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
  (Cl A),(Int A`) constitute_a_decomposition &
  (Cl A`),(Int A) constitute_a_decomposition &
  (Int A),(Cl A`) constitute_a_decomposition &
  (Int A`),(Cl A) constitute_a_decomposition
proof
A1: A, A` constitute_a_decomposition by Th5;
  hence (Cl A),(Int A`) constitute_a_decomposition by Th9;
A2: A`,A constitute_a_decomposition by Th5;
  hence (Cl A`),(Int A) constitute_a_decomposition by Th9;
  thus (Int A),(Cl A`) constitute_a_decomposition by A2,Th9;
  thus thesis by A1,Th9;
end;
