let X be non empty TopSpace; :: thesis: for A, B being Subset of X st A is everywhere_dense & B is dense holds
A /\ B is dense
let A, B be Subset of X; :: thesis: ( A is everywhere_dense & B is dense implies A /\ B is dense )
assume A is everywhere_dense ; :: thesis: ( not B is dense or A /\ B is dense )
then A1: A ` is nowhere_dense by Th39;
assume B is dense ; :: thesis: A /\ B is dense
then B ` is boundary by Th18;
then ( (A `) \/ (B `) = (A /\ B) ` & (A `) \/ (B `) is boundary ) by A1, Th30, XBOOLE_1:54;
hence A /\ B is dense by Th18; :: thesis: verum