
theorem Th80:
  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;
  set C = E/\A;
A5: C = dom(f|A) by A3,RELAT_1:61;
A6: dom(f|A) = C by A3,RELAT_1:61
    .= dom f /\ C by A3,XBOOLE_1:17,28
    .= dom(f|C) by RELAT_1:61;
A7: for x be object st x in dom(f|A) holds (f|A).x = (f|C).x
  proof
    let x be object;
    assume
A8: x in dom(f|A);
    then (f|A).x = f.x by FUNCT_1:47;
    hence thesis by A6,A8,FUNCT_1:47;
  end;
A9: dom f /\ C = C by A3,XBOOLE_1:17,28;
  f is C-measurable by A4,MESFUNC1:30,XBOOLE_1:17;
  then f|C is C-measurable by A9,Th42;
  then f|A is C-measurable by A6,A7,FUNCT_1:2;
  hence thesis by A2,A5,Th15,Th79;
end;
