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
  A is everywhere_dense & B is boundary implies A \ B is dense
proof
  assume
A1: A is everywhere_dense;
A2: A \ B = A /\ B` by SUBSET_1:13;
  assume B is boundary;
  then B` is dense by TOPS_1:def 4;
  hence thesis by A1,A2,Th45;
end;
