
theorem
  for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
f be PartFunc of X,ExtREAL, A,B be Element of S st (ex E be Element of S st E =
dom f & f is E-measurable ) & f is nonnegative & A c= B holds Integral(M,f|A
  ) <= Integral(M,f|B)
proof
  let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be
  PartFunc of X,ExtREAL, A,B be Element of S;
  assume that
A1: ex E be Element of S st E = dom f & f is E-measurable and
A2: f is nonnegative and
A3: A c= B;
  consider E be Element of S such that
A4: E = dom f and
A5: f is E-measurable by A1;
A6: ex C be Element of S st C = dom(f|A) & f|A is C-measurable
  proof
    take C = E /\ A;
    thus dom(f|A) = C by A4,RELAT_1:61;
A7: C = dom f /\ C by A4,XBOOLE_1:17,28;
A8: dom(f|A) = C by A4,RELAT_1:61
      .= dom(f|C) by A7,RELAT_1:61;
A9: for x be object st x in dom(f|A) holds (f|A).x = (f|C).x
    proof
      let x be object;
      assume
A10:  x in dom(f|A);
      then (f|A).x = f.x by FUNCT_1:47;
      hence thesis by A8,A10,FUNCT_1:47;
    end;
    f is C-measurable by A5,MESFUNC1:30,XBOOLE_1:17;
    then f|C is C-measurable by A7,Th42;
    hence thesis by A8,A9,FUNCT_1:2;
  end;
A11: ex C be Element of S st C = dom(f|B) & f|B is C-measurable
  proof
    take C = E /\ B;
    thus dom(f|B) = C by A4,RELAT_1:61;
A12: C = dom f /\ C by A4,XBOOLE_1:17,28;
A13: dom(f|B) = C by A4,RELAT_1:61
      .= dom(f|C) by A12,RELAT_1:61;
A14: for x be object st x in dom(f|B) holds (f|B).x = (f|C).x
    proof
      let x be object;
      assume
A15:  x in dom(f|B);
      then (f|B).x = f.x by FUNCT_1:47;
      hence thesis by A13,A15,FUNCT_1:47;
    end;
    f is C-measurable by A5,MESFUNC1:30,XBOOLE_1:17;
    then f|C is C-measurable by A12,Th42;
    hence thesis by A13,A14,FUNCT_1:2;
  end;
  integral+(M,f|A) <= integral+(M,f|B) by A1,A2,A3,Th83;
  then Integral(M,f|A) <= integral+(M,f|B) by A2,A6,Th15,Th88;
  hence thesis by A2,A11,Th15,Th88;
end;
