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
  for X,Y being set, P,R being Relation st X misses Y holds P|X misses R|Y
proof
  let X,Y be set, P,R be Relation;
  assume
A1: X misses Y;
A2: dom (P|X) = dom P /\ X by Th55;
  dom (R|Y) = dom R /\ Y by Th55;
  then dom (P|X) /\ dom (R|Y) = dom P /\ (X /\ (dom R /\ Y)) by A2,XBOOLE_1:16
    .= dom P /\ (X /\ Y /\ dom R) by XBOOLE_1:16
    .= {} by A1;
  then dom (P|X) misses dom (R|Y);
  hence thesis by Th169;
end;
