reserve X for non empty TopSpace;
reserve Y for non empty TopStruct;
reserve x for Point of Y;
reserve Y for non empty TopStruct;
reserve X for non empty TopSpace;
reserve x,y for Point of X;
reserve A, B for Subset of X;
reserve P, Q for Subset of X;
reserve Y for non empty TopStruct;
reserve X for non empty TopSpace,
  Y0 for non empty SubSpace of X;

theorem
  for Y0 being anti-discrete SubSpace of X for X0 being open 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 open SubSpace of X;
  reconsider G = the carrier of X0 as Subset of X by TSEP_1:1;
A1: G is open by TSEP_1:16;
  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;
