
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 be Element of S st (ex E be Element of S st E =
  dom f & f is E-measurable ) & f is nonnegative holds 0<= Integral(M,f|A)
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 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;
  consider E be Element of S such that
A3: E = dom f and
A4: f is E-measurable by A1;
A5: 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 A3,RELAT_1:61;
A6: C = dom f /\ C by A3,XBOOLE_1:17,28;
A7: dom(f|A) = C by A3,RELAT_1:61
      .= dom(f|C) by A6,RELAT_1:61;
A8: for x be object st x in dom(f|A) holds (f|A).x = (f|C).x
    proof
      let x be object;
      assume
A9:   x in dom(f|A);
      then (f|A).x = f.x by FUNCT_1:47;
      hence thesis by A7,A9,FUNCT_1:47;
    end;
    f is C-measurable by A4,MESFUNC1:30,XBOOLE_1:17;
    then f|C is C-measurable by A6,Th42;
    hence thesis by A7,A8,FUNCT_1:2;
  end;
  then 0<= integral+(M,f|A) by A2,Th15,Th79;
  hence thesis by A2,A5,Th15,Th88;
end;
