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;
reserve X for non empty TopSpace;

theorem Th37:
  for X1,X2 being SubSpace of X st X1,X2
constitute_a_decomposition holds X1 is everywhere_dense iff X2 is nowhere_dense
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 everywhere_dense implies X2 is nowhere_dense
  proof
    assume
A2: for A1 being Subset of X st A1 = the carrier of X1 holds A1 is
    everywhere_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 everywhere_dense by A2;
      hence A2 is nowhere_dense by A3,Th4;
    end;
    hence thesis;
  end;
  assume
A4: for A2 being Subset of X st A2 = the carrier of X2 holds A2 is
  nowhere_dense;
  now
    let A1 be Subset of X;
    assume A1 = the carrier of X1;
    then
A5: A1,A2 constitute_a_decomposition by A1;
    A2 is nowhere_dense by A4;
    hence A1 is everywhere_dense by A5,Th4;
  end;
  hence thesis;
end;
