let X be non empty TopSpace; :: thesis: for A0 being non empty Subset of X st A0 is closed holds
ex X0 being non empty strict closed SubSpace of X st A0 = the carrier of X0
let A0 be non empty Subset of X; :: thesis: ( A0 is closed implies ex X0 being non empty strict closed SubSpace of X st A0 = the carrier of X0 )
assume A1: A0 is closed ; :: thesis: ex X0 being non empty strict closed SubSpace of X st A0 = the carrier of X0
consider X0 being non empty strict SubSpace of X such that
A2: A0 = the carrier of X0 by Th10;
reconsider Y0 = X0 as non empty strict closed SubSpace of X by A1, A2, Th11;
take Y0 ; :: thesis: A0 = the carrier of Y0
thus A0 = the carrier of Y0 by A2; :: thesis: verum