theorem
  f|{x} is constant
proof
  now
    per cases;
    suppose
      {x} /\ dom f = {};
      then {x} misses dom f;
      hence thesis by Th39;
    end;
    suppose
A1:   {x} /\ dom f <> {};
      set y = the Element of {x} /\ dom f;
      y in {x} & y in dom f by A1,XBOOLE_0:def 4;
      then reconsider x1=x as Element of C by TARSKI:def 1;
      now
        take d = f/.x1;
        let c;
        assume c in {x} /\ dom f;
        then c in {x} by XBOOLE_0:def 4;
        hence f/.c = f/.x1 by TARSKI:def 1;
      end;
      hence thesis by Th35;
    end;
  end;
  hence thesis;
end;
