reserve X for non empty TopSpace,
  A,B for Subset of X;
reserve Y1,Y2 for non empty SubSpace of X;
reserve X1, X2 for non empty SubSpace of X;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;
reserve X for non discrete non empty TopSpace;
reserve X for non almost_discrete non empty TopSpace;

theorem
  for A0 being non empty proper Subset of X st A0 is everywhere_dense ex
X0 being everywhere_dense proper strict SubSpace of X st A0 = the carrier of X0
proof
  let A0 be non empty proper Subset of X;
  assume A0 is everywhere_dense;
  then consider
  X0 being everywhere_dense strict non empty SubSpace of X such that
A1: A0 = the carrier of X0 by Th17;
  reconsider X0 as everywhere_dense proper strict SubSpace of X by A1,TEX_2:8;
  take X0;
  thus thesis by A1;
end;
