
theorem Th65:
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,
 x be Element of X1, y be Element of X2
 st E = [:A,B:] holds
  Integral(M2,ProjMap1(chi(E,[:X1,X2:]),x))
    = M2.(Measurable-X-section(E,x)) * chi(A,X1).x
& Integral(M1,ProjMap2(chi(E,[:X1,X2:]),y))
    = M1.(Measurable-Y-section(E,y)) * chi(B,X2).y
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,
   x be Element of X1, y be Element of X2;
   assume A1: E = [:A,B:]; then
A2:Integral(M2,ProjMap1(chi(E,[:X1,X2:]),x)) = M2.B * chi(A,X1).x by Th47;
A3:M2.B * chi(A,X1).x = M2.(Measurable-X-section(E,x)) by A1,Th48;
A4:Integral(M1,ProjMap2(chi(E,[:X1,X2:]),y)) = M1.A * chi(B,X2).y
      by A1,Th49;
A5:M1.A * chi(B,X2).y = M1.(Measurable-Y-section(E,y)) by A1,Th50;
   thus Integral(M2,ProjMap1(chi(E,[:X1,X2:]),x))
         = M2.(Measurable-X-section(E,x)) * chi(A,X1).x
   proof
    per cases;
    suppose x in A; then
     chi(A,X1).x = 1 by FUNCT_3:def 3;
     hence thesis by A2,A3,XXREAL_3:81;
    end;
    suppose not x in A; then
     chi(A,X1).x = 0 by FUNCT_3:def 3;
     hence thesis by A2;
    end;
   end;
    per cases;
    suppose y in B; then
     chi(B,X2).y = 1 by FUNCT_3:def 3;
     hence thesis by A4,A5,XXREAL_3:81;
    end;
    suppose not y in B; then
     chi(B,X2).y = 0 by FUNCT_3:def 3;
     hence thesis by A4;
    end;
end;
