theorem
  (X /\ Y)|`R = X|`R /\ Y|`R
proof
  let x,y;
A1: y in X /\ Y iff y in X & y in Y by XBOOLE_0:def 4;
A2: [x,y] in X|`R /\ Y|`R iff [x,y] in X|`R & [x,y] in Y|`R by XBOOLE_0:def 4;
  [x,y] in (X /\ Y)|`R iff y in X /\ Y & [x,y] in R by Def10;
  hence thesis by A1,A2,Def10;
end;
