let X0 be SubSpace of X; :: thesis: ( X0 is nowhere_dense implies X0 is boundary )
assume A1: X0 is nowhere_dense ; :: thesis: X0 is boundary

now

let A be Subset of X; :: thesis: ( A = the carrier of X0 implies A is boundary )
assume A = the carrier of X0 ; :: thesis: A is boundary
then A is nowhere_dense by A1, Def4;
hence A is boundary ; :: thesis: verum

end;

hence X0 is boundary by Def3; :: thesis: verum