reserve X for non empty 1-sorted;
reserve A, A1, A2, B1, B2 for Subset of X;
reserve X for non empty TopSpace;
reserve A, A1, A2, B1, B2 for Subset of X;

theorem Th9:
  A1,A2 constitute_a_decomposition implies (Cl A1),(Int A2)
  constitute_a_decomposition & (Int A1),(Cl A2) constitute_a_decomposition
proof
  assume
A1: A1,A2 constitute_a_decomposition;
  Cl A1 = (Int A1`)` by TDLAT_3:1;
  then Cl A1 = (Int A2)` by A1,Th3;
  hence (Cl A1),(Int A2) constitute_a_decomposition by Th4;
  Int A1 = (Cl A1`)` by TOPS_1:def 1;
  then Int A1 = (Cl A2)` by A1,Th3;
  hence (Int A1),(Cl A2) constitute_a_decomposition by Th4;
end;
