reserve x,y,X,Y for set;
reserve C,D,E for non empty set;
reserve SC for Subset of C;
reserve SD for Subset of D;
reserve SE for Subset of E;
reserve c,c1,c2 for Element of C;
reserve d,d1,d2 for Element of D;
reserve e for Element of E;
reserve f,f1,g for PartFunc of C,D;
reserve t for PartFunc of D,C;
reserve s for PartFunc of D,E;
reserve h for PartFunc of C,E;
reserve F for PartFunc of D,D;

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;
