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;

theorem Th11:
  for A1, A2 being Subset of X st A1,A2 constitute_a_decomposition
  holds (A1 is open iff A2 is closed)
proof
  let A1, A2 be Subset of X;
  assume
A1: A1,A2 constitute_a_decomposition;
  then A2 = A1` by Th3;
  hence A1 is open implies A2 is closed by TOPS_1:4;
  assume
A2: A2 is closed;
  A1 = A2` by A1,Th3;
  hence thesis by A2,TOPS_1:3;
end;
