theorem Th51:
  for X being set, f being PartFunc of X,REAL holds f is
  nonnegative iff for x being object holds 0 <= f.x
proof
  let X be set, f be PartFunc of X,REAL;
  hereby
    assume f is nonnegative;
    then
A1: rng f is nonnegative;
    hereby
      let x be object;
      now
        assume x in dom f;
        then
A2:     f.x in rng f by FUNCT_1:def 3;
        thus 0 <= f.x by A1,A2;
      end;
      hence 0 <= f.x by FUNCT_1:def 2;
    end;
  end;
  assume
A3: for x being object holds 0 <= f.x;
  let y be ExtReal;
  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;
