
theorem Th78:
for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
 M1 be sigma_Measure of S1, M2 be sigma_Measure of S2,
 E be Element of sigma measurable_rectangles(S1,S2),
 f be PartFunc of [:X1,X2:],ExtREAL, r be Real st
 E = dom f & (f is nonnegative or f is nonpositive) & f is E-measurable
 holds
  Integral1(M1,r(#)f) = r(#)Integral1(M1,f)
& Integral2(M2,r(#)f) = r(#)Integral2(M2,f)
proof
    let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
    M1 be sigma_Measure of S1, M2 be sigma_Measure of S2,
    E be Element of sigma measurable_rectangles(S1,S2),
    f be PartFunc of [:X1,X2:],ExtREAL, r be Real;
    assume that
A1:  E = dom f and
A2:  (f is nonnegative or f is nonpositive) and
A3:  f is E-measurable;

A4: dom(r(#)Integral1(M1,f)) = X2 & dom(r(#)Integral2(M2,f)) = X1
     by FUNCT_2:def 1;

    now let y be Element of X2;
     dom(ProjPMap2(f,y)) = Y-section(E,y) by A1,Def4; then
A5:  dom(ProjPMap2(f,y)) = Measurable-Y-section(E,y) by MEASUR11:def 7;
A6:  ProjPMap2(f,y) is nonnegative or ProjPMap2(f,y) is nonpositive
       by A2,Th32,Th33;

     Integral1(M1,r(#)f).y
      = Integral(M1,ProjPMap2(r(#)f,y)) by Def7
     .= Integral(M1,r(#)ProjPMap2(f,y)) by Th29
     .= r * Integral(M1,ProjPMap2(f,y)) by A5,A6,A1,A3,Th47,Lm1,Lm2
     .= r * Integral1(M1,f).y by Def7;
     hence Integral1(M1,r(#)f).y = (r(#)Integral1(M1,f)).y
       by A4,MESFUNC1:def 6;
    end;
    hence Integral1(M1,r(#)f) = r(#)Integral1(M1,f) by FUNCT_2:def 8;

    now let x be Element of X1;
     dom(ProjPMap1(f,x)) = X-section(E,x) by A1,Def3; then
B5:  dom(ProjPMap1(f,x)) = Measurable-X-section(E,x) by MEASUR11:def 6;
B6:  ProjPMap1(f,x) is nonnegative or ProjPMap1(f,x) is nonpositive
       by A2,Th32,Th33;

     Integral2(M2,r(#)f).x
      = Integral(M2,ProjPMap1(r(#)f,x)) by Def8
     .= Integral(M2,r(#)ProjPMap1(f,x)) by Th29
     .= r * Integral(M2,ProjPMap1(f,x)) by B6,B5,A1,A3,Th47,Lm1,Lm2
     .= r * Integral2(M2,f).x by Def8;
     hence Integral2(M2,r(#)f).x = (r(#)Integral2(M2,f)).x
       by A4,MESFUNC1:def 6;
    end;
    hence Integral2(M2,r(#)f) = r(#)Integral2(M2,f) by FUNCT_2:def 8;
end;
