theorem
  for A0 being non empty Subset of X st A0 is nowhere_dense ex X0 being
  strict SubSpace of X st X0 is nowhere_dense & A0 = the carrier of X0
proof
  let A0 be non empty Subset of X;
  assume
A1: A0 is nowhere_dense;
  consider X0 being strict non empty SubSpace of X such that
A2: A0 = the carrier of X0 by TSEP_1:10;
  take X0;
  thus thesis by A1,A2;
end;
