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;

theorem
  D is nowhere_dense implies ex C,B being Subset of X st C is closed & C
is boundary & B is open & B is dense & C /\ (D \/ B) = D & C misses B & C \/ B
  = the carrier of X
proof
  assume D is nowhere_dense;
  then consider C being Subset of X such that
A1: D c= C and
A2: C is closed & C is boundary by Th27;
  take C;
  take B = C`;
  thus C is closed & C is boundary & B is open & B is dense by A2;
A3: C misses C` by XBOOLE_1:79;
  thus C /\ (D \/ B) = (C /\ D) \/ (C /\ C`) by XBOOLE_1:23
    .= (C /\ D) \/ {}X by A3
    .= D by A1,XBOOLE_1:28;
  C misses B by XBOOLE_1:79;
  hence C /\ B = {};
  C \/ B = [#]X by PRE_TOPC:2;
  hence thesis;
end;
