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;
reserve X for non empty TopSpace,
  A1, A2 for Subset of X;

theorem
  for A1, A2, C1, C2 being Subset of X st A1,C1
constitute_a_decomposition & A2,C2 constitute_a_decomposition holds C1 \/ C2 =
  the carrier of X & C1,C2 are_weakly_separated implies A1,A2 are_separated
proof
  let A1, A2, C1, C2 be Subset of X;
  assume
A1: A1,C1 constitute_a_decomposition & A2,C2 constitute_a_decomposition;
  assume C1 \/ C2 = the carrier of X;
  then
A2: (C1 \/ C2)` = {}X by XBOOLE_1:37;
  A1 = C1` & A2 = C2` by A1,Th3;
  then A1 /\ A2 = {} by A2,XBOOLE_1:53;
  then
A3: A1 misses A2;
  assume C1,C2 are_weakly_separated;
  hence thesis by A1,A3,Th18;
end;
