theorem
  for Y0 being anti-discrete SubSpace of X for X0 being closed SubSpace
  of X holds Y0 misses X0 or Y0 is SubSpace of X0
proof
  let Y0 be anti-discrete SubSpace of X;
  reconsider A = the carrier of Y0 as Subset of X by TSEP_1:1;
  let X0 be closed SubSpace of X;
  reconsider G = the carrier of X0 as Subset of X by TSEP_1:1;
A1: G is closed by TSEP_1:11;
  A is anti-discrete by Th66;
  then A misses G or A c= G by A1;
  hence thesis by TSEP_1:4,def 3;
end;
