
theorem
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
 F be Functional_Sequence of X,ExtREAL, E be Element of S
  st E = dom(F.0) & F is additive & F is with_the_same_dom
   & (for n be Nat holds F.n is E-measurable & F.n is nonpositive)
 holds
  ex I be ExtREAL_sequence st for n be Nat holds
   I.n = Integral(M,F.n) & Integral(M,(Partial_Sums F).n) = (Partial_Sums I).n
proof
    let X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
     F be Functional_Sequence of X,ExtREAL, E be Element of S;
     assume that
A1:   E = dom(F.0) and
A2:   F is additive and
A3:   F is with_the_same_dom and
A4:   for n be Nat holds F.n is E-measurable & F.n is nonpositive;
     set G = -F;
     G.0 = -(F.0) by Th37; then
A5:  E = dom(G.0) by A1,MESFUNC1:def 7;
A7:  G is with_the_same_dom by A3,Th40;
     for n be Nat holds G.n is E-measurable & G.n is nonnegative
     proof
      let n be Nat;
      E = dom(F.n) & F.n is E-measurable by A4,A1,A3,MESFUNC8:def 2; then
      -(F.n) is E-measurable by MEASUR11:63;
      hence G.n is E-measurable by Th37;
      F.n is nonpositive by A4; then
      -(F.n) is nonnegative;
      hence G.n is nonnegative by Th37;
     end; then
     consider J be ExtREAL_sequence such that
A8:   for n be Nat holds J.n = Integral(M,G.n)
      & Integral(M,(Partial_Sums G).n) = (Partial_Sums J).n
        by A5,A7,A2,Th41,MESFUNC9:50;
     set I = -J;
     take I;
A10: for n be Nat holds
      F.n is E-measurable & F.n is without+infty
     proof
      let n be Nat;
      thus F.n is E-measurable by A4;
      F.n is nonpositive by A4;
      hence F.n is without+infty;
     end;
     hereby let n be Nat;
      dom I = NAT by FUNCT_2:def 1; then
      n in dom I by ORDINAL1:def 12; then
      I.n = -(J.n) by MESFUNC1:def 7; then
A9:   I.n = - Integral(M,G.n) by A8;
      E = dom(F.n) & F.n is E-measurable & (G.n) = -(F.n)
        by A4,A1,A3,Th37,MESFUNC8:def 2; then
      Integral(M,G.n) = - Integral(M,F.n) by Th52;
      hence I.n = Integral(M,F.n) by A9;
A11:  E = dom((Partial_Sums F).n) by A1,A2,A3,MESFUNC9:29;
      (Partial_Sums G).n = (-(Partial_Sums F)).n by Th42
        .= -((Partial_Sums F).n) by Th37; then
A13:  Integral(M,(Partial_Sums G).n)
        = - Integral(M,(Partial_Sums F).n) by A10,A1,A3,A11,Th52,Th67;
      (Partial_Sums I).n = -((Partial_Sums J).n) by Th43
       .= - Integral(M,(Partial_Sums G).n) by A8;
      hence Integral(M,(Partial_Sums F).n) = (Partial_Sums I).n by A13;
     end;
end;
