let P, R be Relation; :: thesis: (P /\ R) ~ = (P ~ ) /\ (R ~ )
for x, y being set holds
( [x,y] in (P /\ R) ~ iff [x,y] in (P ~ ) /\ (R ~ ) )

proof

let x, y be set ; :: thesis: ( [x,y] in (P /\ R) ~ iff [x,y] in (P ~ ) /\ (R ~ ) )
( [x,y] in (P /\ R) ~ iff [y,x] in P /\ R ) by Def7;
then ( [x,y] in (P /\ R) ~ iff ( [y,x] in P & [y,x] in R ) ) by XBOOLE_0:def 4;
then ( [x,y] in (P /\ R) ~ iff ( [x,y] in P ~ & [x,y] in R ~ ) ) by Def7;
hence ( [x,y] in (P /\ R) ~ iff [x,y] in (P ~ ) /\ (R ~ ) ) by XBOOLE_0:def 4; :: thesis: verum

end;

hence (P /\ R) ~ = (P ~ ) /\ (R ~ ) by Def2; :: thesis: verum