let X be non empty TopSpace; :: thesis: for A0 being non empty Subset of X st A0 is discrete holds
ex X0 being non empty strict discrete SubSpace of X st A0 = the carrier of X0
let A0 be non empty Subset of X; :: thesis: ( A0 is discrete implies ex X0 being non empty strict discrete 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 discrete ; :: thesis: ex X0 being non empty strict discrete SubSpace of X st A0 = the carrier of X0
then reconsider X0 = X0 as non empty strict discrete SubSpace of X by A1, Th20;
take X0 ; :: thesis: A0 = the carrier of X0
thus A0 = the carrier of X0 by A1; :: thesis: verum