reserve X for non empty TopSpace,
  A,B for Subset of X;
reserve Y1,Y2 for non empty SubSpace of X;
reserve X1, X2 for non empty SubSpace of X;

theorem Th31:
  for X1,X2 being SubSpace of X st X1,X2
  constitute_a_decomposition holds X1 is dense iff X2 is boundary
proof
  let X1,X2 be SubSpace of X;
  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 dense implies X2 is boundary
  proof
    assume
A2: for A1 being Subset of X st A1 = the carrier of X1 holds A1 is dense;
    now
      let A2 be Subset of X;
      assume A2 = the carrier of X2;
      then
A3:   A1,A2 constitute_a_decomposition by A1;
      A1 is dense by A2;
      hence A2 is boundary by A3,Th2;
    end;
    hence thesis;
  end;
  assume
A4: for A2 being Subset of X st A2 = the carrier of X2 holds A2 is boundary;
  now
    let A1 be Subset of X;
    assume A1 = the carrier of X1;
    then
A5: A1,A2 constitute_a_decomposition by A1;
    A2 is boundary by A4;
    hence A1 is dense by A5,Th2;
  end;
  hence thesis;
end;
