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 Th15:
  for A1, A2, C1, C2 being Subset of X st A1,C1
  constitute_a_decomposition & A2,C2 constitute_a_decomposition holds A1,A2
  are_weakly_separated iff C1,C2 are_weakly_separated
proof
  let A1, A2, C1, C2 be Subset of X;
  assume A1,C1 constitute_a_decomposition & A2,C2 constitute_a_decomposition;
  then
A1: C1 = A1` & C2 = A2` by Th3;
  thus A1,A2 are_weakly_separated implies C1,C2 are_weakly_separated
  proof
    assume A1 \ A2,A2 \ A1 are_separated;
    then C2 \ C1, C2` \ C1` are_separated by A1,Th1;
    then C2 \ C1,C1 \ C2 are_separated by Th1;
    hence thesis;
  end;
  assume C1,C2 are_weakly_separated;
  then C1 \ C2,C2 \ C1 are_separated;
  then C2` \ C1`,C2 \ C1 are_separated by Th1;
  then A2 \ A1,A1 \ A2 are_separated by A1,Th1;
  hence thesis;
end;
