:: FIB_NUM3 semantic presentation
begin
theorem
:: FIB_NUM3:1
for
a
being ( (
real
) (
V24
()
ext-real
real
)
number
)
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
a
: ( (
real
) (
V24
()
ext-real
real
)
number
)
to_power
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( (
real
) (
V24
()
ext-real
real
)
set
)
=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V24
()
ext-real
non
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
a
: ( (
real
) (
V24
()
ext-real
real
)
number
)
=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V24
()
ext-real
non
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:2
for
a
being ( ( non
negative
real
) (
V24
()
ext-real
non
negative
real
)
number
) holds
(
sqrt
a
: ( ( non
negative
real
) (
V24
()
ext-real
non
negative
real
)
number
)
)
: ( (
real
) (
V24
()
ext-real
real
)
set
)
*
(
sqrt
a
: ( ( non
negative
real
) (
V24
()
ext-real
non
negative
real
)
number
)
)
: ( (
real
) (
V24
()
ext-real
real
)
set
) : ( ( ) (
V24
()
ext-real
real
)
set
)
=
a
: ( ( non
negative
real
) (
V24
()
ext-real
non
negative
real
)
number
) ;
theorem
:: FIB_NUM3:3
for
a
being ( (
real
) (
V24
()
ext-real
real
)
number
) holds
a
: ( (
real
) (
V24
()
ext-real
real
)
number
)
to_power
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( (
real
) (
V24
()
ext-real
real
)
set
)
=
(
-
a
: ( (
real
) (
V24
()
ext-real
real
)
number
)
)
: ( (
V24
() ) (
V24
()
ext-real
real
)
set
)
to_power
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( (
real
) (
V24
()
ext-real
real
)
set
) ;
theorem
:: FIB_NUM3:4
for
k
being ( ( non
empty
natural
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
)
Nat
) holds
(
k
: ( ( non
empty
natural
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
)
Nat
)
-'
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
(
k
: ( ( non
empty
natural
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
)
Nat
)
+
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
-'
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:5
for
a
,
b
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
|^
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
(
(
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:6
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
for
a
being ( ( non
empty
real
) ( non
empty
V24
()
ext-real
real
)
number
) holds
(
a
: ( ( non
empty
real
) ( non
empty
V24
()
ext-real
real
)
number
)
to_power
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
real
) (
V24
()
ext-real
real
)
set
)
to_power
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( (
real
) (
V24
()
ext-real
real
)
set
)
=
a
: ( ( non
empty
real
) ( non
empty
V24
()
ext-real
real
)
number
)
to_power
(
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( (
real
) (
V24
()
ext-real
real
)
set
) ;
theorem
:: FIB_NUM3:7
for
a
,
b
being ( (
real
) (
V24
()
ext-real
real
)
number
) holds
(
a
: ( (
real
) (
V24
()
ext-real
real
)
number
)
+
b
: ( (
real
) (
V24
()
ext-real
real
)
number
)
)
: ( ( ) (
V24
()
ext-real
real
)
set
)
*
(
a
: ( (
real
) (
V24
()
ext-real
real
)
number
)
-
b
: ( (
real
) (
V24
()
ext-real
real
)
number
)
)
: ( ( ) (
V24
()
ext-real
real
)
set
) : ( ( ) (
V24
()
ext-real
real
)
set
)
=
(
a
: ( (
real
) (
V24
()
ext-real
real
)
number
)
to_power
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
real
) (
V24
()
ext-real
real
)
set
)
-
(
b
: ( (
real
) (
V24
()
ext-real
real
)
number
)
to_power
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
real
) (
V24
()
ext-real
real
)
set
) : ( ( ) (
V24
()
ext-real
real
)
set
) ;
theorem
:: FIB_NUM3:8
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
for
a
,
b
being ( ( non
empty
real
) ( non
empty
V24
()
ext-real
real
)
number
) holds
(
a
: ( ( non
empty
real
) ( non
empty
V24
()
ext-real
real
)
number
)
*
b
: ( ( non
empty
real
) ( non
empty
V24
()
ext-real
real
)
number
)
)
: ( ( ) ( non
empty
V24
()
ext-real
real
)
set
)
to_power
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( (
real
) (
V24
()
ext-real
real
)
set
)
=
(
a
: ( ( non
empty
real
) ( non
empty
V24
()
ext-real
real
)
number
)
to_power
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
real
) (
V24
()
ext-real
real
)
set
)
*
(
b
: ( ( non
empty
real
) ( non
empty
V24
()
ext-real
real
)
number
)
to_power
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
real
) (
V24
()
ext-real
real
)
set
) : ( ( ) (
V24
()
ext-real
real
)
set
) ;
registration
cluster
tau
: ( (
real
) (
V24
()
ext-real
real
)
set
)
->
