:: FIB_NUM2 semantic presentation

begin

theorem :: FIB_NUM2:1
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) holds (n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) = n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

theorem :: FIB_NUM2:2
for n being ( ( integer odd ) ( non empty V11() real integer ext-real odd ) Integer) holds (- 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( V11() ) ( V11() real integer ext-real non positive ) set ) to_power n : ( ( integer odd ) ( non empty V11() real integer ext-real odd ) Integer) : ( ( real ) ( V11() real ext-real ) set ) = - 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( V11() ) ( V11() real integer ext-real non positive ) set ) ;

theorem :: FIB_NUM2:3
for n being ( ( integer even ) ( V11() real integer ext-real even ) Integer) holds (- 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( V11() ) ( V11() real integer ext-real non positive ) set ) to_power n : ( ( integer even ) ( V11() real integer ext-real even ) Integer) : ( ( real ) ( V11() real ext-real ) set ) = 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

theorem :: FIB_NUM2:4
for m being ( ( non empty real ) ( non empty V11() real ext-real ) number )
for n being ( ( integer ) ( V11() real integer ext-real ) Integer) holds ((- 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( V11() ) ( V11() real integer ext-real non positive ) set ) * m : ( ( non empty real ) ( non empty V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) to_power n : ( ( integer ) ( V11() real integer ext-real ) Integer) : ( ( real ) ( V11() real ext-real ) set ) = ((- 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( V11() ) ( V11() real integer ext-real non positive ) set ) to_power n : ( ( integer ) ( V11() real integer ext-real ) Integer) ) : ( ( real ) ( V11() real ext-real ) set ) * (m : ( ( non empty real ) ( non empty V11() real ext-real ) number ) to_power n : ( ( integer ) ( V11() real integer ext-real ) Integer) ) : ( ( real ) ( V11() real ext-real ) set ) : ( ( ) ( V11() real ext-real ) set ) ;

theorem :: FIB_NUM2:5
for k, m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat)
for a being ( ( real ) ( V11() real ext-real ) number ) holds a : ( ( real ) ( V11() real ext-real ) number ) to_power (k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) + m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) set ) : ( ( real ) ( V11() real ext-real ) set ) = (a : ( ( real ) ( V11() real ext-real ) number ) to_power k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( real ) ( V11() real ext-real ) set ) * (a : ( ( real ) ( V11() real ext-real ) number ) to_power m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( real ) ( V11() real ext-real ) set ) : ( ( ) ( V11() real ext-real ) set ) ;

theorem :: FIB_NUM2:6
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat)
for k being ( ( non empty real ) ( non empty V11() real ext-real ) number )
for m being ( ( integer odd ) ( non empty V11() real integer ext-real odd ) Integer) holds (k : ( ( non empty real ) ( non empty V11() real ext-real ) number ) to_power m : ( ( integer odd ) ( non empty V11() real integer ext-real odd ) Integer) ) : ( ( real ) ( V11() real ext-real ) set ) to_power n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) : ( ( real ) ( V11() real ext-real ) set ) = k : ( ( non empty real ) ( non empty V11() real ext-real ) number ) to_power (m : ( ( integer odd ) ( non empty V11() real integer ext-real odd ) Integer) * n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( V11() real integer ext-real ) set ) : ( ( real ) ( V11() real ext-real ) set ) ;

theorem :: FIB_NUM2:7
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) holds ((- 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( V11() ) ( V11() real integer ext-real non positive ) set ) to_power (- n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( V11() ) ( V11() real integer ext-real non positive ) set ) ) : ( ( real ) ( V11() real ext-real ) set ) ^2 : ( ( ) ( V11() real ext-real ) set ) = 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

theorem :: FIB_NUM2:8
for k, m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat)
for a being ( ( non empty real ) ( non empty V11() real ext-real ) number ) holds (a : ( ( non empty real ) ( non empty V11() real ext-real ) number ) to_power (- k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( V11() ) ( V11() real integer ext-real non positive ) set ) ) : ( ( real ) ( V11() real ext-real ) set ) * (a : ( ( non empty real ) ( non empty V11() real ext-real ) number ) to_power (- m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( V11() ) ( V11() real integer ext-real non positive ) set ) ) : ( ( real ) ( V11() real ext-real ) set ) : ( ( ) ( V11() real ext-real ) set ) = a : ( ( non empty real ) ( non empty V11() real ext-real ) number ) to_power ((- k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( V11() ) ( V11() real integer ext-real non positive ) set ) - m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( V11() real integer ext-real non positive ) set ) : ( ( real ) ( V11() real ext-real ) set ) ;

