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;

theorem Th45:
  A is everywhere_dense & B is dense implies A /\ B is dense
proof
  assume A is everywhere_dense;
  then
A1: A` is nowhere_dense by Th39;
  assume B is dense;
  then B` is boundary by Th18;
  then A` \/ B` = (A /\ B)` & A` \/ B` is boundary by A1,Th30,XBOOLE_1:54;
  hence thesis by Th18;
end;
