
theorem Th67:
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),
  x be Element of X1, y be Element of X2,
  U be Element of S1, V be Element of S2
holds
   M1.(Measurable-Y-section(E,y) /\ U)
        = Integral(M1,ProjMap2(chi(E /\ [:U,X2:],[:X1,X2:]),y))
 & M2.(Measurable-X-section(E,x) /\ V)
        = Integral(M2,ProjMap1(chi(E /\ [:X1,V:],[:X1,X2:]),x))
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),
       x be Element of X1, y be Element of X2,
       U be Element of S1, V be Element of S2;
   for x be Element of X1 holds (ProjMap2(chi(E /\ [:U,X2:],[:X1,X2:]),y)).x
      = chi(Measurable-Y-section(E,y) /\ U,X1).x
   proof
    let x be Element of X1;
A1: X2 = [#]X2 by SUBSET_1:def 3;

    (ProjMap2(chi(E /\ [:U,X2:],[:X1,X2:]),y)).x
      = chi(E /\ [:U,X2:],[:X1,X2:]).(x,y) by MESFUNC9:def 7
     .= chi(Y-section(E /\ [:U,X2:],y),X1).x by Th28
     .= chi(Y-section(E,y) /\ Y-section([:U,[#]X2:],y),X1).x by A1,Th21;
    hence (ProjMap2(chi(E /\ [:U,X2:],[:X1,X2:]),y)).x
              = chi(Measurable-Y-section(E,y) /\ U,X1).x by A1,Th16;
   end; then
   ProjMap2(chi(E /\ [:U,X2:],[:X1,X2:]),y)
     = chi(Measurable-Y-section(E,y) /\ U,X1)
      by FUNCT_2:def 8;
   hence M1.(Measurable-Y-section(E,y) /\ U)
          = Integral(M1,ProjMap2(chi(E /\ [:U,X2:],[:X1,X2:]),y))
      by MESFUNC9:14;

   for y be Element of X2 holds (ProjMap1(chi(E /\ [:X1,V:],[:X1,X2:]),x)).y
      = chi(Measurable-X-section(E,x) /\ V,X2).y
   proof
    let y be Element of X2;
A3: X1 = [#]X1 by SUBSET_1:def 3;

    (ProjMap1(chi(E /\ [:X1,V:],[:X1,X2:]),x)).y
      = chi(E /\ [:X1,V:],[:X1,X2:]).(x,y) by MESFUNC9:def 6
     .= chi(X-section(E /\ [:X1,V:],x),X2).y by Th28
     .= chi(X-section(E,x) /\ X-section([:[#]X1,V:],x),X2).y by A3,Th21;
    hence (ProjMap1(chi(E /\ [:X1,V:],[:X1,X2:]),x)).y
      = chi(Measurable-X-section(E,x) /\ V,X2).y by A3,Th16;
   end; then
   ProjMap1(chi(E /\ [:X1,V:],[:X1,X2:]),x)
     = chi(Measurable-X-section(E,x) /\ V,X2) by FUNCT_2:def 8;
   hence M2.(Measurable-X-section(E,x) /\ V)
          = Integral(M2,ProjMap1(chi(E /\ [:X1,V:],[:X1,X2:]),x))
      by MESFUNC9:14;
end;
