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

theorem
  for X being non empty TopSpace, A0 being non empty Subset of X st A0
  is T_0 ex X0 being strict Kolmogorov_subspace of X st A0 = the carrier of X0
proof
  let X be non empty TopSpace, A0 be non empty Subset of X;
  consider X0 being strict non empty SubSpace of X such that
A1: A0 = the carrier of X0 by TSEP_1:10;
  assume A0 is T_0;
  then reconsider X0 as strict Kolmogorov_subspace of X by A1,Th13;
  take X0;
  thus thesis by A1;
end;
