let X be non empty non discrete TopSpace; :: thesis: for A0 being non empty proper Subset of X st A0 is dense holds
ex X0 being strict proper dense SubSpace of X st A0 = the carrier of X0
let A0 be non empty proper Subset of X; :: thesis: ( A0 is dense implies ex X0 being strict proper dense SubSpace of X st A0 = the carrier of X0 )
consider X0 being non empty strict SubSpace of X such that
A1: A0 = the carrier of X0 by TSEP_1:10;
assume A0 is dense ; :: thesis: ex X0 being strict proper dense SubSpace of X st A0 = the carrier of X0
then reconsider X0 = X0 as strict proper dense SubSpace of X by A1, Th9, TEX_2:8;
take X0 ; :: thesis: A0 = the carrier of X0
thus A0 = the carrier of X0 by A1; :: thesis: verum