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 Th64:
  c_n(r).(n+1)*c_d(r).n - c_n(r).n*c_d(r).(n+1) = (-1)|^n
proof
  set s=scf(r), s1=c_n(r),s2=c_d(r);
  defpred G[Nat] means s1.($1+1)*s2.$1-s1.$1*s2.($1+1)=(-1)|^$1;
A1: s2.0 = 1 & s2.1 = s.1 by Def6;
A2: for n st G[n] holds G[n+1]
  proof
    let l;
    assume
A3: s1.(l+1)*s2.l-s1.l*s2.(l+1)=(-1)|^l;
    s1.(l+2)*s2.(l+1)-s1.(l+1)*s2.(l+2) =(s.(l+2) * s1.(l+1) + s1.l)*s2.(l
    +1)-s1.(l+1)*s2.(l+2) by Def5
      .=(s.(l+2) * s1.(l+1)*s2.(l+1) + s1.l*s2.(l+1)) -s1.(l+1)*(s.(l+2) *
    s2.(l+1) + s2.l) by Def6
      .=(-1)*(s1.(l+1)*s2.l-s1.l*s2.(l+1))
      .=(-1)|^(l+1) by A3,NEWTON:6;
    hence thesis;
  end;
  s1.0 = s.0 & s1.1 = s.1 * s.0 + 1 by Def5;
  then
A4: G[0] by A1,NEWTON:4;
  for n holds G[n] from NAT_1:sch 2(A4,A2);
  hence thesis;
end;
