
theorem Th14:
for A be non empty closed_interval Subset of REAL,
f being Function of A,COMPLEX,
    T being DivSequence of A,
    S be middle_volume_Sequence of f,T
      st f is bounded & f is integrable
       & delta(T) is convergent & lim delta(T)=0
  holds
    middle_sum(f,S) is convergent & lim (middle_sum(f,S))=integral(f)
proof
let A be non empty closed_interval Subset of REAL,
    f be Function of A,COMPLEX,
    T be DivSequence of A,
    S be middle_volume_Sequence of f,T;
assume that
A1: f is bounded & f is integrable and
A2: delta(T) is convergent & lim delta(T)=0;
set seq=middle_sum(f,S);
set xs=integral(f);
A3: Re f is bounded & Re f is integrable by A1,Th13;
A4: Im f is bounded & Im f is integrable by A1,Th13;
reconsider xseq=seq as sequence of  COMPLEX;
ex rseqi be Real_Sequence st for k be Nat holds
    rseqi.k = Re (xseq.k) & rseqi is convergent & Re xs = lim rseqi
  proof
    reconsider pjinf= Re f as Function of A,REAL;
    defpred P[Element of NAT,set] means $2= Re (S.$1);
  A5:for x being Element of NAT ex y being Element of (REAL)* st P[x, y]
      proof
        let x be Element of NAT;
        Re (S.x) is Element of (REAL)* by FINSEQ_1:def 11;
        hence thesis;
      end;
    consider F being sequence of  (REAL)* such that
  A6:for x being Element of NAT holds P[x, F.x] from FUNCT_2:sch 3(A5);
  A7:for x being Element of NAT holds dom(Re (S.x)) = Seg len(S.x)
      proof
        let x be Element of NAT;
        thus dom(Re (S.x)) = dom (S.x) by COMSEQ_3:def 3
                          .= Seg len(S.x) by FINSEQ_1:def 3;
      end;
    for k be Element of NAT holds F.k is middle_volume of pjinf,T.k
      proof
        let k be Element of NAT;
        set Tk=T.k;
        reconsider Fk=F.k as FinSequence of REAL;
      A8:F.k= Re (S.k) by A6; then
      A9:dom (Fk) = Seg len(S.k) by A7
                 .= Seg len(Tk) by Def1; then
      A10:dom (Fk) = dom (Tk) by FINSEQ_1:def 3;
      A11:now
          let j be Nat;
          assume A12:j in dom (Tk); then
          consider r be Element of COMPLEX such that
        A13:r in rng (f|divset((T.k),j)) and
        A14:(S.k).j=r * vol divset((T.k),j) by Def1;
          reconsider v=Re r as Element of REAL;
          take v;
          consider t be object such that
        A15:t in dom (f|divset((T.k),j)) and
        A16:r=(f|divset((T.k),j)).t by A13,FUNCT_1:def 3;
          t in dom(f) /\ divset((T.k),j) by A15,RELAT_1:61; then
          t in dom(f) by XBOOLE_0:def 4; then
        A17:t in dom (Re f) by COMSEQ_3:def 3;
        A18:dom (f|divset((T.k),j)) =dom (f) /\ divset((T.k),j) by RELAT_1:61
          .=dom (pjinf) /\ divset((T.k),j) by COMSEQ_3:def 3
          .=dom (pjinf|divset((T.k),j)) by RELAT_1:61;
          Re r = Re (f.t) by A15,A16,FUNCT_1:47
              .=(Re f).t by A17,COMSEQ_3:def 3
              .=(pjinf|divset((T.k),j)).t by A15,A18,FUNCT_1:47;
          hence v in rng (pjinf|divset((T.k),j)) by A15,A18,FUNCT_1:3;
          thus (Fk).j = Re ((S.k).j) by A8,A10,A12,COMSEQ_3:def 3
            .= v*vol divset((T.k),j) by A14,Th1;
        end;
        len (Fk) = len(Tk) by A9,FINSEQ_1:def 3;
        hence thesis by A11,INTEGR15:def 1;
      end; then
    reconsider pjis = F as middle_volume_Sequence of pjinf,T by INTEGR15:def 3;
    reconsider rseqi = middle_sum(pjinf,pjis) as Real_Sequence;
  A19:for k be Nat holds rseqi.k =Re (xseq.k)
    proof
      let k be Nat;
       reconsider kk=k as Element of NAT by ORDINAL1:def 12;
    A20: Re (S.kk) is middle_volume of Re f,T.kk by Th4;
      thus rseqi.k = middle_sum(pjinf,pjis.kk) by INTEGR15:def 4
                  .= Re (middle_sum(f,S.kk)) by A6,A20,Th7
                  .= Re (xseq.k) by Def4;
    end;
  take rseqi;
