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 everywhere_dense implies ex C,B being Subset of X st C is open &
  C is dense & B is closed & B is boundary & C \/ (D /\ B) = D & C misses B & C
  \/ B = the carrier of X
proof
  assume D is everywhere_dense;
  then consider C being Subset of X such that
A1: C c= D and
A2: C is open & C is dense by Th41;
  take C;
  take B = C`;
  thus C is open & C is dense & B is closed & B is boundary by A2,Th18;
  thus C \/ (D /\ B) = (C \/ D) /\ (C \/ C`) by XBOOLE_1:24
    .= (C \/ D) /\ [#]X by PRE_TOPC:2
    .= C \/ D by XBOOLE_1:28
    .= D by A1,XBOOLE_1:12;
  C misses B by XBOOLE_1:79;
  hence C /\ B = {};
  C \/ B = [#]X by PRE_TOPC:2;
  hence thesis;
end;
