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 nowhere_dense & C \/ B = D & C misses B
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 = D \ C;
  thus C is open & C is dense by A2;
  C is everywhere_dense by A2,Th36;
  then C` is nowhere_dense by Th39;
  hence B is nowhere_dense by Th26,XBOOLE_1:33;
  thus thesis by A1,XBOOLE_1:45,79;
end;
