
theorem Th50:
for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
  M1 be sigma_Measure of S1,
  E be Element of sigma(measurable_rectangles(S1,S2)),
  A be Element of S1, B be Element of S2, y be Element of X2
  st E = [:A,B:] holds M1.(Measurable-Y-section(E,y)) = M1.A * chi(B,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,
       E be Element of sigma(measurable_rectangles(S1,S2)),
       A be Element of S1, B be Element of S2,
       y be Element of X2;
   assume A1: E = [:A,B:];
   per cases;
   suppose A4: y in B; then
A2: M1.(Measurable-Y-section(E,y)) = M1.A by A1,Th16;
    chi(B,X2).y = 1 by A4,FUNCT_3:def 3;
    hence M1.(Measurable-Y-section(E,y)) = M1.A * chi(B,X2).y
      by A2,XXREAL_3:81;
   end;
   suppose A5: not y in B; then
    Measurable-Y-section(E,y) = {} by A1,Th16; then
A3: M1.(Measurable-Y-section(E,y)) = 0 by VALUED_0:def 19;
    chi(B,X2).y = 0 by A5,FUNCT_3:def 3;
    hence M1.(Measurable-Y-section(E,y)) = M1.A * chi(B,X2).y by A3;
   end;
end;
