
theorem Th48:
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)),
  A be Element of S1, B be Element of S2, x be Element of X1
  st E = [:A,B:] holds M2.(Measurable-X-section(E,x)) = M2.B * chi(A,X1).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,
       E be Element of sigma(measurable_rectangles(S1,S2)),
       A be Element of S1, B be Element of S2, x be Element of X1;
   assume A1: E = [:A,B:];
   per cases;
   suppose A4: x in A; then
A2: M2.(Measurable-X-section(E,x)) = M2.B by A1,Th16;
    chi(A,X1).x = 1 by A4,FUNCT_3:def 3;
    hence M2.(Measurable-X-section(E,x)) = M2.B * chi(A,X1).x
      by A2,XXREAL_3:81;
   end;
   suppose A5: not x in A; then
    Measurable-X-section(E,x) = {} by A1,Th16; then
A3: M2.(Measurable-X-section(E,x)) = 0 by VALUED_0:def 19;
    chi(A,X1).x = 0 by A5,FUNCT_3:def 3;
    hence M2.(Measurable-X-section(E,x)) = M2.B * chi(A,X1).x by A3;
   end;
end;
