reserve X for TopStruct,
  A for Subset of X;
reserve X for TopSpace,
  A,B for Subset of X;
reserve X for non empty TopSpace,
  A for Subset of X;
reserve X for TopSpace,
  A,B for Subset of X;
reserve X for non empty TopSpace,
  A, B for Subset of X;
reserve D for Subset of X;
reserve Y0 for SubSpace of X;
reserve X0 for SubSpace of X;
reserve X0 for non empty SubSpace of X;

theorem Th68:
  for A being Subset of X, B being Subset of X0 st A c= B holds B
  is nowhere_dense implies A is nowhere_dense
proof
  let A be Subset of X, B be Subset of X0;
  reconsider D = the carrier of X0 as Subset of X by TSEP_1:1;
  reconsider G = (Int Cl A) /\ [#]X0 as Subset of X0;
  assume
A1: A c= B;
  then reconsider C = A as Subset of X0 by XBOOLE_1:1;
  assume B is nowhere_dense;
  then C is nowhere_dense by A1,Th26;
  then
A2: G is open & Int Cl C = {} by TOPS_2:24;
  (Int Cl A) /\ [#]X0 c= (Cl A) /\ [#]X0 by TOPS_1:16,XBOOLE_1:26;
  then
A3: (Int Cl A) /\ [#]X0 c= Cl C by PRE_TOPC:17;
  now
    assume Int Cl A <> {};
    then A meets Int Cl A by Th7;
    then
A4: A /\ Int Cl A <> {};
    C c= D;
    then (Int Cl A) /\ D <> {} by A4,XBOOLE_1:3,26;
    hence contradiction by A3,A2,TOPS_1:24,XBOOLE_1:3;
  end;
  hence thesis;
end;
