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 holds c_n(r).(n+2) > scf(r).(n+2) * c_n(r). (n+1)
proof
  assume
A1: scf(r).0 > 0;
  let n;
  set s=scf(r);
  set s1=c_n(r);
  set m=s.(n+2)*s1.(n+1);
  s1.(n+2)-m=m+s1.n-m by Def5;
  then s1.(n+2)-s.(n+2) * s1.(n+1)>0 by A1,Th45;
  then s1.(n+2)-m+m>0+m by XREAL_1:6;
  hence thesis;
end;
