
theorem Th96:
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),
  A be Element of S1, B be Element of S2
   st E = [:A,B:] & M2 is sigma_finite holds
   (M2.B = +infty implies Y-vol(E,M2) = Xchi(A,X1))
 & (M2.B <> +infty implies
      ex r be Real st r = M2.B & Y-vol(E,M2) = r(#)chi(A,X1))
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),
       A be Element of S1, B be Element of S2;
   assume that
A1: E = [:A,B:] and
A2: M2 is sigma_finite;
   hereby assume
A3: M2.B = +infty;
    for x be Element of X1 holds (Y-vol(E,M2)).x = Xchi(A,X1).x
    proof
     let x be Element of X1;
A4:  (Y-vol(E,M2)).x = M2.(Measurable-X-section(E,x)) by A2,DefYvol
      .= M2.B * chi(A,X1).x by A1,Th48;
     per cases;
     suppose
A5:   x in A; then
      chi(A,X1).x = 1 by FUNCT_3:def 3; then
      (Y-vol(E,M2)).x = +infty by A3,A4,XXREAL_3:81;
      hence (Y-vol(E,M2)).x = Xchi(A,X1).x by A5,MEASUR10:def 7;
     end;
     suppose
A6:   not x in A; then
      chi(A,X1).x = 0 by FUNCT_3:def 3; then
      (Y-vol(E,M2)).x = 0 by A4;
      hence (Y-vol(E,M2)).x = Xchi(A,X1).x by A6,MEASUR10:def 7;
     end;
    end;
    hence Y-vol(E,M2) = Xchi(A,X1) by FUNCT_2:def 8;
   end;
   assume
P1: M2.B <> +infty;
   M2.B >= 0 by SUPINF_2:51; then
   M2.B in REAL by P1,XXREAL_0:14; then
   reconsider r = M2.B as Real;
   take r;
   dom(r(#)chi(A,X1)) = dom(chi(A,X1)) by MESFUNC1:def 6; then
A8:dom(r(#)chi(A,X1)) = X1 by FUNCT_3:def 3; then
P2:dom(Y-vol(E,M2)) = dom(r(#)chi(A,X1)) by FUNCT_2:def 1;
   for x be Element of X1 st x in dom(Y-vol(E,M2)) holds
    (Y-vol(E,M2)).x = (r(#)chi(A,X1)).x
   proof
    let x be Element of X1;
    assume x in dom(Y-vol(E,M2));
    (Y-vol(E,M2)).x = M2.(Measurable-X-section(E,x)) by A2,DefYvol
     .= r * chi(A,X1).x by A1,Th48;
    hence (Y-vol(E,M2)).x = (r(#)chi(A,X1)).x by A8,MESFUNC1:def 6;
   end;
   hence r = M2.B & Y-vol(E,M2) = r(#)chi(A,X1) by P2,PARTFUN1:5;
end;
