theorem
  for X being set, f being PartFunc of X,REAL holds f is nonpositive iff
  for x being set holds f.x <= 0
proof
  let X be set, f be PartFunc of X,REAL;
  hereby
    assume f is nonpositive;
    then
A1: rng f is nonpositive;
    hereby
      let x be set;
      now
        assume x in dom f;
        then
A2:     f.x in rng f by FUNCT_1:def 3;
        thus f.x <= 0 by A1,A2;
      end;
      hence f.x <= 0 by FUNCT_1:def 2;
    end;
  end;
  assume
A3: for x being set holds f.x <= 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 A3;
end;