theorem :: FIB_NUM2:9
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) holds (- 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( V11() ) ( V11() real integer ext-real non positive ) set ) to_power (- (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( V11() ) ( V11() real integer ext-real non positive ) set ) : ( ( real ) ( V11() real ext-real ) set ) = 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

theorem :: FIB_NUM2:10
for k being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat)
for a being ( ( non empty real ) ( non empty V11() real ext-real ) number ) holds (a : ( ( non empty real ) ( non empty V11() real ext-real ) number ) to_power k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( real ) ( V11() real ext-real ) set ) * (a : ( ( non empty real ) ( non empty V11() real ext-real ) number ) to_power (- k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( V11() ) ( V11() real integer ext-real non positive ) set ) ) : ( ( real ) ( V11() real ext-real ) set ) : ( ( ) ( V11() real ext-real ) set ) = 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

registration
let n be ( ( integer odd ) ( non empty V11() real integer ext-real odd ) Integer) ;
cluster - n : ( ( integer odd ) ( non empty V11() real integer ext-real odd ) set ) : ( ( V11() ) ( V11() real integer ext-real ) set ) -> V11() odd ;
end;

registration
let n be ( ( integer even ) ( V11() real integer ext-real even ) Integer) ;
cluster - n : ( ( integer even ) ( V11() real integer ext-real even ) set ) : ( ( V11() ) ( V11() real integer ext-real ) set ) -> V11() even ;
end;

theorem :: FIB_NUM2:11
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) holds (- 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( V11() ) ( V11() real integer ext-real non positive ) set ) to_power (- n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( V11() ) ( V11() real integer ext-real non positive ) set ) : ( ( real ) ( V11() real ext-real ) set ) = (- 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( V11() ) ( V11() real integer ext-real non positive ) set ) to_power n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) : ( ( real ) ( V11() real ext-real ) set ) ;

theorem :: FIB_NUM2:12
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat)
for k, m, m1, n1 being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) st k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) divides m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) & k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) divides n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) holds
k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) divides (m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * m1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) * n1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

registration
cluster non empty finite with_non-empty_elements natural-membered for ( ( ) ( ) set ) ;
end;

registration
let f be ( ( Function-like V29( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -valued Function-like V29( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) V76() V77() V78() V79() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;
let A be ( ( finite with_non-empty_elements natural-membered ) ( finite with_non-empty_elements complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) set ) ;
cluster f : ( ( Function-like V29( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -valued Function-like V29( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) V76() V77() V78() V79() ) Element of bool [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ,NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) :] : ( ( ) ( Relation-like RAT : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered rational-membered V92() ) set ) -valued INT : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered rational-membered integer-membered V92() ) set ) -valued V76() V77() V78() V79() ) set ) : ( ( ) ( ) set ) ) | A : ( ( finite with_non-empty_elements natural-membered ) ( finite with_non-empty_elements complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) set ) : ( ( Relation-like ) ( Relation-like A : ( ( finite with_non-empty_elements natural-membered ) ( finite with_non-empty_elements complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) set ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined RAT : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered rational-membered V92() ) set ) -valued NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -valued Function-like finite V76() V77() V78() V79() ) set ) -> Relation-like FinSubsequence-like ;
end;

theorem :: FIB_NUM2:13
for p being ( ( Relation-like Function-like FinSubsequence-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined Function-like FinSubsequence-like ) FinSubsequence) holds rng (Seq p : ( ( Relation-like Function-like FinSubsequence-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined Function-like FinSubsequence-like ) FinSubsequence) ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined Function-like finite FinSequence-like FinSubsequence-like ) set ) : ( ( ) ( finite ) set ) c= rng p : ( ( Relation-like Function-like FinSubsequence-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined Function-like FinSubsequence-like ) FinSubsequence) : ( ( ) ( ) set ) ;

definition
let f be ( ( Function-like V29( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -valued Function-like V29( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) V76() V77() V78() V79() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;
let A be ( ( finite with_non-empty_elements natural-membered ) ( finite with_non-empty_elements complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) set ) ;
func Prefix (f,A) -> ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V76() V77() V78() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) equals :: FIB_NUM2:def 1
 Seq (f : ( ( ) ( ) set ) | A : ( ( ) ( ) set ) ) : ( ( Relation-like ) ( Relation-like ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ;
end;

theorem :: FIB_NUM2:14
for m, n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat)
for k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) st k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) <> 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real integer Relation-like non-empty empty-yielding NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined RAT : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered rational-membered V92() ) set ) -valued Function-like one-to-one constant functional finite finite-yielding V40() FinSequence-like FinSubsequence-like FinSequence-membered V47() ext-real non positive non negative V76() V77() V78() V79() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) & k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) <= n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) holds
m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) < n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ;