positive
real
;
cluster
tau_bar
: ( (
real
) (
V24
()
ext-real
real
)
set
)
->
negative
real
;
end;
theorem
:: FIB_NUM3:9
for
n
being ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) holds
(
tau
: ( (
real
) ( non
empty
V24
()
ext-real
positive
non
negative
real
)
set
)
to_power
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
)
: ( (
real
) (
V24
()
ext-real
real
)
set
)
+
(
tau
: ( (
real
) ( non
empty
V24
()
ext-real
positive
non
negative
real
)
set
)
to_power
(
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
real
) (
V24
()
ext-real
real
)
set
) : ( ( ) (
V24
()
ext-real
real
)
set
)
=
tau
: ( (
real
) ( non
empty
V24
()
ext-real
positive
non
negative
real
)
set
)
to_power
(
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( (
real
) (
V24
()
ext-real
real
)
set
) ;
theorem
:: FIB_NUM3:10
for
n
being ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) holds
(
tau_bar
: ( (
real
) ( non
empty
V24
()
ext-real
non
positive
negative
real
)
set
)
to_power
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
)
: ( (
real
) (
V24
()
ext-real
real
)
set
)
+
(
tau_bar
: ( (
real
) ( non
empty
V24
()
ext-real
non
positive
negative
real
)
set
)
to_power
(
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
real
) (
V24
()
ext-real
real
)
set
) : ( ( ) (
V24
()
ext-real
real
)
set
)
=
tau_bar
: ( (
real
) ( non
empty
V24
()
ext-real
non
positive
negative
real
)
set
)
to_power
(
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( (
real
) (
V24
()
ext-real
real
)
set
) ;
begin
definition
let
n
be ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) ;
func
Lucas
n
->
( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
means
:: FIB_NUM3:def 1
ex
L
being ( (
V12
()
V50
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) (
V12
()
V50
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
Function
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
(
it
: ( (
V12
()
V50
(
n
: ( ( ) ( )
set
) ,
n
: ( ( ) ( )
set
) ) ) (
V12
()
V50
(
n
: ( ( ) ( )
set
) ,
n
: ( ( ) ( )
set
) ) )
Element
of
K32
(
K33
(
n
: ( ( ) ( )
set
) ,
n
: ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) )
=
(
L
: ( (
V12
()
V50
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) (
V12
()
V50
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
Function
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
.
n
: ( ( ) ( )
set
)
)
: ( ( ) ( )
Element
of
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
`1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) &
L
: ( (
V12
()
V50
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) (
V12
()
V50
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
Function
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
.
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V24
()
ext-real
non
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( )
Element
of
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
[
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
]
: ( ( ) ( )
Element
of
K33
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
)
set
) ) & ( for
n
being ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) holds
L
: ( (
V12
()
V50
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) (
V12
()
V50
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
Function
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
.
(
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( )
Element
of
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
[
(
(
L
: ( (
V12
()
V50
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) (
V12
()
V50
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
Function
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
.
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
)
: ( ( ) ( )
Element
of
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
`2
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
(
(
L
: ( (
V12
()
V50
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) (
V12
()
V50
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
Function
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
.
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
)
: ( ( ) ( )
Element
of
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
`1
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
(
L
: ( (
V12
()
V50
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) (
V12
()
V50
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
Function
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
.
