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 non-Kolmogorov_space, A0 being Subset of X st A0
is not T_0 ex X0 being strict non-Kolmogorov_subspace of X st A0 = the carrier
  of X0
proof
  let X be non empty non-Kolmogorov_space, A0 be Subset of X;
  assume
A1: A0 is not T_0;
  then A0 is non empty;
  then consider X0 being strict non empty SubSpace of X such that
A2: A0 = the carrier of X0 by TSEP_1:10;
  reconsider X0 as non T_0 strict SubSpace of X by A1,A2,Th13;
  take X0;
  thus thesis by A2;
end;
