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 proper SubSpace of X ex X1 being dense
open proper strict SubSpace of X, X2 being boundary closed strict SubSpace of X
  st X1,X2 constitute_a_decomposition & X1 is SubSpace of X0
proof
  let X0 be everywhere_dense proper SubSpace of X;
  reconsider D = the carrier of X0 as 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 closed boundary and
A3: C \/ (D /\ B) = D and
A4: C misses B and
A5: C \/ B = the carrier of X by TOPS_3:50;
  C <> {} by A1,TOPS_3:17;
  then consider X1 being dense open strict non empty SubSpace of X such that
A6: C = the carrier of X1 by A1,Th23;
A7: now
    assume C is non proper;
    then C = the carrier of X by SUBSET_1:def 6;
    then the carrier of X c= D by A3,XBOOLE_1:7;
    then D = the carrier of X by XBOOLE_0:def 10;
    then D is non proper by SUBSET_1:def 6;
    hence contradiction by TEX_2:8;
  end;
  then B is non empty by A5,SUBSET_1:def 6;
  then consider
  X2 being boundary closed strict non empty SubSpace of X such that
A8: B = the carrier of X2 by A2,Th67;
  reconsider X1 as dense open proper strict SubSpace of X by A6,A7,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;
  C c= D by A3,XBOOLE_1:7;
  hence thesis by A6,TSEP_1:4;
end;
