theorem
  for A0 being non empty Subset of X st A0 is boundary ex X0 being
  boundary strict SubSpace of X st A0 = the carrier of X0
proof
  let A0 be non empty Subset of X;
  assume A0 is boundary;
  then
  ex X0 being strict SubSpace of X st X0 is boundary & A0 = the carrier of
  X0 by Th30;
  hence thesis;
end;
