theorem
  for A0 being non empty proper Subset of X st A0 is dense ex X0 being
  dense proper strict SubSpace of X st A0 = the carrier of X0
proof
  let A0 be non empty proper 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 dense;
  then reconsider X0 as dense proper strict SubSpace of X by A1,Th9,TEX_2:8;
  take X0;
  thus thesis by A1;
end;