registration
cluster omega : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) set ) -> bounded_below ;
end;

theorem :: FIB_NUM2:15
for i, j being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat)
for x, y being ( ( ) ( ) set ) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real integer Relation-like non-empty empty-yielding NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined RAT : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered rational-membered V92() ) set ) -valued Function-like one-to-one constant functional finite finite-yielding V40() FinSequence-like FinSubsequence-like FinSequence-membered V47() ext-real non positive non negative V76() V77() V78() V79() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) < i : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) & i : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) < j : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) holds
{[i : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ,x : ( ( ) ( ) set ) ] : ( ( ) ( ) set ) ,[j : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ,y : ( ( ) ( ) set ) ] : ( ( ) ( ) set ) } : ( ( ) ( non empty Relation-like finite ) set ) is ( ( Relation-like Function-like FinSubsequence-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined Function-like FinSubsequence-like ) FinSubsequence) ;

theorem :: FIB_NUM2:16
for i, j being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat)
for x, y being ( ( ) ( ) set )
for q being ( ( Relation-like Function-like FinSubsequence-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined Function-like FinSubsequence-like ) FinSubsequence) st i : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) < j : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) & q : ( ( Relation-like Function-like FinSubsequence-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined Function-like FinSubsequence-like ) FinSubsequence) = {[i : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ,x : ( ( ) ( ) set ) ] : ( ( ) ( ) set ) ,[j : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ,y : ( ( ) ( ) set ) ] : ( ( ) ( ) set ) } : ( ( ) ( non empty Relation-like finite ) set ) holds
Seq q : ( ( Relation-like Function-like FinSubsequence-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined Function-like FinSubsequence-like ) FinSubsequence) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined Function-like finite FinSequence-like FinSubsequence-like ) set ) = <*x : ( ( ) ( ) set ) ,y : ( ( ) ( ) set ) *> : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like ) set ) ;

registration
let n be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;
cluster Seg n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( finite n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) -element ) set ) -> with_non-empty_elements ;
end;

registration
let A be ( ( with_non-empty_elements ) ( with_non-empty_elements ) set ) ;
cluster -> with_non-empty_elements for ( ( ) ( ) Element of bool A : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ;
end;

registration
let A be ( ( with_non-empty_elements ) ( with_non-empty_elements ) set ) ;
let B be ( ( ) ( ) set ) ;
cluster A : ( ( with_non-empty_elements ) ( with_non-empty_elements ) set ) /\ B : ( ( ) ( ) set ) : ( ( ) ( ) set ) -> with_non-empty_elements ;
cluster B : ( ( ) ( ) set ) /\ A : ( ( with_non-empty_elements ) ( with_non-empty_elements ) set ) : ( ( ) ( ) set ) -> with_non-empty_elements ;
end;

theorem :: FIB_NUM2:17
for k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) )
for a being ( ( ) ( ) set ) st k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) >= 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) holds
{[k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ,a : ( ( ) ( ) set ) ] : ( ( ) ( ) set ) } : ( ( ) ( non empty Relation-like Function-like finite ) set ) is ( ( Relation-like Function-like FinSubsequence-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined Function-like FinSubsequence-like ) FinSubsequence) ;

theorem :: FIB_NUM2:18
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) )
for y being ( ( ) ( ) set )
for f being ( ( Relation-like Function-like FinSubsequence-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined Function-like FinSubsequence-like ) FinSubsequence) st f : ( ( Relation-like Function-like FinSubsequence-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined Function-like FinSubsequence-like ) FinSubsequence) = {[1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ,y : ( ( ) ( ) set ) ] : ( ( ) ( ) set ) } : ( ( ) ( non empty Relation-like Function-like finite ) set ) holds
i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) Shift f : ( ( Relation-like Function-like FinSubsequence-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined Function-like FinSubsequence-like ) FinSubsequence) : ( ( Relation-like Function-like FinSubsequence-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined Function-like FinSubsequence-like ) set ) = {[(1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ,y : ( ( ) ( ) set ) ] : ( ( ) ( ) set ) } : ( ( ) ( non empty Relation-like Function-like finite ) set ) ;

