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
  a > 0 & (for n holds scf(r).n >= a) implies for n holds c_d(r).(n+1)
  >= ((a+sqrt (a^2+4))/2)|^n
proof
  assume that
A1: a > 0 and
A2: for n holds scf(r).n >= a;
  set s=scf(r);
  set s2=c_d(r);
  defpred P[Nat] means s2.($1+1) >= ((a+sqrt (a^2+4))/2)|^$1;
A3: s.1>0 by A1,A2;
  then s.1>=1 by Th40;
  then s2.(0+1)>=1 by Def6;
  then
A4: P[0] by NEWTON:4;
  s.2>0 by A1,A2;
  then s.2*s.1 >= 1*s.1 by A3,Th40,XREAL_1:64;
  then
A5: s.2*s.1+1 >= s.1+1 by XREAL_1:6;
  s.1 >= a by A2;
  then s.1+1 >= a+1 by XREAL_1:6;
  then
A6: s.2*s.1+1 >= a+1 by A5,XXREAL_0:2;
  4*a > 0 by A1,XREAL_1:129;
  then a^2+4 < a^2+4 + 4*a by XREAL_1:39;
  then sqrt (a^2+4) < sqrt (a+2)^2 by SQUARE_1:27;
  then sqrt (a^2+4) < a+2 by SQUARE_1:22;
  then a+sqrt (a^2+4) < a+(a+2) by XREAL_1:8;
  then
A7: ((a+sqrt (a^2+4))/2)|^1 = (a+sqrt (a^2+4))/2 & (a+sqrt (a^2+4))/2 < (2*
  a+2*1 )/2 by XREAL_1:74;
  let n;
A8: for n being Nat st P[n] & P[n+1] holds P[n+2]
  proof
    let n be Nat;
    assume that
A9: s2.(n+1) >= ((a+sqrt (a^2+4))/2)|^n and
A10: s2.(n+1+1) >= ((a+sqrt (a^2+4))/2)|^(n+1);
A11: a*((a+sqrt (a^2+4))/2)|^(n+1)+((a+sqrt (a^2+4))/2)|^n =a*(((a+sqrt (a
    ^2+4))/2)|^n * ((a+sqrt (a^2+4))/2)) + ((a+sqrt (a^2+4))/2)|^n by NEWTON:6
      .=((a+sqrt (a^2+4))/2)|^n * ((a^2+a*sqrt (a^2+4)+2)/2);
    sqrt (a^2+4) > 0 by SQUARE_1:25;
    then (a+sqrt (a^2+4))/2 > 0 by XREAL_1:139;
    then
A12: ((a+sqrt (a^2+4))/2)|^(n+1) > 0 by PREPOWER:6;
A13: ((a+sqrt (a^2+4))/2)|^(n+2) =((a+sqrt (a^2+4))/2)|^n * ((a+sqrt (a^2+
    4))/2)|^2 by NEWTON:8
      .=((a+sqrt (a^2+4))/2)|^n * ((a+sqrt (a^2+4))/2)^2 by WSIERP_1:1
      .=((a+sqrt (a^2+4))/2)|^n * ((a^2+2*a*sqrt (a^2+4)+(sqrt (a^2+4))^2)/(
    2*2))
      .=((a+sqrt (a^2+4))/2)|^n * ((a^2+2*a*sqrt (a^2+4)+(a^2+4))/(2*2)) by
SQUARE_1:def 2
      .=((a+sqrt (a^2+4))/2)|^n * ((a^2+a*sqrt (a^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) >= a by A2;
    then s.(n+3)*s2.(n+1+1) >= a*((a+sqrt (a^2+4))/2)|^(n+1) by A10,A12,
XREAL_1:66;
    hence thesis by A9,A14,A11,A13,XREAL_1:7;
  end;
  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
A15: P[1] by A6,A7,XXREAL_0:2;
  for n being Nat holds P[n] from FIB_NUM:sch 1(A4,A15,A8);
  hence thesis;
end;
