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 nowhere_dense non empty SubSpace of X ex X1 being dense
open proper strict non empty SubSpace of X, X2 being boundary closed strict non
empty SubSpace of X st X1,X2 constitute_a_decomposition & X0 is SubSpace of X2
proof
  let X0 be nowhere_dense non empty SubSpace of X;
  reconsider D = the carrier of X0 as non empty Subset of X by TSEP_1:1;
  D is nowhere_dense by Th35;
  then consider C, B being Subset of X such that
A1: C is closed boundary and
A2: B is open dense and
A3: C /\ (D \/ B) = D and
A4: C misses B and
A5: C \/ B = the carrier of X by TOPS_3:52;
  B <> {} by A2,TOPS_3:17;
  then consider X1 being dense open strict non empty SubSpace of X such that
A6: B = the carrier of X1 by A2,Th23;
A7: C <> {} by A3;
  then consider
  X2 being boundary closed strict non empty SubSpace of X such that
A8: C = the carrier of X2 by A1,Th67;
A9: C /\ B = {} by A4,XBOOLE_0:def 7;
  now
    assume B is non proper;
    then B = the carrier of X by SUBSET_1:def 6;
    hence contradiction by A7,A9,XBOOLE_1:28;
  end;
  then reconsider
  X1 as dense open proper strict non empty SubSpace of X by A6,TEX_2:8;
  take X1, X2;
  for P,Q be Subset of X st P = the carrier of X1 & Q = the carrier of X2
  holds P,Q constitute_a_decomposition by A4,A5,A6,A8;
  hence X1,X2 constitute_a_decomposition;
  D c= C by A3,XBOOLE_1:17;
  hence thesis by A8,TSEP_1:4;
end;
