
theorem Th56:
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), r be Real
st M2 is sigma_finite holds
  (r(#)Y-vol(E,M2)).x = Integral(M2,ProjPMap1(chi(r,E,[:X1,X2:]),x))
 & (r >= 0 implies
      (r(#)Y-vol(E,M2)).x = integral+(M2,ProjPMap1(chi(r,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),
   r be Real;
   assume A1: M2 is sigma_finite;

   set p2 = ProjPMap1(chi(E,[:X1,X2:]),x);

   chi(r,E,[:X1,X2:]) = r(#)chi(E,[:X1,X2:]) by Th1; then
A2:ProjPMap1(chi(r,E,[:X1,X2:]),x) = r(#)p2 by Th29;

A3:p2 is nonnegative by Th32;

A4:dom(r(#)Y-vol(E,M2)) = X1 by FUNCT_2:def 1;

A5:chi(E,[:X1,X2:]) is_simple_func_in sigma measurable_rectangles(S1,S2)
     by Th12; then
   Integral(M2,ProjPMap1(chi(r,E,[:X1,X2:]),x))
     = r * integral'(M2,p2) by A2,A3,Th31,MESFUN11:59
    .= r * Y-vol(E,M2).x by A1,Th52;
   hence A7: (r(#)Y-vol(E,M2)).x = Integral(M2,ProjPMap1(chi(r,E,[:X1,X2:]),x))
      by A4,MESFUNC1:def 6;
   thus (r >= 0 implies
      (r(#)Y-vol(E,M2)).x = integral+(M2,ProjPMap1(chi(r,E,[:X1,X2:]),x)))
   proof
    assume r >= 0; then
A8: r(#)p2 is nonnegative by A3,MESFUNC5:20;
    r(#)p2 is_simple_func_in S2 by A5,Th31,MESFUNC5:39;
    hence thesis by A2,A7,A8,MESFUNC5:89;
   end;
end;