theorem :: FIB_NUM2:19
for q being ( ( Relation-like Function-like FinSubsequence-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined Function-like FinSubsequence-like ) FinSubsequence)
for k, n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) st dom q : ( ( Relation-like Function-like FinSubsequence-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined Function-like FinSubsequence-like ) FinSubsequence) : ( ( ) ( ) set ) c= Seg k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( finite b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) -element with_non-empty_elements complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) : ( ( ) ( non empty non empty-membered ) set ) ) & n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) > k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) holds
ex p being ( ( Relation-like Function-like FinSequence-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined Function-like finite FinSequence-like FinSubsequence-like ) FinSequence) st
( q : ( ( Relation-like Function-like FinSubsequence-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined Function-like FinSubsequence-like ) FinSubsequence) c= p : ( ( Relation-like Function-like FinSequence-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined Function-like finite FinSequence-like FinSubsequence-like ) FinSequence) & dom p : ( ( Relation-like Function-like FinSequence-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined Function-like finite FinSequence-like FinSubsequence-like ) FinSequence) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) : ( ( ) ( non empty non empty-membered ) set ) ) = Seg n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( finite b3 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) -element with_non-empty_elements complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

theorem :: FIB_NUM2:20
for q being ( ( Relation-like Function-like FinSubsequence-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined Function-like FinSubsequence-like ) FinSubsequence) ex p being ( ( Relation-like Function-like FinSequence-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined Function-like finite FinSequence-like FinSubsequence-like ) FinSequence) st q : ( ( Relation-like Function-like FinSubsequence-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined Function-like FinSubsequence-like ) FinSubsequence) c= p : ( ( Relation-like Function-like FinSequence-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined Function-like finite FinSequence-like FinSubsequence-like ) FinSequence) ;

begin

scheme :: FIB_NUM2:sch 1
FibInd1{ P1[ ( ( ) ( ) set ) ] } :
for k being ( ( non empty natural ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real positive non negative ) Nat) holds P1[k : ( ( ) ( ) set ) ]
provided
P1[1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ] and
P1[2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ] and
for k being ( ( non empty natural ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real positive non negative ) Nat) st P1[k : ( ( non empty natural ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real positive non negative ) Nat) ] & P1[k : ( ( non empty natural ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real positive non negative ) Nat) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ] holds
P1[k : ( ( non empty natural ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real positive non negative ) Nat) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ]
proof end;

scheme :: FIB_NUM2:sch 2
FibInd2{ P1[ ( ( ) ( ) set ) ] } :
for k being ( ( non trivial natural ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real positive non negative non empty-membered ) Nat) holds P1[k : ( ( ) ( ) set ) ]
provided
P1[2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ] and
P1[3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ] and
for k being ( ( non trivial natural ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real positive non negative non empty-membered ) Nat) st P1[k : ( ( non trivial natural ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real positive non negative non empty-membered ) Nat) ] & P1[k : ( ( non trivial natural ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real positive non negative non empty-membered ) Nat) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ] holds
P1[k : ( ( non trivial natural ) ( non empty non trivial epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real positive non negative non empty-membered ) Nat) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ]
proof end;

theorem :: FIB_NUM2:21
Fib 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) = 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

theorem :: FIB_NUM2:22
Fib 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) = 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

theorem :: FIB_NUM2:23
Fib 4 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) = 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

theorem :: FIB_NUM2:24
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) holds Fib (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) = (Fib n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + (Fib (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

theorem :: FIB_NUM2:25
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) holds Fib (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) + 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) = (Fib (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + (Fib (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

theorem :: FIB_NUM2:26
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) holds Fib (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) + 4 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) = (Fib (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + (Fib (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) + 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

theorem :: FIB_NUM2:27
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) holds Fib (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) + 5 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) = (Fib (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) + 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + (Fib (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) + 4 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

theorem :: FIB_NUM2:28
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) holds Fib (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) = (Fib (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) + 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) - (Fib (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( V11() real integer ext-real ) set ) ;

theorem :: FIB_NUM2:29
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) holds Fib (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) = (Fib (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) - (Fib n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( V11() real integer ext-real ) set ) ;

theorem :: FIB_NUM2:30
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) holds Fib n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) = (Fib (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) - (Fib (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( V11() real integer ext-real ) set ) ;

begin

theorem :: FIB_NUM2:31
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) holds ((Fib n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * (Fib (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) - ((Fib (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ^2) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( V11() real integer ext-real ) set ) = (- 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( V11() ) ( V11() real integer ext-real non positive ) set ) |^ (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( V11() real ext-real ) set ) ;

theorem :: FIB_NUM2:32
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) holds ((Fib (n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * (Fib (n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) - ((Fib n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ^2) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( V11() real integer ext-real ) set ) = (- 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( V11() ) ( V11() real integer ext-real non positive ) set ) |^ n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( V11() real ext-real ) set ) ;