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
)
: ( ( ) ( )
Element
of
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
`2
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
]
: ( ( ) ( )
Element
of
K33
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
)
set
) ) ) );
end;
theorem
:: FIB_NUM3:11
(
Lucas
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V24
()
ext-real
non
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) &
Lucas
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) & ( for
n
being ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) holds
Lucas
(
(
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
(
Lucas
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
Lucas
(
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) ) ;
theorem
:: FIB_NUM3:12
for
n
being ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) holds
Lucas
(
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
(
Lucas
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
Lucas
(
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:13
for
n
being ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) holds
(
Lucas
(
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
Lucas
(
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
Lucas
(
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:14
Lucas
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:15
Lucas
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:16
Lucas
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
7 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:17
for
k
being ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) holds
Lucas
k
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
k
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) ;
theorem
:: FIB_NUM3:18
for
m
being ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
Lucas
(
m
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
>=
Lucas
m
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
registration
let
n
be ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
cluster
Lucas
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
->
positive
;
end;
theorem
:: FIB_NUM3:19
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds 2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
Lucas
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
(
Lucas
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
Lucas
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:20
for
n
being ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) holds
Lucas
(
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
(
Fib
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
Fib
(
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:21
for
n
being ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) holds
Lucas
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
(
tau
: ( (
real
) ( non
empty
V24
()
ext-real
positive
non
negative
real
)
set
)
to_power
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
)
: ( (
real
) (
V24
()
ext-real
real
)
set
)
+
(
tau_bar
: ( (
real
) ( non
empty
V24
()
ext-real
non
positive
negative
real
)
set
)
to_power
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
)
: ( (
real
) (
V24
()
ext-real
real
)
set
) : ( ( ) (
V24
()
ext-real
real
)
set
) ;
theorem
:: FIB_NUM3:22
for
n
being ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) holds
(
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
Lucas
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
Lucas
(
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
5 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
Fib
(
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:23
for
n
being ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) holds
(
Lucas
(
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
-
(
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
Lucas
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V24
()
ext-real
real
)
set
)
=
5 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
Fib
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:24
for
n
being ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) holds
(
Lucas
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
Fib
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
Fib
(
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:25
for
n
being ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) holds
(
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
Fib
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
Lucas
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
Fib
(
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:26
for
n
,
m
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds 2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
Lucas
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
m
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
(
(
Lucas
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
Lucas
m
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
(
5 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
Fib
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
Fib
m
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:27
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
(
Lucas
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
Lucas
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
(
(
Lucas
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
|^
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
-
(
(
Lucas
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
|^
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V24
()
ext-real
real
)
set
) ;
theorem
:: FIB_NUM3:28
for
n
being ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) holds
Fib
(
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
(
Fib
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
Lucas
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:29
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds 2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
Fib
(
(
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
(
(
Lucas
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
Fib
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
(
Lucas
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
Fib
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:30
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
(
5 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
(
Fib
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
|^
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
-
(
(
Lucas
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
|^
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V24
()
ext-real
real
)
set
)
=
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
V24
() ) ( non
empty
V24
()
ext-real
non
positive
negative
real
)
set
)
to_power
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
real
) (
V24
()
ext-real
real
)
set
) : ( ( ) (
V24
()
ext-real
real
)
set
) ;
theorem
:: FIB_NUM3:31
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
Fib
(
(
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
(
(
Fib
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
Lucas
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
-
(
(
Fib
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
Lucas
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V24
()
ext-real
real
)
set
) ;
begin
definition
let
a
,
b
be ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) ;
:: original:
[
redefine
func
[
a
,
b
]
->
( ( ) ( )
Element
of
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
end;
definition
let
a
,
b
,
n
be ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) ;
func
GenFib
(
a
,
b
,
n
)
->
( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
means
:: FIB_NUM3:def 2
ex
L
being ( (
V12
()
V50
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) (
V12
()
V50
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
Function
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) st
(
it
: ( ( ) ( )
Element
of
K32
(
K32
(
a
: ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
=
(
L
: ( (
V12
()
V50
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) (
V12
()
V50
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
Function
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
.
