
theorem Th81:
for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
  M1 be sigma_Measure of S1, M2 be sigma_Measure of S2,
  E be Element of sigma measurable_rectangles(S1,S2), er be ExtReal holds
  Integral1(M1,chi(er,E,[:X1,X2:])|E) = Integral1(M1,chi(er,E,[:X1,X2:]))
& Integral2(M2,chi(er,E,[:X1,X2:])|E) = Integral2(M2,chi(er,E,[:X1,X2:]))
proof
    let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
    M1 be sigma_Measure of S1, M2 be sigma_Measure of S2,
    E be Element of sigma measurable_rectangles(S1,S2), er be ExtReal;
    reconsider XX12 = [:X1,X2:] as Element of
      sigma measurable_rectangles(S1,S2) by MEASURE1:7;
    reconsider C = chi(E,[:X1,X2:])|E as PartFunc of [:X1,X2:],ExtREAL;
    per cases by XXREAL_0:14;
    suppose er in REAL; then
     reconsider r = er as Real;
A1: chi(r,E,[:X1,X2:]) = r(#)chi(E,[:X1,X2:]) by Th1;
A2: chi(E,[:X1,X2:]) is XX12-measurable by MESFUNC2:29;
A3: dom(chi(E,[:X1,X2:])) = XX12 by FUNCT_2:def 1;

A4: dom( chi(E,[:X1,X2:])|E ) = dom(chi(E,[:X1,X2:])) /\ E by RELAT_1:61
     .= [:X1,X2:] /\ E by FUNCT_2:def 1
     .= E by XBOOLE_1:28;
A5: chi(E,[:X1,X2:])|E is nonnegative by MESFUNC5:15;
    E = dom(chi(E,[:X1,X2:])) /\ E by A3,XBOOLE_1:28; then
A6: chi(E,[:X1,X2:])|E is E-measurable by MESFUNC2:29,MESFUNC5:42;

    Integral1(M1,chi(r,E,[:X1,X2:])|E)
     = Integral1(M1,r(#)C) by A1,MESFUN11:2
    .= r(#)Integral1(M1,C) by A4,A5,A6,Th78
    .= r(#)Integral1(M1,chi(E,[:X1,X2:])) by Th79
    .= Integral1(M1,r(#)chi(E,[:X1,X2:])) by A2,A3,Th78;
    hence
     Integral1(M1,chi(er,E,[:X1,X2:])|E) = Integral1(M1,chi(er,E,[:X1,X2:]))
      by Th1;

    Integral2(M2,chi(r,E,[:X1,X2:])|E)
     = Integral2(M2,r(#)C) by A1,MESFUN11:2
    .= r(#)Integral2(M2,C) by A4,A5,A6,Th78
    .= r(#)Integral2(M2,chi(E,[:X1,X2:])) by Th79
    .= Integral2(M2,r(#)chi(E,[:X1,X2:])) by A2,A3,Th78;
    hence
     Integral2(M2,chi(er,E,[:X1,X2:])|E) = Integral2(M2,chi(er,E,[:X1,X2:]))
      by Th1;
    end;
    suppose er = +infty; then
     chi(er,E,[:X1,X2:]) = Xchi(E,[:X1,X2:]) by Th2;
     hence
      Integral1(M1,chi(er,E,[:X1,X2:])|E) = Integral1(M1,chi(er,E,[:X1,X2:]))
    & Integral2(M2,chi(er,E,[:X1,X2:])|E) = Integral2(M2,chi(er,E,[:X1,X2:]))
     by Th80;
    end;
    suppose d0: er = -infty;
     reconsider XX12 = [:X1,X2:]
      as Element of sigma measurable_rectangles(S1,S2) by MEASURE1:7;
     reconsider XE = Xchi(E,[:X1,X2:])|E as PartFunc of [:X1,X2:],ExtREAL;
d3:  Xchi(E,[:X1,X2:]) is XX12-measurable by MEASUR10:32;
e2:  XE is nonnegative by MESFUNC5:15;
d4:  dom Xchi(E,[:X1,X2:]) = XX12 by FUNCT_2:def 1; then
e1:  dom XE = E by RELAT_1:62; then
     E = dom Xchi(E,[:X1,X2:]) /\ E by RELAT_1:61; then
e3:  XE is E-measurable by MESFUNC5:42;
d1:  chi(er,E,[:X1,X2:]) = -Xchi(E,[:X1,X2:]) by d0,Th2
      .= (-1)(#)Xchi(E,[:X1,X2:]) by MESFUNC2:9;
     Integral1(M1,chi(er,E,[:X1,X2:]))
      = (-1) (#) Integral1(M1,Xchi(E,[:X1,X2:])) by d1,d3,d4,Th78
     .= (-1) (#) Integral1(M1,Xchi(E,[:X1,X2:])|E) by Th80
     .= Integral1(M1,(-1)(#)XE) by e1,e2,e3,Th78
     .= Integral1(M1,chi(er,E,[:X1,X2:])|E) by d1,MESFUN11:2;
     hence
  Integral1(M1,chi(er,E,[:X1,X2:])|E) = Integral1(M1,chi(er,E,[:X1,X2:]));
     Integral2(M2,chi(er,E,[:X1,X2:]))
      = (-1) (#) Integral2(M2,Xchi(E,[:X1,X2:])) by d1,d3,d4,Th78
     .= (-1) (#) Integral2(M2,Xchi(E,[:X1,X2:])|E) by Th80
     .= Integral2(M2,(-1)(#)XE) by e1,e2,e3,Th78
     .= Integral2(M2,chi(er,E,[:X1,X2:])|E) by d1,MESFUN11:2;

     hence
  Integral2(M2,chi(er,E,[:X1,X2:])|E) = Integral2(M2,chi(er,E,[:X1,X2:]));
    end;
end;
