
theorem Th8:
  for X being set, F being PartFunc of X,ExtREAL holds F is
  nonpositive iff for n being object holds F.n <= 0.
proof
  let X be set, F be PartFunc of X,ExtREAL;
  hereby
    assume F is nonpositive;
    then
A1: rng F is nonpositive;
    let n be object;
    per cases;
    suppose
      n in dom F;
      then F.n in rng F by FUNCT_1:def 3;
      hence F.n <= 0. by A1;
    end;
    suppose
      not n in dom F;
      hence F.n <= 0. by FUNCT_1:def 2;
    end;
  end;
  assume
A2: for n being object holds F.n <= 0.;
  let y be R_eal;
  assume y in rng F;
  then ex x being object st x in dom F & y = F.x by FUNCT_1:def 3;
  hence thesis by A2;
end;
