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

theorem Th13:
  A1,A2 constitute_a_decomposition & B1,B2
  constitute_a_decomposition implies (A1 /\ B1),(A2 \/ B2)
  constitute_a_decomposition
proof
  assume
A1: A1,A2 constitute_a_decomposition;
  then A1 misses A2;
  then
A2: A1 /\ A2 = {};
  assume
A3: B1,B2 constitute_a_decomposition;
  then
A4: B1 \/ B2 = [#]X;
  B1 misses B2 by A3;
  then
A5: B1 /\ B2 = {}X;
  (A1 /\ B1) /\ (A2 \/ B2) =((B1 /\ A1) /\ A2) \/ ((A1 /\ B1) /\ B2) by
XBOOLE_1:23
    .= (B1 /\ (A1 /\ A2)) \/ ((A1 /\ B1) /\ B2) by XBOOLE_1:16
    .= (B1 /\ (A1 /\ A2)) \/ (A1 /\ (B1 /\ B2)) by XBOOLE_1:16
    .= {}X by A5,A2;
  then
A6: (A1 /\ B1) misses (A2 \/ B2);
  (A1 /\ B1) \/ (A2 \/ B2) = (A1 \/ (A2 \/ B2)) /\ (B1 \/ (A2 \/ B2)) by
XBOOLE_1:24
    .= ((A1 \/ A2) \/ B2) /\ (B1 \/ (B2 \/ A2)) by XBOOLE_1:4
    .= ((A1 \/ A2) \/ B2) /\ ((B1 \/ B2) \/ A2) by XBOOLE_1:4
    .= ([#]X \/ B2) /\ ([#]X \/ A2) by A1,A4
    .= ([#]X \/ B2) /\ [#]X by XBOOLE_1:12
    .= [#]X /\ [#]X by XBOOLE_1:12
    .= [#]X;
  hence thesis by A6;
end;
