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 & (for n holds scf(r).n <= b) implies for n holds c_n(r).
  n <= ((b+sqrt (b^2+4))/2)|^(n+1)
proof
  assume that
A1: scf(r).0 > 0 and
A2: for n holds scf(r).n <= b;
  set s1=c_n(r);
  set s=scf(r);
A3: s.0 <= b by A2;
  defpred P[Nat] means s1.$1 <= ((b+sqrt (b^2+4))/2)|^($1+1);
A4: for n being Nat st P[n] & P[n+1] holds P[n+2]
  proof
    let n be Nat;
    assume that
A5: s1.n <= ((b+sqrt (b^2+4))/2)|^(n+1) and
A6: s1.(n+1) <= ((b+sqrt (b^2+4))/2)|^(n+1+1);
    n+2 >= 0+1 by XREAL_1:7;
    then
A7: s.(n+2) >= 0 by Th38;
    s.(n+2) <= b & s1.(n+1) > 0 by A1,A2,Th45;
    then
A8: s.(n+2)*s1.(n+1) <= b*((b+sqrt (b^2+4))/2)|^(n+1+1) by A6,A7,XREAL_1:66;
A9: b*((b+sqrt (b^2+4))/2)|^(n+1+1)+((b+sqrt (b^2+4))/2)|^(n+1) =b*(((b+
sqrt (b^2+4))/2)|^(n+1) * ((b+sqrt (b^2+4))/2)) + ((b+sqrt (b^2+4))/2)|^(n+1)
    by NEWTON:6
      .=((b+sqrt (b^2+4))/2)|^(n+1) * ((b^2+b*sqrt (b^2+4)+2)/2);
A10: ((b+sqrt (b^2+4))/2)|^(n+2+1) =((b+sqrt (b^2+4))/2)|^(n+1+2)
      .=((b+sqrt (b^2+4))/2)|^(n+1) * ((b+sqrt (b^2+4))/2)|^2 by NEWTON:8
      .=((b+sqrt (b^2+4))/2)|^(n+1) * ((b+sqrt (b^2+4))/2)^2 by WSIERP_1:1
      .=((b+sqrt (b^2+4))/2)|^(n+1) * ((b^2+2*b*sqrt (b^2+4)+(sqrt (b^2+4))
    ^2)/(2*2))
      .=((b+sqrt (b^2+4))/2)|^(n+1) * ((b^2+2*b*sqrt (b^2+4)+(b^2+4))/(2*2))
    by SQUARE_1:def 2
      .=((b+sqrt (b^2+4))/2)|^(n+1) * ((b^2+b*sqrt (b^2+4)+2)/2);
    s1.(n+1+1) = s.(n+2)*s1.(n+1)+s1.n by Def5;
    hence thesis by A5,A8,A9,A10,XREAL_1:7;
  end;
  let n;
  b^2+4 > b^2 by XREAL_1:39;
  then sqrt (b^2+4) > sqrt b^2 by SQUARE_1:27;
  then
A11: sqrt (b^2+4) > b by SQUARE_1:22;
  then b+sqrt (b^2+4) > b+b by XREAL_1:8;
  then (b+sqrt (b^2+4))/2 > (2*b)/2 by XREAL_1:74;
  then
A12: ((b+sqrt (b^2+4))/2)|^(0+1) > b;
A13: s.1 >= 0 by Th38;
A14: s1.1 = s.1*s.0+1 by Def5;
  s.1 <= b by A2;
  then s.1*s.0 <= b^2 by A1,A3,A13,XREAL_1:66;
  then
A15: s1.1 <= b^2+1 by A14,XREAL_1:6;
  b*sqrt (b^2+4) >= b*b by A11,XREAL_1:64;
  then b^2+b*sqrt (b^2+4) >= b^2+b*b by XREAL_1:6;
  then b^2+b*sqrt (b^2+4)+2 >= b^2+b^2+2 by XREAL_1:6;
  then
A16: (b^2+b*sqrt (b^2+4)+2)/2 >= (2*(b^2+1))/2 by XREAL_1:72;
  ((b+sqrt (b^2+4))/2)|^(1+1) =((b+sqrt (b^2+4))/2)^2 by WSIERP_1:1
    .=(b^2+2*b*sqrt (b^2+4)+(sqrt (b^2+4))^2)/(2*2)
    .=(b^2+2*b*sqrt (b^2+4)+(b^2+4))/(2*2) by SQUARE_1:def 2
    .=(b^2+b*sqrt (b^2+4)+2)/2;
  then
A17: P[1] by A15,A16,XXREAL_0:2;
  s1.0=s.0 by Def5;
  then
A18: P[0] by A3,A12,XXREAL_0:2;
  for n being Nat holds P[n] from FIB_NUM:sch 1(A18,A17,A4);
  hence thesis;
end;
