
theorem Th68:
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,
  E1,E2 be Element of sigma measurable_rectangles(S1,S2), V be Element of S2
 st M1 is sigma_finite
  & (f is nonnegative or f is nonpositive) & E1 = dom f & f is E1-measurable
 holds
  Integral1(M1,f|E2) is V-measurable
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,
    E,A be Element of sigma measurable_rectangles(S1,S2), V be Element of S2;
    assume that
A1:  M1 is sigma_finite and
A2:  f is nonnegative or f is nonpositive and
A3:  E = dom f and
A4:  f is E-measurable;

A5: dom(f|A) = E /\ A by A3,RELAT_1:61;
A6: dom f /\ (E /\ A) = E /\ A by A3,XBOOLE_1:17,28;
    f is (E/\A)-measurable by A4,XBOOLE_1:17,MESFUNC1:30; then
    f|(E/\A) is (E/\A)-measurable by A6,MESFUNC5:42; then
    (f|E)|A is (E/\A)-measurable by RELAT_1:71;
    hence Integral1(M1,f|A) is V-measurable
      by A1,A2,A3,A5,MESFUNC5:15,MESFUN11:1,Th59;
end;
