 reserve n for Nat;
 reserve s1 for sequence of Euclid n,
         s2 for sequence of REAL-NS n;

theorem Th13:
  for S2 being sequence of Euclid n st S2 is Cauchy holds S2 is convergent
  proof
    let S2 be sequence of Euclid n;
    assume
A1: S2 is Cauchy;
    reconsider S2NS = S2 as sequence of REAL-NS n by Th10;
    S2NS is Cauchy_sequence_by_Norm by A1,Th11;
    then S2NS is convergent by LOPBAN_1:def 15;
    hence S2 is convergent by Th12;
  end;
