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 holds X0 is
  boundary closed or ex X1 being everywhere_dense proper strict non empty
  SubSpace of X, X2 being boundary closed strict non empty SubSpace of X st X1
  meet X2 = the TopStruct of X0 & X1 union X2 = the TopStruct of X
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;
A1: X is SubSpace of X by TSEP_1:2;
  D is nowhere_dense by Th35;
  then consider C, B being Subset of X such that
A2: C is closed boundary and
A3: B is everywhere_dense and
A4: C /\ B = D and
A5: C \/ B = the carrier of X by TOPS_3:51;
  B <> {} by A4;
  then consider
  X1 being everywhere_dense strict non empty SubSpace of X such that
A6: B = the carrier of X1 by A3,Th17;
  assume
A7: X0 is non boundary or X0 is non closed;
  now
    assume B is non proper;
    then B = the carrier of X by SUBSET_1:def 6;
    then D = C by A4,XBOOLE_1:28;
    hence contradiction by A7,A2,TSEP_1:11;
  end;
  then reconsider
  X1 as everywhere_dense proper strict non empty SubSpace of X by A6,TEX_2:8;
  C <> {} by A4;
  then consider
  X2 being boundary closed strict non empty SubSpace of X such that
A8: C = the carrier of X2 by A2,Th67;
  take X1, X2;
  C meets B by A4,XBOOLE_0:def 7;
  then X1 meets X2 by A8,A6,TSEP_1:def 3;
  then the carrier of X1 meet X2 = D by A4,A8,A6,TSEP_1:def 4;
  hence X1 meet X2 = the TopStruct of X0 by TSEP_1:5;
  the carrier of X1 union X2 = the carrier of X by A5,A8,A6,TSEP_1:def 2;
  hence thesis by A1,TSEP_1:5;
end;