  lim (middle_sum(pjinf,pjis))=integral(pjinf) by A2,A3,INTEGR15:9;
  hence thesis by A2,A3,A19,COMPLEX1:12,INTEGR15:9;
  end; then
consider rseqi be Real_Sequence such that
A21:for k be Nat holds
      rseqi.k = Re (xseq.k) & rseqi is convergent & Re xs = lim rseqi;
ex iseqi be Real_Sequence st for k be Nat holds
    iseqi.k = Im (xseq.k) & iseqi is convergent & Im xs = lim iseqi
  proof
    reconsider pjinf= Im f as Function of A,REAL;
    defpred P[Element of NAT,set] means $2= Im (S.$1);
  A22:for x being Element of NAT ex y being Element of (REAL)* st P[x, y]
      proof
        let x be Element of NAT;
        Im (S.x) is Element of (REAL)* by FINSEQ_1:def 11;
        hence thesis;
      end;
    consider F being sequence of  (REAL)* such that
  A23:for x being Element of NAT holds P[x, F.x] from FUNCT_2:sch 3(A22);
  A24:for x being Element of NAT holds dom(Im (S.x)) = Seg len(S.x)
      proof
        let x be Element of NAT;
        dom(Im (S.x)) = dom (S.x) by COMSEQ_3:def 4
                     .= Seg len(S.x) by FINSEQ_1:def 3;
        hence thesis;
      end;
    for k be Element of NAT holds F.k is middle_volume of pjinf,T.k
      proof
        let k be Element of NAT;
        reconsider Tk=T.k as FinSequence;
        reconsider Fk=F.k as FinSequence of REAL;
      A25:F.k= Im (S.k) by A23; then
      A26:dom (Fk) = Seg len(S.k) by A24
                 .= Seg len(Tk) by Def1; then
      A27:dom (Fk) = dom (Tk) by FINSEQ_1:def 3;
      A28:now let j be Nat;
          assume A29: j in dom (Tk); then
          consider r be Element of COMPLEX such that
        A30:r in rng (f|divset((T.k),j)) and
        A31:(S.k).j=r * vol divset((T.k),j) by Def1;
          reconsider v=Im r as Element of REAL;
          take v;
          consider t be object such that
        A32:t in dom (f|divset((T.k),j)) and
        A33:r=(f|divset((T.k),j)).t by A30,FUNCT_1:def 3;
          t in dom(f) /\ divset((T.k),j) by A32,RELAT_1:61; then
          t in dom(f) by XBOOLE_0:def 4; then
        A34:t in dom (Im f) by COMSEQ_3:def 4;
        A35:dom (f|divset((T.k),j)) =dom (f) /\ divset((T.k),j) by RELAT_1:61
          .=dom (pjinf) /\ divset((T.k),j) by COMSEQ_3:def 4
          .=dom (pjinf|divset((T.k),j)) by RELAT_1:61;
          Im r = Im (f.t) by A32,A33,FUNCT_1:47
              .= (Im f).t by A34,COMSEQ_3:def 4
              .= (pjinf|divset((T.k),j)).t by A32,A35,FUNCT_1:47;
          hence v in rng (pjinf|divset((T.k),j)) by A32,A35,FUNCT_1:3;
          thus (Fk).j = Im ((S.k).j) by A25,A27,A29,COMSEQ_3:def 4
                     .= v*vol divset((T.k),j) by A31,Th1;
        end;
        len (Fk) = len(Tk) by A26,FINSEQ_1:def 3;
        hence thesis by A28,INTEGR15:def 1;
      end; then
    reconsider pjis = F as middle_volume_Sequence of pjinf,T by INTEGR15:def 3;
    reconsider iseqi = middle_sum(pjinf,pjis) as Real_Sequence;
  A36:for k be Nat holds iseqi.k =Im (xseq.k)
    proof
      let k be Nat;
       reconsider kk=k as Element of NAT by ORDINAL1:def 12;
    A37:Im (S.kk) is middle_volume of Im f,T.kk by Th4;
      thus iseqi.k = middle_sum(pjinf,pjis.kk) by INTEGR15:def 4
                  .= Im (middle_sum(f,S.kk)) by A23,A37,Th8
                  .= Im (xseq.k) by Def4;
    end;
    take iseqi;
    lim (middle_sum(pjinf,pjis))=integral(pjinf) by A2,A4,INTEGR15:9;
    hence thesis by A2,A4,A36,COMPLEX1:12,INTEGR15:9;
  end; then
consider iseqi be Real_Sequence such that
A38:for k be Nat holds
      iseqi.k = Im (xseq.k) & iseqi is convergent & Im xs = lim iseqi;
thus middle_sum(f,S) is convergent by A21,A38,COMSEQ_3:38;
thus lim middle_sum(f,S) = lim(rseqi) + lim(iseqi)*<i> by A21,A38,COMSEQ_3:39
   .= integral(f) by A21,A38,COMPLEX1:13;
end;
