
theorem
for X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
  M1 be sigma_Measure of S1, f be PartFunc of [:X1,X2:],ExtREAL,
  E be Element of sigma measurable_rectangles(S1,S2), y be Element of X2
 st E = dom f & (f is nonnegative or f is nonpositive) & f is E-measurable &
    (for x be Element of X1 st x in dom(ProjPMap2(f,y)) holds
     ProjPMap2(f,y).x = 0)
 holds Integral1(M1,f).y = 0
proof
    let X1,X2 be non empty set, S1 be SigmaField of X1, S2 be SigmaField of X2,
    M1 be sigma_Measure of S1, f be PartFunc of [:X1,X2:],ExtREAL,
    A be Element of sigma measurable_rectangles(S1,S2), y be Element of X2;
    assume that
A1:  A = dom f and
A2:  f is nonnegative or f is nonpositive and
A3:  f is A-measurable and
A4:  for x be Element of X1 st x in dom(ProjPMap2(f,y)) holds
       ProjPMap2(f,y).x = 0;
A5: dom(ProjPMap2(f,y)) = Y-section(A,y) by A1,Def4
     .= Measurable-Y-section(A,y) by MEASUR11:def 7;
A6: ProjPMap2(f,y) is (Measurable-Y-section(A,y))-measurable by A1,A3,Th47;

    per cases by A2;
    suppose A7:f is nonnegative;
     integral+(M1,ProjPMap2(f,y)) = 0 by A1,A3,A4,A5,Th47,MESFUNC5:87; then
     Integral(M1,ProjPMap2(f,y)) = 0 by A5,A6,A7,Th32,MESFUNC5:88;
     hence Integral1(M1,f).y = 0 by Def7;
    end;
    suppose f is nonpositive; then
A8:  ProjPMap2(f,y) is nonpositive by Th33;
A9:  dom(-ProjPMap2(f,y)) = Measurable-Y-section(A,y) by A5,MESFUNC1:def 7;
     for x be Element of X1 st x in dom(-ProjPMap2(f,y)) holds
      (-ProjPMap2(f,y)).x = 0
     proof
      let x be Element of X1;
      assume A10: x in dom(-ProjPMap2(f,y)); then
      (-ProjPMap2(f,y)).x = -(ProjPMap2(f,y).x) by MESFUNC1:def 7; then
      (-ProjPMap2(f,y)).x = -0 by A4,A5,A9,A10;
      hence (-ProjPMap2(f,y)).x = 0;
     end; then
     integral+(M1,-ProjPMap2(f,y)) = 0
       by A5,A6,A9,MEASUR11:63,MESFUNC5:87; then
     -integral+(M1,-ProjPMap2(f,y)) = 0; then
     Integral(M1,ProjPMap2(f,y)) = 0 by A5,A6,A8,MESFUN11:57;
     hence Integral1(M1,f).y = 0 by Def7;
    end;
end;
