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 Th72:
  for Y0 being non empty SubSpace of X, A being Subset of X st A =
  the carrier of Y0 holds Y0 is maximal_anti-discrete iff A is
  maximal_anti-discrete
proof
  let Y0 be non empty SubSpace of X, A be Subset of X;
  assume
A1: A = the carrier of Y0;
  thus Y0 is maximal_anti-discrete implies A is maximal_anti-discrete
  proof
    assume
A2: Y0 is maximal_anti-discrete;
A3: now
      let D be Subset of X;
      assume
A4:   D is anti-discrete;
      assume
A5:   A c= D;
      then D <> {} by A1;
      then consider X0 being strict non empty SubSpace of X such that
A6:   D = the carrier of X0 by TSEP_1:10;
      X0 is anti-discrete by A4,A6,Th67;
      hence A = D by A1,A2,A5,A6;
    end;
    A is anti-discrete by A1,A2,Th66;
    hence thesis by A3;
  end;
  assume
A7: A is maximal_anti-discrete;
A8: now
    let X0 be SubSpace of X;
    reconsider D = the carrier of X0 as Subset of X by TSEP_1:1;
    assume X0 is anti-discrete;
    then
A9: D is anti-discrete by Th66;
    assume the carrier of Y0 c= the carrier of X0;
    hence the carrier of Y0 = the carrier of X0 by A1,A7,A9;
  end;
  A is anti-discrete by A7;
  then Y0 is anti-discrete by A1,Th67;
  hence thesis by A8;
end;
