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

theorem Th11:
  s1 = s2 implies (s1 is Cauchy iff s2 is Cauchy_sequence_by_Norm)
  proof
    assume
A1: s1 = s2;
    thus s1 is Cauchy implies s2 is Cauchy_sequence_by_Norm
    proof
      assume
A3:   s1 is Cauchy;
      now
        let r be Real;
        assume r > 0;
        then consider p be Nat such that
A4:     for n,m being Nat st p <= n & p <= m holds dist(s1.n,s1.m) < r by A3;
        take p;
        hereby
          let n0,m0 be Nat;
          assume
A5:       n0 >= p & m0 >= p;
          dist(s1.n0,s1.m0) = ||.s2.n0-s2.m0.|| by A1,Th9;
          hence ||.s2.n0-s2.m0.|| < r by A5,A4;
        end;
      end;
      hence s2 is Cauchy_sequence_by_Norm by RSSPACE3:8;
    end;
      assume
A9:   s2 is Cauchy_sequence_by_Norm;
      now
        let r be Real;
        assume r > 0;
        then consider k be Nat such that
A10:    for n, m being Nat st n >= k & m >= k holds
          ||. s2.n - s2.m .|| < r by A9,RSSPACE3:8;
        hereby
          take k;
          hereby
            let n0,m0 be Nat;
            assume
A11:        k <= n0 & k <= m0;
            ||.s2.n0-s2.m0.|| = dist(s1.n0,s1.m0) by A1,Th9;
            hence dist(s1.n0,s1.m0) < r by A11,A10;
          end;
        end;
        hence ex p be Nat st for n,m be Nat st p <= n & p <= m holds
          dist(s1.n,s1.m) < r;
      end;
      hence s1 is Cauchy;
  end;
