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
  n1=c_d(r).(n+1) & n2=c_d(r).n implies n1 gcd n2 = 1
proof
  set s=scf(r);
  set s2=c_d(r);
  defpred X[Nat] means for n1,n2 st n1=s2.($1+1) & n2=s2.$1 holds n1 gcd n2 =
  1;
A1: for k st X[k] holds X[k+1]
  proof
    let k;
    k+2>=0+1 by XREAL_1:7;
    then reconsider n4 = s.(k+2) as Element of NAT by Th38,INT_1:3;
    reconsider n3 = s2.(k+2) as Element of NAT by Th50;
    reconsider n2=s2.k as Element of NAT by Th50;
    reconsider n1=s2.(k+1) as Element of NAT by Th50;
    assume for n1,n2 st n1=s2.(k+1) & n2=s2.k holds n1 gcd n2 = 1;
    then
A2: n1 gcd n2 = 1;
    n3 = n4 * n1 + n2 by Def6;
    hence thesis by A2,EULER_1:8;
  end;
A3: X[0]
  proof
    let n1,n2 such that
    n1=s2.(0+1) and
A4: n2=s2.0;
    n1 gcd 1 = 1 by NEWTON:51;
    hence thesis by A4,Def6;
  end;
  for n holds X[n] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
