
theorem Th61:
for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
  M1 be sigma_Measure of S1, y be Element of X2,
  E be Element of sigma measurable_rectangles(S1,S2)
st M1 is sigma_finite holds
   X-vol(E,M1).y = Integral(M1,chi(Measurable-Y-section(E,y),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, y be Element of X2,
    A be Element of sigma measurable_rectangles(S1,S2);
    assume M1 is sigma_finite; then
    X-vol(A,M1).y = M1.(Measurable-Y-section(A,y)) by MEASUR11:def 14;
    hence X-vol(A,M1).y
     = Integral(M1,chi(Measurable-Y-section(A,y),X1)) by MESFUNC9:14;
end;
