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
  for MS being PseudoMetricSpace holds
    TopSpace_induced_by(uniformity_induced_by(MS)) = TopSpaceMetr(MS)
  proof
    let MS be PseudoMetricSpace;
    the topology of FMT2TopSpace(FMT_induced_by(uniformity_induced_by(MS)))
      = Family_open_set(FMT_induced_by(uniformity_induced_by(MS)))
      by FINTOPO7:def 16;
    hence thesis by FINTOPO7:def 16,Th11;
  end;
