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 A1,A2
  are_separated implies C1,C2 are_weakly_separated
proof
  let A1, A2, C1, C2 be Subset of X;
  assume
A1: A1,C1 constitute_a_decomposition & A2,C2 constitute_a_decomposition;
  assume A1,A2 are_separated;
  then A1,A2 are_weakly_separated by TSEP_1:46;
  hence thesis by A1,Th15;
end;
