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 Th8:
  for US being non empty UniformSpace,
       S being Subset of FMT_induced_by(US) holds
  S is open iff for x being Element of US st x in S holds
  S in Neighborhood x
  proof
    let US be non empty UniformSpace, S be Subset of FMT_induced_by(US);
    hereby
      assume
A1:   S is open;
      hereby
        let x be Element of US;
        assume
A2:     x in S;
        reconsider x1 = x as Element of FMT_induced_by(US);
        U_FMT x1 = Neighborhood x by UNIFORM2:def 14;
        hence S in Neighborhood x by A1,A2;
      end;
    end;
    assume
A3: for x being Element of US st x in S holds
      S in Neighborhood x;
    now
      let x be Element of FMT_induced_by(US);
      assume
A4:   x in S;
      consider y be Element of US such that
A5:   x = y and
A6:   U_FMT x = (Neighborhood(US)).y;
      U_FMT x = Neighborhood y by A6, UNIFORM2:def 14;
      hence S in U_FMT x by A3,A4,A5;
     end;
    hence S is open;
  end;
