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 A2: [x,y] in (R | X) /\ (R | Y) ; :: thesis: [x,y] in R | (X /\ Y)
then [x,y] in R | Y by XBOOLE_0:def 4;
then A3: x in Y by Def11;
A4: [x,y] in R | X by A2, XBOOLE_0:def 4;
then x in X by Def11;
then A5: x in X /\ Y by A3, XBOOLE_0:def 4;
[x,y] in R by A4, Def11;
hence [x,y] in R | (X /\ Y) by A5, Def11; :: thesis: verum

end;

now

assume A6: [x,y] in R | (X /\ Y) ; :: thesis: [x,y] in (R | X) /\ (R | Y)
then A7: x in X /\ Y by Def11;
A8: [x,y] in R by A6, Def11;
x in Y by A7, XBOOLE_0:def 4;
then A9: [x,y] in R | Y by A8, Def11;
x in X by A7, XBOOLE_0:def 4;
then [x,y] in R | X by A8, Def11;
hence [x,y] in (R | X) /\ (R | Y) by A9, XBOOLE_0:def 4; :: 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