let b be Nat; :: thesis: for r being real number st (scf r) . 0 > 0 & ( for n being Nat holds (scf r) . n <= b ) holds
for n being Nat holds (c_n r) . n <= ((b + (sqrt ((b ^2 ) + 4))) / 2) |^ (n + 1)
let r be real number ; :: thesis: ( (scf r) . 0 > 0 & ( for n being Nat holds (scf r) . n <= b ) implies for n being Nat holds (c_n r) . n <= ((b + (sqrt ((b ^2 ) + 4))) / 2) |^ (n + 1) )
assume that
A1: (scf r) . 0 > 0 and
A2: for n being Nat holds (scf r) . n <= b ; :: thesis: for n being Nat holds (c_n r) . n <= ((b + (sqrt ((b ^2 ) + 4))) / 2) |^ (n + 1)
set s = scf r;
set s1 = c_n r;
defpred S₂[ Nat] means (c_n r) . $1 <= ((b + (sqrt ((b ^2 ) + 4))) / 2) |^ ($1 + 1);
A3: (c_n r) . 0 = (scf r) . 0 by Def6;
A4: (scf r) . 0 <= b by A2;
(b ^2 ) + 4 > b ^2 by XREAL_1:41;
then sqrt ((b ^2 ) + 4) > sqrt (b ^2 ) by SQUARE_1:95;
then A5: sqrt ((b ^2 ) + 4) > b by SQUARE_1:89;
then b + (sqrt ((b ^2 ) + 4)) > b + b by XREAL_1:10;
then (b + (sqrt ((b ^2 ) + 4))) / 2 > (2 * b) / 2 by XREAL_1:76;
then ((b + (sqrt ((b ^2 ) + 4))) / 2) |^ (0 + 1) > b by NEWTON:10;
then A6: S₂[ 0 ] by A3, A4, XXREAL_0:2;
A7: (c_n r) . 1 = (((scf r) . 1) * ((scf r) . 0 )) + 1 by Def6;
A8: ( (scf r) . 1 >= 0 & (scf r) . 0 > 0 ) by A1, Th38;
(scf r) . 1 <= b by A2;
then ((scf r) . 1) * ((scf r) . 0 ) <= b ^2 by A4, A8, XREAL_1:68;
then A10: (c_n r) . 1 <= (b ^2 ) + 1 by A7, XREAL_1:8;
A11: ((b + (sqrt ((b ^2 ) + 4))) / 2) |^ (1 + 1) = ((b + (sqrt ((b ^2 ) + 4))) / 2) ^2 by WSIERP_1:2
.= (((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 4
.= (((b ^2 ) + (b * (sqrt ((b ^2 ) + 4)))) + 2) / 2 ;
b * (sqrt ((b ^2 ) + 4)) >= b * b by A5, XREAL_1:66;
then (b ^2 ) + (b * (sqrt ((b ^2 ) + 4))) >= (b ^2 ) + (b * b) by XREAL_1:8;
then ((b ^2 ) + (b * (sqrt ((b ^2 ) + 4)))) + 2 >= ((b ^2 ) + (b ^2 )) + 2 by XREAL_1:8;
then (((b ^2 ) + (b * (sqrt ((b ^2 ) + 4)))) + 2) / 2 >= (2 * ((b ^2 ) + 1)) / 2 by XREAL_1:74;
then A12: S₂[1] by A10, A11, XXREAL_0:2;
A13: for n being Nat st S₂[n] & S₂[n + 1] holds
S₂[n + 2]

proof

let n be Nat; :: thesis: ( S₂[n] & S₂[n + 1] implies S₂[n + 2] )
assume that
A14: (c_n r) . n <= ((b + (sqrt ((b ^2 ) + 4))) / 2) |^ (n + 1) and
A15: (c_n r) . (n + 1) <= ((b + (sqrt ((b ^2 ) + 4))) / 2) |^ ((n + 1) + 1) ; :: thesis: S₂[n + 2]
A16: (c_n r) . ((n + 1) + 1) = (((scf r) . (n + 2)) * ((c_n r) . (n + 1))) + ((c_n r) . n) by Def6;
n + 2 >= 0 + 1 by XREAL_1:9;
then A17: (scf r) . (n + 2) >= 0 by Th38;
A18: (scf r) . (n + 2) <= b by A2;
(c_n r) . (n + 1) > 0 by A1, Th45;
then A19: ((scf r) . (n + 2)) * ((c_n r) . (n + 1)) <= b * (((b + (sqrt ((b ^2 ) + 4))) / 2) |^ ((n + 1) + 1)) by A15, A17, A18, XREAL_1:68;
A20: (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:11
.= (((b + (sqrt ((b ^2 ) + 4))) / 2) |^ (n + 1)) * ((((b ^2 ) + (b * (sqrt ((b ^2 ) + 4)))) + 2) / 2) ;
((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:13
.= (((b + (sqrt ((b ^2 ) + 4))) / 2) |^ (n + 1)) * (((b + (sqrt ((b ^2 ) + 4))) / 2) ^2 ) by WSIERP_1:2
.= (((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 4
.= (((b + (sqrt ((b ^2 ) + 4))) / 2) |^ (n + 1)) * ((((b ^2 ) + (b * (sqrt ((b ^2 ) + 4)))) + 2) / 2) ;
hence S₂[n + 2] by A14, A16, A19, A20, XREAL_1:9; :: thesis: verum

end;

A21: for n being Nat holds S₂[n] from FIB_NUM:sch 1(A6, A12, A13);
let n be Nat; :: thesis: (c_n r) . n <= ((b + (sqrt ((b ^2 ) + 4))) / 2) |^ (n + 1)
thus (c_n r) . n <= ((b + (sqrt ((b ^2 ) + 4))) / 2) |^ (n + 1) by A21; :: thesis: verum