theorem Th12:
  W1 = W \/ {F} implies len(W1) <= len(W) + len F
proof
  assume
A1: W1 = W \/ {F};
  now
    per cases;
    suppose
      F in W;
      then {F} c= W by ZFMISC_1:31;
      then W1 = W by A1,XBOOLE_1:12;
      hence thesis by XREAL_1:31;
    end;
    suppose
      not F in W;
      hence thesis by A1,Th11;
    end;
  end;
  hence thesis;
end;