n
: ( (
V12
()
V50
(
a
: ( ( ) ( )
set
) ,
b
: ( (
V12
()
V50
(
a
: ( ( ) ( )
set
) ,
a
: ( ( ) ( )
set
) ) ) (
V12
()
V50
(
a
: ( ( ) ( )
set
) ,
a
: ( ( ) ( )
set
) ) )
Element
of
K32
(
K33
(
a
: ( ( ) ( )
set
) ,
a
: ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) (
V12
()
V50
(
a
: ( ( ) ( )
set
) ,
b
: ( (
V12
()
V50
(
a
: ( ( ) ( )
set
) ,
a
: ( ( ) ( )
set
) ) ) (
V12
()
V50
(
a
: ( ( ) ( )
set
) ,
a
: ( ( ) ( )
set
) ) )
Element
of
K32
(
K33
(
a
: ( ( ) ( )
set
) ,
a
: ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
Element
of
K32
(
K33
(
a
: ( ( ) ( )
set
) ,
b
: ( (
V12
()
V50
(
a
: ( ( ) ( )
set
) ,
a
: ( ( ) ( )
set
) ) ) (
V12
()
V50
(
a
: ( ( ) ( )
set
) ,
a
: ( ( ) ( )
set
) ) )
Element
of
K32
(
K33
(
a
: ( ( ) ( )
set
) ,
a
: ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( )
Element
of
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
`1
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) &
L
: ( (
V12
()
V50
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) (
V12
()
V50
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
Function
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
.
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V24
()
ext-real
non
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( )
Element
of
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
[
a
: ( ( ) ( )
set
) ,
b
: ( (
V12
()
V50
(
a
: ( ( ) ( )
set
) ,
a
: ( ( ) ( )
set
) ) ) (
V12
()
V50
(
a
: ( ( ) ( )
set
) ,
a
: ( ( ) ( )
set
) ) )
Element
of
K32
(
K33
(
a
: ( ( ) ( )
set
) ,
a
: ( ( ) ( )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) )
]
: ( ( ) ( )
Element
of
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) & ( for
n
being ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) holds
L
: ( (
V12
()
V50
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) (
V12
()
V50
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
Function
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
.
(
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( )
Element
of
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
[
(
(
L
: ( (
V12
()
V50
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) (
V12
()
V50
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
Function
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
.
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
)
: ( ( ) ( )
Element
of
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
`2
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
(
(
L
: ( (
V12
()
V50
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) (
V12
()
V50
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
Function
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
.
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
)
: ( ( ) ( )
Element
of
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
`1
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
(
L
: ( (
V12
()
V50
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) (
V12
()
V50
(
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
Function
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
.
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
)
: ( ( ) ( )
Element
of
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
`2
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
]
: ( ( ) ( )
Element
of
[:
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ,
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) )
:]
: ( ( ) ( non
empty
)
Element
of
K32
(
K33
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ,
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) );
end;
theorem
:: FIB_NUM3:32
for
a
,
b
being ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) holds
(
GenFib
(
a
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) ,
b
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) ,
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V24
()
ext-real
non
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
a
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) &
GenFib
(
a
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) ,
b
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) ,1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
b
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) & ( for
n
being ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) holds
GenFib
(
a
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) ,
b
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) ,
(
(
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
(
GenFib
(
a
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) ,
b
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) ,
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
GenFib
(
a
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) ,
b
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) ,
(
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) ) ;
theorem
:: FIB_NUM3:33
for
a
,
b
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
for
k
being ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) holds
(
(
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
k
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
(
k
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
|^
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
(
(
(
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
k
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
|^
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
(
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
k
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
(
k
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
(
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
(
k
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
|^
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:34
for
a
,
b
,
n
being ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) holds
(
GenFib
(
a
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) ,
b
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) ,
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
GenFib
(
a
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) ,
b
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) ,
(
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
GenFib
(
a
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) ,
b
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) ,
(
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:35
for
a
,
b
,
n
being ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) holds
(
GenFib
(
a
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) ,
b
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) ,
(
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
GenFib
(
a
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) ,
b
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) ,
(
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
GenFib
(
a
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) ,
b
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) ,
(
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:36
for
a
,
b
,
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
(
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:37
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
GenFib
(
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V24
()
ext-real
non
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
Fib
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:38
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
GenFib
(2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
Lucas
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:39
