
theorem  Th53:
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,ProjPMap2(chi(E,[:X1,X2:]),y)) &
   X-vol(E,M1).y = integral+(M1,ProjPMap2(chi(E,[:X1,X2:]),y)) &
   X-vol(E,M1).y = integral'(M1,ProjPMap2(chi(E,[:X1,X2:]),y))
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,
   E be Element of sigma measurable_rectangles(S1,S2);
   assume A1: M1 is sigma_finite;

A2:ProjPMap2(chi(E,[:X1,X2:]),y) = chi(Y-section(E,y),X1) by Th48; then
   ProjPMap2(chi(E,[:X1,X2:]),y) = chi(Measurable-Y-section(E,y),X1)
     by MEASUR11:def 7; then
A4:ProjPMap2(chi(E,[:X1,X2:]),y) is_simple_func_in S1 by Th12;
   X-vol(E,M1).y = M1.(Measurable-Y-section(E,y)) by A1,MEASUR11:def 14; then
   X-vol(E,M1).y = Integral(M1,ProjMap2(chi(E,[:X1,X2:]),y))
     by MEASUR11:72;
   hence X-vol(E,M1).y = Integral(M1,ProjPMap2(chi(E,[:X1,X2:]),y))
     by Th27;
   hence X-vol(E,M1).y = integral+(M1,ProjPMap2(chi(E,[:X1,X2:]),y)) &
   X-vol(E,M1).y = integral'(M1,ProjPMap2(chi(E,[:X1,X2:]),y))
     by A2,A4,MESFUNC5:89;
end;
