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
  (for n holds scf(r).n <= b) implies for n holds c_d(r).(n+1) <= ((b+
  sqrt (b^2+4))/2)|^(n+1)
proof
  set s=scf(r);
  set s2=c_d(r);
  defpred P[Nat] means s2.($1+1) <= ((b+sqrt (b^2+4))/2)|^($1+1);
  assume
A1: for n holds scf(r).n <= b;
  then
A2: s.1 <= b;
A3: s.2 <= b by A1;
  s.2 >= 0 & s.1 >= 0 by Th38;
  then
A4: s.2*s.1 <= b^2 by A2,A3,XREAL_1:66;
  s2.(1+1)=s.(0+2)*s2.(0+1)+s2.0 by Def6
    .=s.2*s2.1+1 by Def6
    .=s.2*s.1+1 by Def6;
  then
A5: s2.(1+1) <= b^2+1 by A4,XREAL_1:6;
  let n;
  b^2+4 > b^2 by XREAL_1:39;
  then sqrt (b^2+4) > sqrt b^2 by SQUARE_1:27;
  then
A6: 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
A7: ((b+sqrt (b^2+4))/2)|^(0+1) > b;
A8: for n be Nat st P[n] & P[n+1] holds P[n+2]
  proof
    let n be Nat;
    assume that
A9: s2.(n+1) <= ((b+sqrt (b^2+4))/2)|^(n+1) and
A10: s2.(n+1+1) <= ((b+sqrt (b^2+4))/2)|^(n+1+1);
A11: 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);
    n+3 >=0+1 by XREAL_1:7;
    then
A12: s.(n+3) >= 0 by Th38;
A13: ((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);
A14: s2.(n+2+1) =s.(n+1+2)*s2.(n+1+1)+s2.(n+1) by Def6
      .=s.(n+3)*s2.(n+1+1)+s2.(n+1);
    s.(n+3) <= b & s2.(n+1+1) >= 0 by A1,Th51;
    then s.(n+3)*s2.(n+1+1) <= b*((b+sqrt (b^2+4))/2)|^(n+1+1) by A10,A12,
XREAL_1:66;
    hence thesis by A9,A14,A11,A13,XREAL_1:7;
  end;
  b*sqrt (b^2+4) >= b*b by A6,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
A15: (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
A16: P[1] by A5,A15,XXREAL_0:2;
  s2.(0+1)=s.1 by Def6;
  then
A17: P[0] by A2,A7,XXREAL_0:2;
  for n being Nat holds P[n] from FIB_NUM:sch 1(A17,A16,A8);
  hence thesis;
end;
