reserve A,X,X1,X2,Y,Y1,Y2 for set, a,b,c,d,x,y,z for object;
reserve P,P1,P2,Q,R,S for Relation;

theorem
  dom (R | (dom R \ X)) = dom R \ X
proof
  thus dom (R | (dom R \ X)) = dom R /\ (dom R \ X) by Th55
    .= (dom R /\ dom R) \ X by XBOOLE_1:49
    .= dom R \ X;
end;
