
theorem
for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
  M2 be sigma_Measure of S2, f be PartFunc of [:X1,X2:],ExtREAL,
  E be Element of sigma measurable_rectangles(S1,S2), x be Element of X1
 st E = dom f & (f is nonnegative or f is nonpositive) & f is E-measurable &
    (for y be Element of X2 st y in dom(ProjPMap1(f,x)) holds
     ProjPMap1(f,x).y = 0)
 holds Integral2(M2,f).x = 0
proof
    let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
    M2 be sigma_Measure of S2, f be PartFunc of [:X1,X2:],ExtREAL,
    A be Element of sigma measurable_rectangles(S1,S2), x be Element of X1;
    assume that
A1:  A = dom f and
A2:  f is nonnegative or f is nonpositive and
A3:  f is A-measurable and
A4:  for y be Element of X2 st y in dom(ProjPMap1(f,x)) holds
       ProjPMap1(f,x).y = 0;
A5: dom(ProjPMap1(f,x)) = X-section(A,x) by A1,Def3
     .= Measurable-X-section(A,x) by MEASUR11:def 6;
A6: ProjPMap1(f,x) is (Measurable-X-section(A,x))-measurable by A1,A3,Th47;

    per cases by A2;
    suppose A7:f is nonnegative;
     integral+(M2,ProjPMap1(f,x)) = 0 by A1,A3,A4,A5,Th47,MESFUNC5:87; then
     Integral(M2,ProjPMap1(f,x)) = 0 by A5,A6,A7,Th32,MESFUNC5:88;
     hence Integral2(M2,f).x = 0 by Def8;
    end;
    suppose f is nonpositive; then
A8:  ProjPMap1(f,x) is nonpositive by Th33;
A9:  dom(-ProjPMap1(f,x)) = Measurable-X-section(A,x) by A5,MESFUNC1:def 7;
     for y be Element of X2 st y in dom(-ProjPMap1(f,x)) holds
      (-ProjPMap1(f,x)).y = 0
     proof
      let y be Element of X2;
      assume A10: y in dom(-ProjPMap1(f,x)); then
      (-ProjPMap1(f,x)).y = -(ProjPMap1(f,x).y) by MESFUNC1:def 7; then
      (-ProjPMap1(f,x)).y = -0 by A4,A5,A9,A10;
      hence (-ProjPMap1(f,x)).y = 0;
     end; then
     integral+(M2,-ProjPMap1(f,x)) = 0
       by A5,A6,A9,MEASUR11:63,MESFUNC5:87; then
     -integral+(M2,-ProjPMap1(f,x)) = 0; then
     Integral(M2,ProjPMap1(f,x)) = 0 by A5,A6,A8,MESFUN11:57;
     hence Integral2(M2,f).x = 0 by Def8;
    end;
end;
