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 X0 being open SubSpace of X holds Y0 misses X0 or Y0 is SubSpace
  of X0) implies Y0 is anti-discrete
proof
  reconsider A = the carrier of Y0 as Subset of X by TSEP_1:1;
  assume
A1: for X0 being open SubSpace of X holds Y0 misses X0 or Y0 is SubSpace of X0;
  now
    let G be Subset of X;
    assume
A2: G is open;
    now
      per cases;
      suppose
        G is empty;
        hence A misses G or A c= G;
      end;
      suppose
        G is non empty;
        then consider X0 being strict open non empty SubSpace of X such that
A3:     G = the carrier of X0 by A2,TSEP_1:20;
        Y0 misses X0 or Y0 is SubSpace of X0 by A1;
        hence A misses G or A c= G by A3,TSEP_1:4,def 3;
      end;
    end;
    hence A misses G or A c= G;
  end;
  then A is anti-discrete;
  hence thesis by Th67;
end;
