
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.0 is nonpositive & F is with_the_same_dom
   & (for n be Nat holds F.n is E-measurable)
   & (for n,m be Nat st n <=m holds for x be Element of X st x in E
        holds (F.n).x >= (F.m).x )
   & (for x be Element of X st x in E holds F#x is convergent)
 holds
  ex I be ExtREAL_sequence st (for n be Nat holds I.n = Integral(M,F.n))
    & I is convergent & Integral(M,lim F) = lim I
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.0 is nonpositive and
A3:  F is with_the_same_dom and
A4:  for n be Nat holds F.n is E-measurable and
A5:  for n,m be Nat st n <=m holds for x be Element of X st x in E
        holds (F.n).x >= (F.m).x and
A6:  for x be Element of X st x in E holds F#x is convergent;
    set G = -F;
A7: G.0 = -(F.0) by Th37; then
A8: E = dom(G.0) by A1,MESFUNC1:def 7;
A11:for n be Nat holds G.n is E-measurable
    proof
     let n be Nat;
     E = dom(F.n) by A1,A3,MESFUNC8:def 2; then
     -(F.n) is E-measurable by A4,MEASUR11:63;
     hence G.n is E-measurable by Th37;
    end;
A13:for n,m be Nat st n <= m holds for x be Element of X st x in E
      holds (G.n).x <= (G.m).x
    proof
     let n,m be Nat;
     assume A14: n <= m;
     let x be Element of X;
     assume A15: x in E; then
A17: x in dom(G.n) & x in dom(G.m) by A8,A3,Th40,MESFUNC8:def 2;
     G.n = -(F.n) & G.m = -(F.m) by Th37; then
     (G.n).x = -((F.n).x) & (G.m).x = -((F.m).x) by A17,MESFUNC1:def 7;
     hence (G.n).x <= (G.m).x by A15,A5,A14,XXREAL_3:38;
    end;
A18:for x be Element of X st x in E holds G#x is convergent
    proof
     let x be Element of X;
     assume x in E; then
     F#x is convergent by A6; then
     -(F#x) is convergent by DBLSEQ_3:17;
     hence G#x is convergent by Th38;
    end;
    consider J be ExtREAL_sequence such that
A19: (for n be Nat holds J.n = Integral(M,G.n))
   & J is convergent & Integral(M,lim G) = lim J
      by A8,A2,A7,A3,Th40,A11,A13,A18,MESFUNC9:52;
    set I = -J;
    take I;
    thus for n be Nat holds I.n = Integral(M,F.n)
    proof
     let n be Nat;
     n in NAT by ORDINAL1:def 12; then
A20: n in dom I by FUNCT_2:def 1;
A21: dom(F.n) = E by A1,A3,MESFUNC8:def 2;
     G.n = -(F.n) by Th37; then
     Integral(M,G.n) = - Integral(M,F.n)
       by A21,A4,Th52; then
     J.n = - Integral(M,F.n) by A19; then
     I.n = -(-Integral(M,F.n)) by A20,MESFUNC1:def 7;
     hence I.n = Integral(M,F.n);
    end;
    thus I is convergent by A19,DBLSEQ_3:17;
A23:lim I = - Integral(M,lim G) by A19,DBLSEQ_3:17;
A24:E = dom(lim F) by A1,MESFUNC8:def 9; then
A25:dom(-(lim F)) = E by MESFUNC1:def 7; then
A26:dom(lim G) = dom(-(lim F)) by A8,MESFUNC8:def 9;
    for x be Element of X st x in dom(lim G) holds (lim G).x = (-(lim F)).x
    proof
     let x be Element of X;
     assume A27: x in dom(lim G); then
A30: (lim G).x = lim(G#x) by MESFUNC8:def 9;
A28: F#x is convergent by A27,A26,A25,A6;
     G#x = -(F#x) by Th38; then
A29: lim(G#x) = - lim(F#x) by A28,DBLSEQ_3:17;
     lim(F#x) = (lim F).x by A27,A26,A25,A24,MESFUNC8:def 9;
     hence (lim G).x = (-(lim F)).x by A29,A30,A27,A26,MESFUNC1:def 7;
    end; then
    lim G = -(lim F) by A26,PARTFUN1:5; then
    Integral(M,lim G) = - Integral(M,lim F)
      by A1,A3,A4,A6,A24,Th52,MESFUNC8:25;
    hence Integral(M,lim F) = lim I by A23;
end;