theorem :: FIB_NUM2:33
tau : ( ( real ) ( V11() real ext-real ) set ) > 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real integer Relation-like non-empty empty-yielding NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined RAT : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered rational-membered V92() ) set ) -valued Function-like one-to-one constant functional finite finite-yielding V40() FinSequence-like FinSubsequence-like FinSequence-membered V47() ext-real non positive non negative V76() V77() V78() V79() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

theorem :: FIB_NUM2:34
tau_bar : ( ( real ) ( V11() real ext-real ) set ) = (- tau : ( ( real ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) to_power (- 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( V11() ) ( V11() real integer ext-real non positive ) set ) : ( ( real ) ( V11() real ext-real ) set ) ;

theorem :: FIB_NUM2:35
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) holds (- tau : ( ( real ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) to_power ((- 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( V11() ) ( V11() real integer ext-real non positive ) set ) * n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( V11() real integer ext-real non positive ) set ) : ( ( real ) ( V11() real ext-real ) set ) = ((- tau : ( ( real ) ( V11() real ext-real ) set ) ) : ( ( V11() ) ( V11() real ext-real ) set ) to_power (- 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( V11() ) ( V11() real integer ext-real non positive ) set ) ) : ( ( real ) ( V11() real ext-real ) set ) to_power n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) : ( ( real ) ( V11() real ext-real ) set ) ;

theorem :: FIB_NUM2:36
- (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) / tau : ( ( real ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) : ( ( V11() ) ( V11() real ext-real ) set ) = tau_bar : ( ( real ) ( V11() real ext-real ) set ) ;

theorem :: FIB_NUM2:37
for r being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) holds (((tau : ( ( real ) ( V11() real ext-real ) set ) to_power r : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( real ) ( V11() real ext-real ) set ) ^2) : ( ( ) ( V11() real ext-real ) set ) - (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * ((- 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( V11() ) ( V11() real integer ext-real non positive ) set ) to_power r : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( real ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) + ((tau : ( ( real ) ( V11() real ext-real ) set ) to_power (- r : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( V11() ) ( V11() real integer ext-real non positive ) set ) ) : ( ( real ) ( V11() real ext-real ) set ) ^2) : ( ( ) ( V11() real ext-real ) set ) : ( ( ) ( V11() real ext-real ) set ) = ((tau : ( ( real ) ( V11() real ext-real ) set ) to_power r : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( real ) ( V11() real ext-real ) set ) - (tau_bar : ( ( real ) ( V11() real ext-real ) set ) to_power r : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( real ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ^2 : ( ( ) ( V11() real ext-real ) set ) ;

theorem :: FIB_NUM2:38
for n, r being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) st r : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) <= n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) holds
((Fib n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ^2) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) - ((Fib (n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + r : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * (Fib (n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) -' r : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( V11() real integer ext-real ) set ) = ((- 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( V11() ) ( V11() real integer ext-real non positive ) set ) |^ (n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) -' r : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) set ) * ((Fib r : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ^2) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( V11() real ext-real ) set ) ;

theorem :: FIB_NUM2:39
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) holds ((Fib n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ^2) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + ((Fib (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ^2) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) = Fib ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered non even left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

theorem :: FIB_NUM2:40
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat)
for k being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) holds Fib (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) + k : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) = ((Fib k : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * (Fib (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + ((Fib (k : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) -' 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * (Fib n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

theorem :: FIB_NUM2:41
for k being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat)
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) holds Fib n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) divides Fib (n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

theorem :: FIB_NUM2:42
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat)
for k being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) st k : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) divides n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) holds
Fib k : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) divides Fib n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

theorem :: FIB_NUM2:43
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) holds Fib n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) <= Fib (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

theorem :: FIB_NUM2:44
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) st n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) > 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) holds
Fib n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) < Fib (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

theorem :: FIB_NUM2:45
for m, n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) st m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) >= n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) holds
Fib m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) >= Fib n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

theorem :: FIB_NUM2:46
for n, k being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) st k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) > 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) & k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) < n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) holds
Fib k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) < Fib n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

theorem :: FIB_NUM2:47
for k being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) holds
( Fib k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) = 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) iff ( k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) = 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) or k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) = 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) ) ;

theorem :: FIB_NUM2:48
for k, n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) st n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) > 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) & k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) <> 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real integer Relation-like non-empty empty-yielding NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined RAT : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered rational-membered V92() ) set ) -valued Function-like one-to-one constant functional finite finite-yielding V40() FinSequence-like FinSubsequence-like FinSequence-membered V47() ext-real non positive non negative V76() V77() V78() V79() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) & k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) <> 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) holds
( Fib k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) = Fib n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) iff k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) = n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) ;

