
theorem Th24:
  for S, T being non empty TopSpace, a being Point of S, b being
  Point of T, A being Subset of S, B being Subset of T st a = b & A = B & the
  TopStruct of S = the TopStruct of T & a is_dense_point_of A holds b
  is_dense_point_of B
proof
  let S, T be non empty TopSpace, a be Point of S, b be Point of T, A be
  Subset of S, B be Subset of T;
  assume a = b & A = B & the TopStruct of S = the TopStruct of T & a in A & A
  c= Cl{a};
  hence b in B & B c= Cl{b} by TOPS_3:80;
end;
