
theorem Th62:
for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
  M2 be sigma_Measure of S2, x be Element of X1,
  E be Element of sigma measurable_rectangles(S1,S2)
 st M2 is sigma_finite holds
   Y-vol(E,M2).x = Integral(M2,chi(Measurable-X-section(E,x),X2))
proof
    let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
    M2 be sigma_Measure of S2, x be Element of X1,
    A be Element of sigma measurable_rectangles(S1,S2);
    assume M2 is sigma_finite; then
    Y-vol(A,M2).x = M2.(Measurable-X-section(A,x)) by MEASUR11:def 13;
    hence Y-vol(A,M2).x
      = Integral(M2,chi(Measurable-X-section(A,x),X2)) by MESFUNC9:14;
end;
