
theorem  Th52:
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,ProjPMap1(chi(E,[:X1,X2:]),x)) &
   Y-vol(E,M2).x = integral+(M2,ProjPMap1(chi(E,[:X1,X2:]),x)) &
   Y-vol(E,M2).x = integral'(M2,ProjPMap1(chi(E,[:X1,X2:]),x))
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,
   E be Element of sigma measurable_rectangles(S1,S2);
   assume A1: M2 is sigma_finite;
A2:ProjPMap1(chi(E,[:X1,X2:]),x) = chi(X-section(E,x),X2) by Th48; then
   ProjPMap1(chi(E,[:X1,X2:]),x) = chi(Measurable-X-section(E,x),X2)
     by MEASUR11:def 6; then
A4:ProjPMap1(chi(E,[:X1,X2:]),x) is_simple_func_in S2 by Th12;
   Y-vol(E,M2).x = M2.(Measurable-X-section(E,x)) by A1,MEASUR11:def 13; then
   Y-vol(E,M2).x = Integral(M2,ProjMap1(chi(E,[:X1,X2:]),x))
     by MEASUR11:72;
   hence Y-vol(E,M2).x = Integral(M2,ProjPMap1(chi(E,[:X1,X2:]),x))
     by Th27;
   hence Y-vol(E,M2).x = integral+(M2,ProjPMap1(chi(E,[:X1,X2:]),x)) &
   Y-vol(E,M2).x = integral'(M2,ProjPMap1(chi(E,[:X1,X2:]),x))
     by A2,A4,MESFUNC5:89;
end;
