let X, Y be set ; :: thesis: for R being Relation holds R | (X /\ Y) = (R | X) /\ (R | Y)
let R be Relation; :: thesis: R | (X /\ Y) = (R | X) /\ (R | Y)

now

let x, y be set ; :: thesis: ( [x,y] in R | (X /\ Y) iff [x,y] in (R | X) /\ (R | Y) )

A1: now

assume [x,y] in R | (X /\ Y) ; :: thesis: [x,y] in (R | X) /\ (R | Y)
then ( x in X /\ Y & [x,y] in R ) by Def11;
then ( x in X & x in Y & [x,y] in R ) by XBOOLE_0:def 4;
then ( [x,y] in R | X & [x,y] in R | Y ) by Def11;
hence [x,y] in (R | X) /\ (R | Y) by XBOOLE_0:def 4; :: thesis: verum

end;

now

assume [x,y] in (R | X) /\ (R | Y) ; :: thesis: [x,y] in R | (X /\ Y)
then A2: ( [x,y] in R | X & [x,y] in R | Y ) by XBOOLE_0:def 4;
then A3: ( x in X & [x,y] in R ) by Def11;
( x in Y & [x,y] in R ) by A2, Def11;
then ( x in X /\ Y & [x,y] in R ) by A3, XBOOLE_0:def 4;
hence [x,y] in R | (X /\ Y) by Def11; :: thesis: verum

end;

hence ( [x,y] in R | (X /\ Y) iff [x,y] in (R | X) /\ (R | Y) ) by A1; :: thesis: verum

end;

hence R | (X /\ Y) = (R | X) /\ (R | Y) by Def2; :: thesis: verum