theorem Th54:
  (for x being object st x in dom f holds f.x <= 0) implies f is nonpositive
proof
  assume
A1: for x being object st x in dom f 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 A1;
end;
