
theorem Th64:
for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
  M1 be sigma_Measure of S1, E be Element of sigma measurable_rectangles(S1,S2)
st M1 is sigma_finite holds X-vol(E,M1) = Integral1(M1,chi(E,[:X1,X2:]))
proof
    let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
    M1 be sigma_Measure of S1,
    A be Element of sigma measurable_rectangles(S1,S2);
    assume
A1:  M1 is sigma_finite;
    now let y be Element of X2;
A2:  X-vol(A,M1).y = Integral(M1,chi(Measurable-Y-section(A,y),X1))
       by A1,Th61;
     ProjPMap2(chi(A,[:X1,X2:]),y) = chi(Measurable-Y-section(A,y),X1)
       by Th63;
     hence X-vol(A,M1).y = Integral1(M1,chi(A,[:X1,X2:])).y by A2,Def7;
    end;
    hence thesis by FUNCT_2:def 8;
end;