theorem :: FIB_NUM2:49
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) st n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) > 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) & n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) <> 4 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) & not n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) is prime holds
ex k being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) st
( k : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) <> 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) & k : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) <> 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) & k : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) <> n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) & k : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) divides n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) ;

theorem :: FIB_NUM2:50
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) st n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) > 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) & n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) <> 4 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) & Fib n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) is prime holds
n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) is prime ;

begin

definition
func FIB -> ( ( Function-like V29( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -valued Function-like V29( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) V76() V77() V78() V79() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) means :: FIB_NUM2:def 2
 for k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) holds it : ( ( ) ( ) set ) . k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) = Fib k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;
end;

definition
func EvenNAT -> ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Subset of ( ( ) ( non empty non empty-membered ) set ) ) equals :: FIB_NUM2:def 3
 { (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) where k is ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : verum } ;
func OddNAT -> ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Subset of ( ( ) ( non empty non empty-membered ) set ) ) equals :: FIB_NUM2:def 4
 { ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered non even left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) where k is ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : verum } ;
end;

theorem :: FIB_NUM2:51
for k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) holds
( 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) in EvenNAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Subset of ( ( ) ( non empty non empty-membered ) set ) ) & not (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered non even left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) in EvenNAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Subset of ( ( ) ( non empty non empty-membered ) set ) ) ) ;

theorem :: FIB_NUM2:52
for k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) holds
( (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered non even left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) in OddNAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Subset of ( ( ) ( non empty non empty-membered ) set ) ) & not 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) in OddNAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Subset of ( ( ) ( non empty non empty-membered ) set ) ) ) ;

definition
let n be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;
func EvenFibs n -> ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V76() V77() V78() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) equals :: FIB_NUM2:def 5
 Prefix (FIB : ( ( Function-like V29( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -valued Function-like V29( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) V76() V77() V78() V79() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ,(EvenNAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Subset of ( ( ) ( non empty non empty-membered ) set ) ) /\ (Seg n : ( ( ) ( ) set ) ) : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of bool NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V76() V77() V78() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;
func OddFibs n -> ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V76() V77() V78() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) equals :: FIB_NUM2:def 6
 Prefix (FIB : ( ( Function-like V29( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -valued Function-like V29( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) V76() V77() V78() V79() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ,(OddNAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Subset of ( ( ) ( non empty non empty-membered ) set ) ) /\ (Seg n : ( ( ) ( ) set ) ) : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of bool NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V76() V77() V78() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;
end;

theorem :: FIB_NUM2:53
EvenFibs 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real integer Relation-like non-empty empty-yielding NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined RAT : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered rational-membered V92() ) set ) -valued Function-like one-to-one constant functional finite finite-yielding V40() FinSequence-like FinSubsequence-like FinSequence-membered V47() ext-real non positive non negative V76() V77() V78() V79() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V76() V77() V78() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) = {} : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real integer Relation-like non-empty empty-yielding NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined RAT : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered rational-membered V92() ) set ) -valued Function-like one-to-one constant functional finite finite-yielding V40() FinSequence-like FinSubsequence-like FinSequence-membered ext-real non positive non negative V76() V77() V78() V79() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() bounded_below bounded_above real-bounded V120() ) set ) ;

theorem :: FIB_NUM2:54
Seq (FIB : ( ( Function-like V29( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -valued Function-like V29( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) V76() V77() V78() V79() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) | {2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) } : ( ( ) ( non empty finite with_non-empty_elements non empty-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( Relation-like ) ( Relation-like {2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) } : ( ( ) ( non empty finite with_non-empty_elements non empty-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined RAT : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered rational-membered V92() ) set ) -valued NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -valued Function-like finite FinSubsequence-like V76() V77() V78() V79() ) set ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined Function-like finite FinSequence-like FinSubsequence-like ) set ) = <*1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) *> : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -valued Function-like finite 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V76() V77() V78() V79() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

theorem :: FIB_NUM2:55
EvenFibs 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V76() V77() V78() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) = <*1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) *> : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -valued Function-like finite 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V76() V77() V78() V79() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

theorem :: FIB_NUM2:56
EvenFibs 4 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V76() V77() V78() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) = <*1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ,3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) *> : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like ) set ) ;

theorem :: FIB_NUM2:57
for k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) holds (EvenNAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Subset of ( ( ) ( non empty non empty-membered ) set ) ) /\ (Seg ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty finite (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) -element with_non-empty_elements non empty-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( finite with_non-empty_elements complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) set ) \/ {((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 4 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) } : ( ( ) ( non empty finite with_non-empty_elements non empty-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) : ( ( ) ( non empty non empty-membered ) set ) ) : ( ( ) ( non empty finite with_non-empty_elements non empty-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded ) set ) = EvenNAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Subset of ( ( ) ( non empty non empty-membered ) set ) ) /\ (Seg ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 4 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty finite (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 4 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) -element with_non-empty_elements non empty-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) : ( ( ) ( non empty non empty-membered ) set ) ) : ( ( ) ( finite with_non-empty_elements complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) set ) ;

