theorem Th1:
  for f being Function, X being set st X c= dom f holds X <> {}
  implies rng (f|X) <> {}
proof
  let f be Function, X be set;
  assume
A1: X c= dom f;
  set x = the Element of X;
  assume X <> {};
  then x in X;
  hence thesis by A1,FUNCT_1:50;
end;
