theorem
  (Y|`R)|X = Y|`(R|X)
proof
  let x,y;
A1: [x,y] in R & x in X iff [x,y] in R|X by Def9;
  [x,y] in (Y|`R) iff [x,y] in R & y in Y by Def10;
  hence thesis by A1,Def9,Def10;
end;
