
theorem
  for M being non empty MetrSpace,
      s being sequence of the carrier of TopSpaceMetr(M),
      x being Point of TopSpaceMetr(M) holds
    x in lim_f s
      iff
    x in Lim s
  proof
    let M be non empty MetrSpace,
    s be sequence of the carrier of TopSpaceMetr(M),
    x be Point of TopSpaceMetr(M);
    hereby
      assume x in lim_f s;
      then for b be Element of Balls(x) ex i be Nat st
      for j be Nat st i <=j holds s.j in b by Th6;
      hence x in Lim s by Th8;
    end;
    assume x in Lim s;
    then for b be Element of Balls(x) ex i be Nat st
    for j be Nat st i <=j holds s.j in b by Th8;
    hence x in lim_f s by Th6;
  end;
