let X0 be non empty SubSpace of X; :: thesis: ( X0 is closed & X0 is anti-discrete implies X0 is maximal_anti-discrete )
reconsider A = the carrier of X0 as Subset of X by TSEP_1:1;
assume X0 is closed ; :: thesis: ( not X0 is anti-discrete or X0 is maximal_anti-discrete )
then A2: A is closed by TSEP_1:11;
assume X0 is anti-discrete ; :: thesis: X0 is maximal_anti-discrete
then A is anti-discrete by Th66;
then A is maximal_anti-discrete by A2, Th17;
hence X0 is maximal_anti-discrete by Th72; :: thesis: verum