theorem :: FIB_NUM2:58
for k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) holds (FIB : ( ( Function-like V29( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -valued Function-like V29( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) V76() V77() V78() V79() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) | (EvenNAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Subset of ( ( ) ( non empty non empty-membered ) set ) ) /\ (Seg ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty finite (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) -element with_non-empty_elements non empty-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( finite with_non-empty_elements complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) set ) ) : ( ( Relation-like ) ( Relation-like EvenNAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Subset of ( ( ) ( non empty non empty-membered ) set ) ) /\ (Seg ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty finite (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) -element with_non-empty_elements non empty-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) : ( ( ) ( non empty non empty-membered ) set ) ) : ( ( ) ( finite with_non-empty_elements complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) set ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined RAT : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered rational-membered V92() ) set ) -valued NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -valued Function-like finite FinSubsequence-like V76() V77() V78() V79() ) set ) \/ {[((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 4 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ,(FIB : ( ( Function-like V29( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -valued Function-like V29( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) V76() V77() V78() V79() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) . ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 4 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ] : ( ( ) ( ) Element of [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ,NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) :] : ( ( ) ( Relation-like RAT : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered rational-membered V92() ) set ) -valued INT : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered rational-membered integer-membered V92() ) set ) -valued V76() V77() V78() V79() ) set ) ) } : ( ( ) ( non empty Relation-like Function-like finite ) set ) : ( ( ) ( non empty Relation-like finite ) set ) = FIB : ( ( Function-like V29( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -valued Function-like V29( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) V76() V77() V78() V79() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) | (EvenNAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Subset of ( ( ) ( non empty non empty-membered ) set ) ) /\ (Seg ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 4 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty finite (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 4 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) -element with_non-empty_elements non empty-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( finite with_non-empty_elements complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) set ) : ( ( Relation-like ) ( Relation-like EvenNAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Subset of ( ( ) ( non empty non empty-membered ) set ) ) /\ (Seg ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 4 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty finite (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * b1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 4 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) -element with_non-empty_elements non empty-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) : ( ( ) ( non empty non empty-membered ) set ) ) : ( ( ) ( finite with_non-empty_elements complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) set ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined RAT : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered rational-membered V92() ) set ) -valued NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -valued Function-like finite FinSubsequence-like V76() V77() V78() V79() ) set ) ;

theorem :: FIB_NUM2:59
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) holds EvenFibs ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V76() V77() V78() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) = (EvenFibs (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V76() V77() V78() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ^ <*(Fib ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) *> : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -valued Function-like finite 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V76() V77() V78() V79() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V76() V77() V78() V79() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

theorem :: FIB_NUM2:60
OddFibs 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V76() V77() V78() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) = <*1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) *> : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -valued Function-like finite 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V76() V77() V78() V79() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

theorem :: FIB_NUM2:61
OddFibs 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V76() V77() V78() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) = <*1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ,2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) *> : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined Function-like finite 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like ) set ) ;

theorem :: FIB_NUM2:62
for k being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) holds (OddNAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Subset of ( ( ) ( non empty non empty-membered ) set ) ) /\ (Seg ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty finite (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) -element with_non-empty_elements non empty-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( finite with_non-empty_elements complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) set ) \/ {((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 5 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) } : ( ( ) ( non empty finite with_non-empty_elements non empty-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) : ( ( ) ( non empty non empty-membered ) set ) ) : ( ( ) ( non empty finite with_non-empty_elements non empty-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded ) set ) = OddNAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Subset of ( ( ) ( non empty non empty-membered ) set ) ) /\ (Seg ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 5 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty finite (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 5 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) -element with_non-empty_elements non empty-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) : ( ( ) ( non empty non empty-membered ) set ) ) : ( ( ) ( finite with_non-empty_elements complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) set ) ;

