
theorem Th47:
for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
 M2 be sigma_Measure of S2,
 A be Element of S1, B be Element of S2, x be Element of X1 holds
  M2.B * chi(A,X1).x = Integral(M2,ProjMap1(chi([:A,B:],[: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,
   A be Element of S1, B be Element of S2, x be Element of X1;
A1:for y be Element of X2 holds
    ProjMap1(chi([:A,B:],[:X1,X2:]),x).y = chi(A,X1).x * chi(B,X2).y
   proof
    let y be Element of X2;
    ProjMap1(chi([:A,B:],[:X1,X2:]),x).y = chi([:A,B:],[:X1,X2:]).(x,y)
      by MESFUNC9:def 6;
    hence thesis by MEASUR10:2;
   end;
   set CAB = chi([:A,B:],[:X1,X2:]);
   per cases;
   suppose x in A; then
A2: chi(A,X1).x = 1 by FUNCT_3:def 3; then
A3: M2.B * chi(A,X1).x = M2.B by XXREAL_3:81;
A4: dom (ProjMap1(chi([:A,B:],[:X1,X2:]),x)) = X2 by FUNCT_2:def 1
       .= dom chi(B,X2) by FUNCT_3:def 3;
    for y be Element of X2 st y in dom(ProjMap1(CAB,x)) holds
      ProjMap1(CAB,x).y = chi(B,X2).y
    proof
     let y be Element of X2;
     assume y in dom(ProjMap1(CAB,x));
     ProjMap1(CAB,x).y = chi(A,X1).x * chi(B,X2).y by A1;
     hence ProjMap1(CAB,x).y = chi(B,X2).y by A2,XXREAL_3:81;
    end; then
    ProjMap1(CAB,x) = chi(B,X2) by A4,PARTFUN1:5;
    hence M2.B * chi(A,X1).x = Integral(M2,ProjMap1(CAB,x))
       by A3,MESFUNC9:14;
   end;
   suppose not x in A; then
A5: chi(A,X1).x = 0 by FUNCT_3:def 3; then
A6: M2.B * chi(A,X1).x = 0;
A7: {} is Element of S2 by PROB_1:4;
A8: dom(ProjMap1(CAB,x)) = X2 by FUNCT_2:def 1
       .= dom chi({},X2) by FUNCT_3:def 3;
    for y be Element of X2 st y in dom(ProjMap1(CAB,x)) holds
      ProjMap1(CAB,x).y = chi({},X2).y
    proof
     let y be Element of X2;
     assume y in dom(ProjMap1(CAB,x));
     ProjMap1(CAB,x).y = chi(A,X1).x * chi(B,X2).y by A1; then
     ProjMap1(CAB,x).y = 0 by A5;
     hence ProjMap1(CAB,x).y = chi({},X2).y by FUNCT_3:def 3;
    end; then
    ProjMap1(CAB,x) = chi({},X2) by A8,PARTFUN1:5; then
    Integral(M2,ProjMap1(CAB,x)) = M2.{} by A7,MESFUNC9:14;
    hence M2.B * chi(A,X1).x = Integral(M2,ProjMap1(CAB,x))
      by A6,VALUED_0:def 19;
   end;
end;
