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

theorem Th4:
  A,B constitute_a_decomposition implies (A is everywhere_dense iff
  B is nowhere_dense)
proof
  assume
A1: A,B constitute_a_decomposition;
  then B = A` by TSEP_2:3;
  hence A is everywhere_dense implies B is nowhere_dense by TOPS_3:39;
  assume
A2: B is nowhere_dense;
  A = B` by A1,TSEP_2:3;
  hence thesis by A2,TOPS_3:40;
end;
