
theorem Th14:
  for X be non empty set, S be SigmaField of X, f be PartFunc of X
,ExtREAL st f is_simple_func_in S holds f is without+infty & f is without-infty
proof
  let X be non empty set, S be SigmaField of X, f be PartFunc of X,ExtREAL;
  assume
A1: f is_simple_func_in S;
  hereby
    assume not f is without+infty;
    then +infty in rng f;
    then f"{+infty} <> {} by FUNCT_1:72;
    then consider x be object such that
A2: x in f"{+infty} by XBOOLE_0:def 1;
A3: f is real-valued by A1,MESFUNC2:def 4;
    f.x in {+infty} by A2,FUNCT_1:def 7;
    hence contradiction by A3,TARSKI:def 1;
  end;
  hereby
    assume not f is without-infty;
    then -infty in rng f;
    then f"{-infty} <> {} by FUNCT_1:72;
    then consider x be object such that
A4: x in f"{-infty} by XBOOLE_0:def 1;
A5: f is real-valued by A1,MESFUNC2:def 4;
    f.x in {-infty} by A4,FUNCT_1:def 7;
    hence contradiction by A5,TARSKI:def 1;
  end;
end;
