let X be non empty TopSpace; :: thesis: for A, B being Subset of X st A is nowhere_dense & B is boundary holds
A \/ B is boundary
let A, B be Subset of X; :: thesis: ( A is nowhere_dense & B is boundary implies A \/ B is boundary )
assume A is nowhere_dense ; :: thesis: ( not B is boundary or A \/ B is boundary )
then A1: Cl A is boundary ;
assume B is boundary ; :: thesis: A \/ B is boundary
then ( A c= Cl A & B \/ (Cl A) is boundary ) by A1, PRE_TOPC:18, TOPS_1:49;
hence A \/ B is boundary by Th11, XBOOLE_1:9; :: thesis: verum