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

theorem Th2:
  A,B constitute_a_decomposition implies (A is dense iff B is boundary)
proof
  assume
A1: A,B constitute_a_decomposition;
  then B = A` by TSEP_2:3;
  hence A is dense implies B is boundary by TOPS_3:18;
  assume
A2: B is boundary;
  A = B` by A1,TSEP_2:3;
  hence thesis by A2,TOPS_1:def 4;
end;
