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

theorem Th12:
  s1 = s2 implies (s1 is convergent iff s2 is convergent)
  proof
    assume
A1: s1 = s2;
    hereby
      assume s1 is convergent;
      then consider x be Element of Euclid n such that
A2:   for r being Real st r > 0 holds
      ex n0 being Nat st for m being Nat st n0 <= m holds
      dist(s1.m,x) < r;
      x is Element of TOP-REAL n by EUCLID:67;
      then x is Element of REAL n by EUCLID:22;
      then reconsider g = x as Point of REAL-NS n by REAL_NS1:def 4;
      now
        take g;
        hereby
          let r be Real;
          assume 0 < r;
          then consider n0 be Nat such that
A3:       for m being Nat st n0 <= m holds dist(s1.m,x) < r by A2;
          hereby
            take n0;
            hereby
              let n1 be Nat;
              assume n0 <= n1;
              then dist(s1.n1,x) < r & s1.n1 = s2.n1 by A1,A3;
              hence ||.s2.n1 - g .|| < r by Th9;
            end;
          end;
        end;
      end;
      hence s2 is convergent;
    end;
    assume s2 is convergent;
    then consider g be Point of REAL-NS n such that
A4: for r be Real st 0 < r holds ex m be Nat st for n0 be Nat st m <= n0 holds
    ||.(s2.n0)-g.|| < r;
    g is Element of REAL n by REAL_NS1:def 4;
    then g is Element of TOP-REAL n by EUCLID:22;
    then reconsider x = g as Element of Euclid n by EUCLID:67;
    now
      take x;
      hereby
        let r be Real;
        assume r > 0;
        then consider m0 be Nat such that
A5:     for n0 be Nat st m0 <= n0 holds ||.(s2.n0)-g.|| < r by A4;
        hereby
          take m0;
          hereby
            let m be Nat;
            assume m0 <= m;
            then ||. s2.m - g .|| < r & s2.m = s1.m by A1,A5;
            hence dist(s1.m,x) < r by Th9;
          end;
        end;
      end;
    end;
    hence s1 is convergent;
  end;
