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 Th18:
  for A1, A2, C1, C2 being Subset of X st A1,C1
  constitute_a_decomposition & A2,C2 constitute_a_decomposition holds A1 misses
  A2 & 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
A2: A1 /\ A2 = {};
  assume C1,C2 are_weakly_separated;
  then
A3: A1,A2 are_weakly_separated by A1,Th15;
  A1 misses A2 by A2;
  hence thesis by A3,TSEP_1:46;
end;
