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

theorem Th1:
  A,B constitute_a_decomposition implies (A is non empty iff B is proper)
proof
  assume
A1: A,B constitute_a_decomposition;
  then
A2: B = A` by TSEP_2:3;
  thus A is non empty implies B is proper
  by A2,TOPS_3:1,SUBSET_1:def 6;
  assume
A3: B is proper;
  A = B` by A1,TSEP_2:3;
  hence thesis by A3;
end;
