theorem Th52:
  for X being set, f being PartFunc of X,REAL st for x being object
  st x in dom f holds 0 <= f.x holds f is nonnegative
proof
  let X be set, f be PartFunc of X,REAL such that
A1: for x being object st x in dom f 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 A1;
end;
