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 or x in Y ) & [x,y] in R ) by XBOOLE_0:def 3;
then ( [x,y] in R | X or [x,y] in R | Y ) by Def11;
hence [x,y] in (R | X) \/ (R | Y) by XBOOLE_0:def 3; :: thesis: verum

end;

now

assume A2: [x,y] in (R | X) \/ (R | Y) ; :: thesis: [x,y] in R | (X \/ Y)

A3: now

assume [x,y] in R | X ; :: thesis: [x,y] in R | (X \/ Y)
then ( x in X & [x,y] in R ) by Def11;
then ( x in X \/ Y & [x,y] in R ) by XBOOLE_0:def 3;
hence [x,y] in R | (X \/ Y) by Def11; :: thesis: verum

end;

now

assume [x,y] in R | Y ; :: thesis: [x,y] in R | (X \/ Y)
then ( x in Y & [x,y] in R ) by Def11;
then ( x in X \/ Y & [x,y] in R ) by XBOOLE_0:def 3;
hence [x,y] in R | (X \/ Y) by Def11; :: thesis: verum

end;

hence [x,y] in R | (X \/ Y) by A2, A3, XBOOLE_0:def 3; :: 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