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;

theorem Th67:
  for Y0 being SubSpace of Y st Y0 is TopSpace-like for A being
  Subset of Y st A = the carrier of Y0 holds A is anti-discrete implies Y0 is
  anti-discrete
proof
  let Y0 be SubSpace of Y;
  assume
A1: Y0 is TopSpace-like;
  then
A2: the carrier of Y0 in the topology of Y0 by PRE_TOPC:def 1;
  let A be Subset of Y;
  assume
A3: A = the carrier of Y0;
  assume
A4: A is anti-discrete;
  now
    let D be object;
    assume
A5: D in the topology of Y0;
    then reconsider C = D as Subset of Y0;
    consider E being Subset of Y such that
A6: E in the topology of Y and
A7: C = E /\ [#]Y0 by A5,PRE_TOPC:def 4;
    reconsider E as Subset of Y;
A8: E is open by A6,PRE_TOPC:def 2;
    now
      per cases by A4,A8;
      suppose
        A misses E;
        hence C = {} or C = A by A3,A7;
      end;
      suppose
        A c= E;
        hence C = {} or C = A by A3,A7,XBOOLE_1:28;
      end;
    end;
    hence D in {{},the carrier of Y0} by A3,TARSKI:def 2;
  end;
  then
A9: the topology of Y0 c= {{},the carrier of Y0};
  {} in the topology of Y0 by A1,PRE_TOPC:1;
  then {{},the carrier of Y0} c= the topology of Y0 by A2,ZFMISC_1:32;
  then the topology of Y0 = {{},the carrier of Y0} by A9;
  hence thesis by TDLAT_3:def 2;
end;
