
theorem Th4:
  for L1,L2 be RelStr for f be Function of L1,L2 st f is isomorphic
  holds (the carrier of L1 <> {} iff the carrier of L2 <> {}) & (the carrier of
  L2 <> {} or the carrier of L1 = {}) & (the carrier of L1 = {} iff the carrier
  of L2 = {})
proof
  let L1,L2 be RelStr;
  let f be Function of L1,L2 such that
A1: f is isomorphic;
  the carrier of L1 = {} iff the carrier of L2 = {}
  proof
    hereby
      assume the carrier of L1 = {};
      then L1 is empty;
      then L2 is empty by A1,WAYBEL_0:def 38;
      hence the carrier of L2 = {};
    end;
    assume the carrier of L2 = {};
    then L2 is empty;
    then L1 is empty by A1,WAYBEL_0:def 38;
    hence thesis;
  end;
  hence thesis;
end;
