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

theorem
  for X being non trivial TopSpace, D being non empty proper
  Subset of X ex Y0 being proper strict non empty SubSpace of X st D = the
  carrier of Y0
proof
  let X be non trivial TopSpace, D be non empty proper Subset of X;
  consider Y0 being strict non empty SubSpace of X such that
A1: D = the carrier of Y0 by TSEP_1:10;
  reconsider Y0 as proper strict non empty SubSpace of X by A1,TEX_2:8;
  take Y0;
  thus thesis by A1;
end;
