
theorem Th99:
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,
  F be FinSequence of sigma measurable_rectangles(S1,S2),
  n be Nat
st M2 is sigma_finite
 & F is FinSequence of measurable_rectangles(S1,S2) holds
   product_sigma_Measure(M1,M2).(F.n) = Integral(M1,Y-vol(F.n,M2))
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,
       F be FinSequence of sigma measurable_rectangles(S1,S2),
       n be Nat;
   assume that
A1: M2 is sigma_finite and
A2: F is FinSequence of measurable_rectangles(S1,S2);
   reconsider XX1 = X1 as Element of S1 by MEASURE1:7;
   not n in dom F implies F.n in measurable_rectangles(S1,S2)
   proof
    assume not n in dom F; then
    F.n = {} by FUNCT_1:def 2;
    hence F.n in measurable_rectangles(S1,S2) by SETFAM_1:def 8;
   end; then
   F.n in measurable_rectangles(S1,S2) by A2,PARTFUN1:4; then
F.n in the set of all [:A,B:] where A is Element of S1, B is Element of S2
       by MEASUR10:def 5; then
   consider P be Element of S1, Q be Element of S2 such that
d4: F.n = [:P,Q:];
d5:product_sigma_Measure(M1,M2).(F.n) = M1.P * M2.Q by d4,Th10;
   per cases;
   suppose d8: M1.P = 0 & M2.Q = +infty; then
    product_sigma_Measure(M1,M2).(F.n) = 0
  & Y-vol(F.n,M2) = Xchi(P,X1) by A1,d4,d5,Th96;
    hence product_sigma_Measure(M1,M2).(F.n) = Integral(M1,Y-vol(F.n,M2))
      by d8,MEASUR10:33;
   end;
   suppose M1.P = 0 & M2.Q <> +infty; then
    ex r be Real st
     r = M2.Q & Y-vol(F.n,M2) = r(#)chi(P,X1) by A1,d4,Th96;
    hence product_sigma_Measure(M1,M2).(F.n) = Integral(M1,Y-vol(F.n,M2))
       by d5,Th98,SUPINF_2:51;
   end;
   suppose d6: M1.P <> 0 & M2.Q = +infty;
    M1.P >= 0 by SUPINF_2:51; then
d7: product_sigma_Measure(M1,M2).(F.n) = +infty by d5,d6,XXREAL_3:def 5;
    Y-vol(F.n,M2) = Xchi(P,X1) by A1,d4,d6,Th96;
    hence product_sigma_Measure(M1,M2).(F.n) = Integral(M1,Y-vol(F.n,M2))
      by d7,d6,MEASUR10:33;
   end;
   suppose M1.P <> 0 & M2.Q <> +infty; then
    ex r be Real st
     r = M2.Q & Y-vol(F.n,M2) = r(#)chi(P,X1) by A1,d4,Th96;
    hence product_sigma_Measure(M1,M2).(F.n) = Integral(M1,Y-vol(F.n,M2))
      by d5,Th98,SUPINF_2:51;
   end;
end;
