
theorem Th68:
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
holds
   M1.(Measurable-Y-section(E,y)) = Integral(M1,ProjMap2(chi(E,[:X1,X2:]),y))
 & M2.(Measurable-X-section(E,x)) = Integral(M2,ProjMap1(chi(E,[: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;
A1:X1 in S1 & X2 in S2 by MEASURE1:7;
   E /\ [:X1,X2:] = E
 & Measurable-Y-section(E,y) /\ X1 = Measurable-Y-section(E,y)
 & Measurable-X-section(E,x) /\ X2 = Measurable-X-section(E,x) by XBOOLE_1:28;
   hence thesis by Th67,A1;
end;
