reserve a, b, k, n, m for Nat,
  i for Integer,
  r for Real,
  p for Rational,
  c for Complex,
  x for object,
  f for Function;
reserve l, n1, n2 for Nat;
reserve s1, s2 for Real_Sequence;

theorem
  scf(r).0 > 0 implies for n st n1=c_n(r).(n+1) & n2=c_n(r).n holds n1
  gcd n2 = 1
proof
  set s1=c_n(r);
  set s=scf(r);
  defpred X[Nat] means for n1,n2 st n1=s1.($1+1) & n2=s1.$1 holds n1 gcd n2 =
  1;
  assume
A1: scf(r).0 > 0;
A2: for k st X[k] holds X[k+1]
  proof
    let k;
    reconsider n3 = s1.(k+2) as Element of NAT by A1,Th44;
    reconsider n2=s1.k as Element of NAT by A1,Th44;
    k+2 >= 0+1 by XREAL_1:7;
    then reconsider n4 = s.(k+2) as Element of NAT by Th38,INT_1:3;
    reconsider n1=s1.(k+1) as Element of NAT by A1,Th44;
    assume for n1,n2 st n1=s1.(k+1) & n2=s1.k holds n1 gcd n2 = 1;
    then
A3: n1 gcd n2 = 1;
    n3 = n4 * n1 + n2 by Def5;
    hence thesis by A3,EULER_1:8;
  end;
A4: X[0]
  proof
    reconsider u = s.1 as Element of NAT by Th38,INT_1:3;
    let n1,n2 such that
A5: n1=s1.(0+1) and
A6: n2=s1.0;
    n1 =u * s.0 +1 by A5,Def5;
    then
A7: n1 =u * n2 +1 by A6,Def5;
    1 gcd n2 = 1 by NEWTON:51;
    hence thesis by A7,EULER_1:8;
  end;
  for n holds X[n] from NAT_1:sch 2(A4,A2);
  hence thesis;
end;
