,FINSEQ_6:122;
                          hence thesis by A520,A525,A527,A532,INTEGRA1:def 4;
                        end;
                      end;
                      divset (MD2,k)=[.lower_bound divset(MD2,k),
                      upper_bound divset(MD2,k).] by INTEGRA1:4;
                      then
A536:                 divset(MD2,k)=divset( D2,k+k2) by A524,INTEGRA1:4;
                      k in dom MD2 by A523,FINSEQ_1:def 3;
                      then divset(D2,k+k2) c= B by A536,INTEGRA1:8;
                      then
A537:                 rng(f|divset(D2,k+k2))=rng(g|divset(D2,k+k2)) by
FUNCT_1:51;
                      k in dom MD2 by A523,FINSEQ_1:def 3;
                      hence thesis by A522,A536,A537,INTEGRA1:def 7;
                    end;
                    lower_bound divset(D1,j)<=D.(i+1) by A237,INTEGRA2:1;
                    then
A538:               D.(i+1) in divset(MD1,len MD1) by A374,A512,INTEGRA2:1;
                    j-1 in dom D1 by A236,A245,A280,INTEGRA1:7;
                    then D1.n1 <D1.(j-1) by A241,A441,SEQM_3:def 1;
                    then D.(i+1) > lower_bound B by A354,A511,XXREAL_0:2;
                    then Sum lower_volume(g,MD2)-Sum lower_volume(g,MD1)<= (
upper_bound rng g- lower_bound rng g)*delta(MD1) by A356,A431,A488,A538,A387
,Th13;
                    then
A539:               Sum lower_volume(g,MD2)-Sum lower_volume(g,MD1)<= (
upper_bound rng f-lower_bound rng f)*delta(MD1) by A440,XXREAL_0:2;
                    indx(D2,D1,n1)+1 <= len LVf(D2) by A386,INTEGRA1:def 7;
                    then len mid(LVf(D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) =
indx(D2,D1,j)-'(indx(D2,D1,n1)+1)+1 by A262,A388,A442,A513,FINSEQ_6:118;
                    then len lower_volume(g,MD2) =len mid(LVf(D2),indx(D2,D1,
n1)+1,indx(D2,D1,j)) by A272,A385,A445,XREAL_1:233,XXREAL_0:2;
                    then
A540:               Sum lower_volume(g,MD2)=Sum mid(LVf(D2),indx(D2,D1,
                    n1)+1,indx(D2,D1,j)) by A514,FINSEQ_1:14;
                    n1+1 <= len LVf(D1) by A267,INTEGRA1:def 7;
                    then len mid(LVf(D1),n1+1,j) =j-'(n1+1)+1 by A257,A265,A244
,A407,FINSEQ_6:118
                      .=j-(n1+1)+1 by A257,XREAL_1:233;
                    then Sum lower_volume(g,MD1)=Sum mid(LVf(D1),n1+1,j) by
A371,A408,FINSEQ_1:14;
                    hence thesis by A539,A510,A540,XXREAL_0:2;
                  end;
                end;
                hence thesis;
              end;
              then
A541:         (PLf(D2,indx(D2,D1,n1))-PLf(D1,n1))+ (Sum mid(LVf(D2),
indx(D2,D1,n1)+1,indx(D2,D1,j)) -Sum mid(LVf(D1),n1+1,j)) <=i*(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,A268,SEQM_3:def 1;
              then indx(D2,D1,n1) < indx(D2,D1,n1+1) by A259,A274,A269,A271,
SEQ_4:137;
              then
A542:         indx(D2,D1,n1)<indx(D2,D1,j) by A272,XXREAL_0:2;
              indx(D2,D1,n1) in Seg len D2 by A259,FINSEQ_1:def 3;
              then indx(D2,D1,n1) in Seg len LVf(D2) by INTEGRA1:def 7;
              then indx(D2,D1,n1) in dom LVf(D2) by FINSEQ_1:def 3;
              then PLf(D2,indx(D2,D1,n1))=Sum(LVf(D2)|indx(D2,D1,n1)) by
INTEGRA1:def 20
                .=Sum mid(LVf(D2),1,indx(D2,D1,n1)) by A260,FINSEQ_6:116;
              then PLf(D2,indx(D2,D1,n1))+Sum mid(LVf(D2),indx(D2,D1,n1)+1,
indx(D2, D1,j)) =Sum(mid(LVf(D2),1,indx(D2,D1,n1))^ mid(LVf(D2),indx(D2,D1,n1)+
              1,indx(D2,D1,j))) by RVSUM_1:75
                .=Sum mid(LVf(D2),1,indx(D2,D1,j)) by A260,A542,A264,INTEGRA2:4
                .=Sum(LVf(D2)|indx(D2,D1,j)) by A262,FINSEQ_6:116;
              then PLf(D2,indx(D2,D1,j))= PLf(D2,indx(D2,D1,n1))+Sum mid(LVf
              (D2),indx(D2,D1,n1)+1,indx(D2,D1,j)) by A279,INTEGRA1:def 20;
              then (PLf(D2,indx(D2,D1,n1))-PLf(D1,n1))+ (Sum mid(LVf(D2),
indx(D2,D1,n1)+1,indx(D2,D1,j)) -Sum mid(LVf(D1),n1+1,j)) =PLf(D2,indx(D2,D1,j)
              )-PLf(D1,j) by A278;
              hence thesis by A236,A237,A541;
            end;
            hence thesis;
          end;
          for k being non zero Nat holds P[k] from NAT_1:sch 10(A40,
          A231);
          then P[i];
          hence thesis by A17;
        end;
A543:   len D1 in dom D1 by FINSEQ_5:6;
        then D1.(len D1) = D2.indx(D2,D1,len D1) by A13,INTEGRA1:def 19;
        then upper_bound A = D2.indx(D2,D1,len D1) by INTEGRA1:def 2;
        then
A544:   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
A545:   j in dom D1 and
A546:   D.(len D) in divset(D1,j) and
A547:   PLf(D2,indx(D2,D1,j))-PLf(D1,j) <= (len D)*(upper_bound(rng
        f)- lower_bound(rng f))*delta(D1) by A16;
A548:   j = len D1
        proof
          assume
A549:     j<>len D1;
          j <= len D1 by A545,FINSEQ_3:25;
          then j < len D1 by A549,XXREAL_0:1;
          then D1.j < D1.(len D1) by A545,A543,SEQM_3:def 1;
          then
A550:     D1.j < upper_bound A by INTEGRA1:def 2;
A551:     upper_bound divset(D1,j) < upper_bound A
          proof
            per cases;
            suppose
              j=1;
              hence thesis by A545,A550,INTEGRA1:def 4;
            end;
            suppose
              j<>1;
              hence thesis by A545,A550,INTEGRA1:def 4;
            end;
          end;
          D.(len D) <= upper_bound divset(D1,j) by A546,INTEGRA2:1;
          hence contradiction by A551,INTEGRA1:def 2;
        end;
        indx(D2,D1,len D1) in dom D2 by A13,A543,INTEGRA1:def 19;
        then indx(D2,D1,len D1)=len D2 by A15,A544,SEQ_4:138;
        then PLf(D2,len D2)-lower_sum(f,D1)<=(len D)* (upper_bound(rng f)-
        lower_bound(rng f))* delta(D1) by A547,A548,INTEGRA1:43;
        hence thesis by INTEGRA1:43;
      end;
      hence thesis by A12,A13,A14;
    end;
    hence thesis;
  end;
  hence thesis;
end;
