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 Th167:
  dom(R \ (R|A)) = dom R \ A
proof
  R \ (R|A) = R|(dom R \ A) by Th166;
  hence dom(R \ (R|A)) = dom R /\ (dom R \ A) by Th55
      .= (dom R /\ dom R) \ A by XBOOLE_1:49
      .= dom R \ A;
end;
