reserve Y for TopStruct;
reserve X for non empty TopSpace;
reserve X for almost_discrete non empty TopSpace;

theorem Th61:
  for Y0 being discrete non empty SubSpace of X ex X0 being strict
  non empty SubSpace of X st Y0 is SubSpace of X0 & X0 is maximal_discrete
proof
  let Y0 be discrete non empty SubSpace of X;
  reconsider A = the carrier of Y0 as Subset of X by TSEP_1:1;
  A is discrete by Th20;
  then consider M being Subset of X such that
A1: A c= M and
A2: M is maximal_discrete by Th59;
  M is non empty by A2,Th40;
  then consider X0 being strict non empty SubSpace of X such that
A3: X0 is maximal_discrete and
A4: M = the carrier of X0 by A2,Th47;
  take X0;
  thus thesis by A1,A3,A4,TSEP_1:4;
end;
