let X0 be SubSpace of X; :: thesis: ( not X0 is proper implies X0 is everywhere_dense )
assume A1: not X0 is proper ; :: thesis: X0 is everywhere_dense

now :: thesis: for A being Subset of X st A = the carrier of X0 holds
A is everywhere_dense

let A be Subset of X; :: thesis: ( A = the carrier of X0 implies A is everywhere_dense )
assume A = the carrier of X0 ; :: thesis: A is everywhere_dense
then not A is proper by A1, TEX_2:8;
then A = [#] X by SUBSET_1:def 6;
hence A is everywhere_dense by TOPS_3:31; :: thesis: verum

end;

hence X0 is everywhere_dense ; :: thesis: verum