
theorem Th65:
for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
  M2 be sigma_Measure of S2, E be Element of sigma measurable_rectangles(S1,S2)
st M2 is sigma_finite holds Y-vol(E,M2) = Integral2(M2,chi(E,[:X1,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,
    A be Element of sigma measurable_rectangles(S1,S2);
    assume
a1:  M2 is sigma_finite;
    now let x be Element of X1;
A1:  Y-vol(A,M2).x = Integral(M2,chi(Measurable-X-section(A,x),X2))
       by a1,Th62;
     ProjPMap1(chi(A,[:X1,X2:]),x) = chi(Measurable-X-section(A,x),X2)
       by Th63;
     hence Y-vol(A,M2).x = Integral2(M2,chi(A,[:X1,X2:])).x by A1,Def8;
    end;
    hence thesis by FUNCT_2:def 8;
end;