theorem :: FIB_NUM2:63
for k being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) holds (FIB : ( ( Function-like V29( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -valued Function-like V29( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) V76() V77() V78() V79() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) | (OddNAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Subset of ( ( ) ( non empty non empty-membered ) set ) ) /\ (Seg ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty finite (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) -element with_non-empty_elements non empty-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( finite with_non-empty_elements complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) set ) ) : ( ( Relation-like ) ( Relation-like OddNAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Subset of ( ( ) ( non empty non empty-membered ) set ) ) /\ (Seg ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty finite (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) -element with_non-empty_elements non empty-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) : ( ( ) ( non empty non empty-membered ) set ) ) : ( ( ) ( finite with_non-empty_elements complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) set ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined RAT : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered rational-membered V92() ) set ) -valued NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -valued Function-like finite FinSubsequence-like V76() V77() V78() V79() ) set ) \/ {[((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 5 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ,(FIB : ( ( Function-like V29( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -valued Function-like V29( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) V76() V77() V78() V79() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) . ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 5 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ] : ( ( ) ( ) Element of [:NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ,NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) :] : ( ( ) ( Relation-like RAT : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered rational-membered V92() ) set ) -valued INT : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered rational-membered integer-membered V92() ) set ) -valued V76() V77() V78() V79() ) set ) ) } : ( ( ) ( non empty Relation-like Function-like finite ) set ) : ( ( ) ( non empty Relation-like finite ) set ) = FIB : ( ( Function-like V29( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -valued Function-like V29( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) V76() V77() V78() V79() ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) , NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) | (OddNAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Subset of ( ( ) ( non empty non empty-membered ) set ) ) /\ (Seg ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 5 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty finite (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 5 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) -element with_non-empty_elements non empty-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( finite with_non-empty_elements complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) set ) : ( ( Relation-like ) ( Relation-like OddNAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Subset of ( ( ) ( non empty non empty-membered ) set ) ) /\ (Seg ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 5 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty finite (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 5 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) -element with_non-empty_elements non empty-membered complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) : ( ( ) ( non empty non empty-membered ) set ) ) : ( ( ) ( finite with_non-empty_elements complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) set ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined RAT : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered rational-membered V92() ) set ) -valued NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -valued Function-like finite FinSubsequence-like V76() V77() V78() V79() ) set ) ;

theorem :: FIB_NUM2:64
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) holds OddFibs ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V76() V77() V78() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) = (OddFibs ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered non even left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V76() V77() V78() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ^ <*(Fib ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) *> : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -valued Function-like finite 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) -element FinSequence-like FinSubsequence-like V76() V77() V78() V79() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like V76() V77() V78() V79() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

theorem :: FIB_NUM2:65
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) holds Sum (EvenFibs ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V76() V77() V78() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) ) = (Fib ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) - 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( V11() real integer ext-real ) set ) ;

theorem :: FIB_NUM2:66
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) holds Sum (OddFibs ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered non even left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V76() V77() V78() ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) ) = Fib ((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

begin

theorem :: FIB_NUM2:67
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) holds Fib n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) , Fib (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) are_relative_prime ;

theorem :: FIB_NUM2:68
for n being ( ( non empty natural ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real positive non negative ) Nat)
for m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) st m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) <> 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) & m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) divides Fib n : ( ( non empty natural ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real positive non negative ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) holds
not m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) divides Fib (n : ( ( non empty natural ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real positive non negative ) Nat) -' 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

theorem :: FIB_NUM2:69
for m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat)
for n being ( ( non empty natural ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real positive non negative ) Nat) st m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) is prime & n : ( ( non empty natural ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real positive non negative ) Nat) is prime & m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) divides Fib n : ( ( non empty natural ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real positive non negative ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) holds
for r being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) st r : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) < n : ( ( non empty natural ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real positive non negative ) Nat) & r : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) <> 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real integer Relation-like non-empty empty-yielding NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) -defined RAT : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered rational-membered V92() ) set ) -valued Function-like one-to-one constant functional finite finite-yielding V40() FinSequence-like FinSubsequence-like FinSequence-membered V47() ext-real non positive non negative V76() V77() V78() V79() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() bounded_below bounded_above real-bounded V120() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) holds
not m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) divides Fib r : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer ext-real non negative ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ;

begin

theorem :: FIB_NUM2:70
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) holds {((Fib n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * (Fib (n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 3 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ,((2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * (Fib (n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) * (Fib (n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered even bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ,(((Fib (n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ^2) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + ((Fib (n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) + 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ^2) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real integer V47() ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) ) } : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below bounded_above real-bounded ) Element of bool NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V92() left_end bounded_below ) Element of bool REAL : ( ( ) ( non empty non trivial non finite non empty-membered complex-membered ext-real-membered real-membered V92() non bounded_below non bounded_above V120() ) set ) : ( ( ) ( non empty non empty-membered ) set ) ) : ( ( ) ( non empty non empty-membered ) set ) ) is ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Pythagorean_triple ) ;