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;

theorem Th41:
  for A0 being non empty Subset of X st A0 is boundary closed ex
  X0 being closed strict non empty SubSpace of X st X0 is boundary & A0 = the
  carrier of X0
proof
  let A0 be non empty 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
A2: A0 is boundary closed;
  then reconsider Y0 = X0 as closed strict non empty SubSpace of X by A1,
TSEP_1:11;
  take Y0;
  thus thesis by A2,A1;
end;
