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;
reserve X1, X2 for non empty SubSpace of X;
reserve X for non discrete non empty TopSpace;
reserve X for non almost_discrete non empty TopSpace;

theorem
  for X0 being everywhere_dense non empty SubSpace of X holds X0 is
dense open or ex X1 being dense open proper strict non empty SubSpace of X, X2
  being nowhere_dense strict non empty SubSpace of X st X1 misses X2 & X1 union
  X2 = the TopStruct of X0
proof
  let X0 be everywhere_dense non empty SubSpace of X;
  reconsider D = the carrier of X0 as non empty Subset of X by TSEP_1:1;
  D is everywhere_dense by Th16;
  then consider C, B being Subset of X such that
A1: C is open dense and
A2: B is nowhere_dense and
A3: C \/ B = D and
A4: C misses B by TOPS_3:49;
  C <> {} by A1,TOPS_3:17;
  then consider X1 being dense open strict non empty SubSpace of X such that
A5: C = the carrier of X1 by A1,Th23;
  assume
A6: X0 is non dense or X0 is non open;
  now
    assume C is non proper;
    then
A7: C = the carrier of X by SUBSET_1:def 6;
    C c= D by A3,XBOOLE_1:7;
    then D = [#]X by A7,XBOOLE_0:def 10;
    then D is dense open;
    hence contradiction by A6,TSEP_1:16;
  end;
  then reconsider
  X1 as dense open proper strict non empty SubSpace of X by A5,TEX_2:8;
  now
    per cases by A6;
    suppose
A8:   X0 is non dense;
      assume B = {};
      thus contradiction by A8;
    end;
    suppose
A9:   X0 is non open;
      assume B = {};
      hence contradiction by A1,A3,A9,TSEP_1:16;
    end;
  end;
  then consider
  X2 being nowhere_dense strict non empty SubSpace of X such that
A10: B = the carrier of X2 by A2,Th62;
  take X1, X2;
  thus X1 misses X2 by A4,A5,A10,TSEP_1:def 3;
  the carrier of X1 union X2 = the carrier of X0 by A3,A5,A10,TSEP_1:def 2;
  hence thesis by TSEP_1:5;
end;
