
theorem Th9:
  for X being set, F being PartFunc of X,ExtREAL st for n being
  set st n in dom F holds F.n <= 0. holds F is nonpositive
proof
  let X be set, F be PartFunc of X,ExtREAL such that
A1: for n being set st n in dom F 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 A1;
end;
