
theorem Th73:
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,
  f be PartFunc of [:X1,X2:],ExtREAL,
  E be Element of sigma measurable_rectangles(S1,S2)
  st E = dom f & f is E-measurable
  holds
  Integral1(M1,-f) = -Integral1(M1,f) & Integral2(M2,-f) = -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,
    f be PartFunc of [:X1,X2:],ExtREAL,
    A be Element of sigma measurable_rectangles(S1,S2);
    assume that
A1:  A = dom f and
A2:  f is A-measurable;

A3: dom(-Integral1(M1,f)) = X2 & dom(-Integral2(M2,f)) = X1 by FUNCT_2:def 1;

    now let y be Element of X2;
     ProjPMap2(-f,y) = ProjPMap2((-1)(#)f,y) by MESFUNC2:9
      .= (-1)(#)ProjPMap2(f,y) by Th29
      .= -(ProjPMap2(f,y)) by MESFUNC2:9; then
A4:  Integral1(M1,-f).y = Integral(M1,-ProjPMap2(f,y)) by Def7;

     dom(ProjPMap2(f,y)) = Y-section(A,y) by A1,Def4; then
A5:  dom(ProjPMap2(f,y)) = Measurable-Y-section(A,y) by MEASUR11:def 7;

     (-Integral1(M1,f)).y = -(Integral1(M1,f).y) by A3,MESFUNC1:def 7; then
     (-Integral1(M1,f)).y = -(Integral(M1,ProjPMap2(f,y))) by Def7;
     hence Integral1(M1,-f).y = (-Integral1(M1,f)).y
       by A1,A2,A4,A5,Th47,MESFUN11:52;
    end;
    hence Integral1(M1,-f) = -Integral1(M1,f) by FUNCT_2:def 8;

    now let x be Element of X1;
     ProjPMap1(-f,x) = ProjPMap1((-1)(#)f,x) by MESFUNC2:9
      .= (-1)(#)ProjPMap1(f,x) by Th29
      .= -(ProjPMap1(f,x)) by MESFUNC2:9; then
A6:  Integral2(M2,-f).x = Integral(M2,-ProjPMap1(f,x)) by Def8;

     dom(ProjPMap1(f,x)) = X-section(A,x) by A1,Def3; then
A7:  dom(ProjPMap1(f,x)) = Measurable-X-section(A,x) by MEASUR11:def 6;

     (-Integral2(M2,f)).x = -(Integral2(M2,f).x) by A3,MESFUNC1:def 7; then
     (-Integral2(M2,f)).x = -(Integral(M2,ProjPMap1(f,x))) by Def8;
     hence Integral2(M2,-f).x = (-Integral2(M2,f)).x
       by A1,A2,A6,A7,Th47,MESFUN11:52;
    end;
    hence Integral2(M2,-f) = -Integral2(M2,f) by FUNCT_2:def 8;
end;
