reserve X for non empty TopSpace,
  A,B for Subset of X;
reserve Y1,Y2 for non empty SubSpace of X;

theorem Th17:
  for A0 being non empty Subset of X st A0 is everywhere_dense ex
X0 being everywhere_dense strict non empty SubSpace of X st A0 = the carrier of
  X0
proof
  let 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 everywhere_dense;
  then reconsider
  Y0 = X0 as everywhere_dense strict non empty SubSpace of X by A1,Th16;
  take Y0;
  thus thesis by A1;
end;
