
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,
 I be ExtREAL_sequence, m be Nat
   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 & I.n = Integral(M,F.n))
   holds Integral(M,(Partial_Sums F).m) = (Partial_Sums I).m
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,
    I be ExtREAL_sequence, m be Nat;
    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 & I.n = Integral(M,F.n);
    set G = -F, J = -I;
    G.0 = -(F.0) by Th37; then
A5: E = dom(G.0) by A1,MESFUNC1:def 7;
A6: G is with_the_same_dom by A3,Th40;
A7: E = dom((Partial_Sums F).m) by A1,A2,A3,MESFUNC9:29;
A8: 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;
    for n be Nat holds
      G.n is E-measurable & G.n is nonnegative & J.n = Integral(M,G.n)
    proof
     let n be Nat;
A9:  F.n is nonpositive & I.n = Integral(M,F.n) by A4;
A10: G.n = -(F.n) by Th37;
     dom J = NAT by FUNCT_2:def 1; then
A11: n in dom J by ORDINAL1:def 12;
A12: dom(F.n) = E by A1,A3,MESFUNC8:def 2;
     hence G.n is E-measurable by A4,A10,MEASUR11:63;
     thus G.n is nonnegative by A9,A10;
     Integral(M,G.n) = -Integral(M,F.n) by A4,A10,A12,Th52;
     hence J.n = Integral(M,G.n) by A9,A11,MESFUNC1:def 7;
    end; then
    Integral(M,(Partial_Sums G).m) = (Partial_Sums J).m
      by A5,A2,A6,Th41,MESFUNC9:46; then
    Integral(M,(-(Partial_Sums F)).m) = (Partial_Sums J).m by Th42; then
    Integral(M,(-(Partial_Sums F)).m) = -((Partial_Sums I).m) by Th43; then
    Integral(M,-((Partial_Sums F).m)) = -((Partial_Sums I).m) by Th37; then
    -Integral(M,(Partial_Sums F).m) = -((Partial_Sums I).m)
      by A1,A3,A7,A8,Th67,Th52;
    hence Integral(M,(Partial_Sums F).m) = (Partial_Sums I).m by XXREAL_3:10;
end;
