reserve X for non empty TopSpace,
  A,B for Subset of X;
reserve Y1,Y2 for non empty SubSpace of X;

theorem Th6:
  Y1,Y2 constitute_a_decomposition implies Y1 is proper & Y2 is proper
proof
  reconsider A1 = the carrier of Y1, A2 = the carrier of Y2 as Subset of X by
TSEP_1:1;
  assume Y1,Y2 constitute_a_decomposition;
  then A1,A2 constitute_a_decomposition;
  then A1 is proper & A2 is proper by Th1;
  hence thesis by TEX_2:8;
end;
