theorem
  (X \/ Y)|`R = (X|`R) \/ (Y|`R)
proof
  let x,y;
A1: y in X \/ Y iff y in X or y in Y by XBOOLE_0:def 3;
A2: [x,y] in (X|`R) \/ (Y|`R) iff [x,y] in X|`R or [x,y] in Y|`R
                 by XBOOLE_0:def 3;
  [x,y] in (X \/ Y)|`R iff y in X \/ Y & [x,y] in R by Def10;
  hence thesis by A1,A2,Def10;
end;
