reserve X for set,
        D for a_partition of X,
        TG for non empty TopologicalGroup;
reserve A for Subset of X;
reserve US for UniformSpace;

theorem Th9:
  for US being non empty UniformSpace,
      S being Subset of FMT_induced_by(US) holds
  S is open iff S in Family_open_set(FMT_induced_by(US))
  proof
    let US be non empty UniformSpace, S be Subset of FMT_induced_by(US);
    thus S is open implies S in Family_open_set(FMT_induced_by(US));
    assume S in Family_open_set(FMT_induced_by(US));
    then ex O be open Subset of FMT_induced_by(US) st S = O;
    hence S is open;
  end;
