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;
reserve X0 for non empty SubSpace of X,
  B1, B2 for Subset of X0;
reserve X0, X1, X2, Y1, Y2 for non empty SubSpace of X;

theorem Th33:
  X1,X2 constitute_a_decomposition implies (X1 is open iff X2 is closed)
proof
  reconsider A1 = the carrier of X1, A2 = the carrier of X2 as Subset of X by
TSEP_1:1;
  assume
A1: for A1, A2 being Subset of X st A1 = the carrier of X1 & A2 = the
  carrier of X2 holds A1,A2 constitute_a_decomposition;
  thus X1 is open implies X2 is closed
  proof
    assume
A2: for A1 being Subset of X st A1 = the carrier of X1 holds A1 is open;
    now
      let A2 be Subset of X;
      assume A2 = the carrier of X2;
      then
A3:   A1,A2 constitute_a_decomposition by A1;
      A1 is open by A2;
      hence A2 is closed by A3,Th11;
    end;
    hence thesis;
  end;
  assume
A4: for A2 being Subset of X st A2 = the carrier of X2 holds A2 is closed;
  now
    let A1 be Subset of X;
    assume A1 = the carrier of X1;
    then
A5: A1,A2 constitute_a_decomposition by A1;
    A2 is closed by A4;
    hence A1 is open by A5,Th11;
  end;
  hence thesis;
end;