for
a
,
b
,
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
(
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:40
for
a
,
b
,
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
(
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:41
for
a
,
b
,
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
(
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
-
(
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V24
()
ext-real
real
)
set
)
=
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:42
for
a
,
b
,
n
being ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
GenFib
(
a
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
(
(
GenFib
(
a
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V24
()
ext-real
non
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
Fib
(
n
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
-'
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
(
GenFib
(
a
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
Fib
n
: ( ( non
empty
) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:43
for
n
,
m
being ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
) holds
(
(
Fib
m
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
Lucas
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
(
Lucas
m
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
Fib
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
GenFib
(
(
Fib
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V24
()
ext-real
non
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
Lucas
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V24
()
ext-real
non
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
n
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
+
m
: ( (
natural
) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
Nat
)
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
)
set
) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:44
for
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
(
Lucas
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
Lucas
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
Lucas
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:45
for
a
,
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
(
(
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V24
()
ext-real
non
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V24
()
ext-real
non
negative
real
)
set
)
*
(
(
Fib
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
Lucas
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V24
()
ext-real
non
negative
real
)
set
) ;
theorem
:: FIB_NUM3:46
for
a
,
b
,
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
GenFib
(
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:47
for
a
,
b
,
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
(
(
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
-
(
(
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
|^
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
V24
()
ext-real
real
)
set
)
=
(
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
V24
() ) ( non
empty
V24
()
ext-real
non
positive
negative
real
)
set
)
to_power
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
real
) (
V24
()
ext-real
real
)
set
)
*
(
(
(
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
|^
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
-
(
(
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V24
()
ext-real
real
)
set
) : ( ( ) (
V24
()
ext-real
real
)
set
) ;
theorem
:: FIB_NUM3:48
for
a
,
b
,
k
,
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
GenFib
(
(
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
k
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
k
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
k
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:49
for
a
,
b
,
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
Fib
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
Fib
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:50
for
b
,
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
GenFib
(
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V24
()
ext-real
non
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
Fib
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:51
for
a
,
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V24
()
ext-real
non
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
Fib
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:52
for
a
,
b
,
c
,
d
,
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
(
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
(
GenFib
(
c
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
d
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
GenFib
(
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
c
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
d
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:53
for
a
,
b
,
k
,
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
GenFib
(
(
k
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
k
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
k
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: FIB_NUM3:54
for
a
,
b
,
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
GenFib
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
(
(
(
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
-
tau_bar
: ( (
real
) ( non
empty
V24
()
ext-real
non
positive
negative
real
)
set
)
)
: ( (
V24
() ) ( non
empty
V24
()
ext-real
positive
non
negative
real
)
set
)
)
: ( ( ) (
V24
()
ext-real
non
negative
real
)
set
)
+
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V24
()
ext-real
non
negative
real
)
set
)
*
(
tau
: ( (
real
) ( non
empty
V24
()
ext-real
positive
non
negative
real
)
set
)
to_power
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
real
) (
V24
()
ext-real
real
)
set
)
)
: ( ( ) (
V24
()
ext-real
real
)
set
)
+
(
(
(
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
tau
: ( (
real
) ( non
empty
V24
()
ext-real
positive
non
negative
real
)
set
)
)
: ( ( ) (
V24
()
ext-real
non
negative
real
)
set
)
-
b
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V24
()
ext-real
real
)
set
)
*
(
tau_bar
: ( (
real
) ( non
empty
V24
()
ext-real
non
positive
negative
real
)
set
)
to_power
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( (
real
) (
V24
()
ext-real
real
)
set
)
)
: ( ( ) (
V24
()
ext-real
real
)
set
)
)
: ( ( ) (
V24
()
ext-real
real
)
set
)
/
(
sqrt
5 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
V24
()
ext-real
real
)
Element
of
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) (
V24
()
ext-real
real
)
set
) ;
theorem
:: FIB_NUM3:55
for
a
,
n
being ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) holds
GenFib
(
(
(
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
(
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ,
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
=
(
(
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
a
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
*
(
Fib
(
n
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
+
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
positive
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) )
)
: ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) : ( ( ) (
epsilon-transitive
epsilon-connected
ordinal
natural
V24
()
ext-real
non
negative
real
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V36
()
V53
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V29
()
V30
()
V31
()
V32
()
V33
()
V34
()
V35
() )
Element
of
K32
(
REAL
: ( ( ) (
V29
()
V30
()
V31
()
V35
() )
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ;