:: SINCOS10 semantic presentation
begin
theorem
:: SINCOS10:1
[.
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
V46
()
V47
()
V48
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
c=
dom
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:2
].
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
V46
()
V47
()
V48
()
V65
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
c=
dom
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:3
[.
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
.[
: ( ( ) (
V46
()
V47
()
V48
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
c=
dom
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:4
].
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
V46
()
V47
()
V48
()
V65
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
c=
dom
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:5
(
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
is_differentiable_on
].
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
open
V46
()
V47
()
V48
()
V65
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) & ( for
x
being ( ( ) (
V11
()
real
ext-real
)
Real
) st
x
: ( ( ) (
V11
()
real
ext-real
)
Real
)
in
].
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
open
V46
()
V47
()
V48
()
V65
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) holds
diff
(
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ,
x
: ( ( ) (
V11
()
real
ext-real
)
Real
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
(
sin
: ( (
Function-like
quasi_total
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
quasi_total
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
x
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
(
(
cos
: ( (
Function-like
quasi_total
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
quasi_total
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
x
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
^2
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ) ) ;
theorem
:: SINCOS10:6
(
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
is_differentiable_on
].
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
open
V46
()
V47
()
V48
()
V65
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) & ( for
x
being ( ( ) (
V11
()
real
ext-real
)
Real
) st
x
: ( ( ) (
V11
()
real
ext-real
)
Real
)
in
].
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
open
V46
()
V47
()
V48
()
V65
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) holds
diff
(
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ,
x
: ( ( ) (
V11
()
real
ext-real
)
Real
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
(
sin
: ( (
Function-like
quasi_total
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
quasi_total
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
x
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
(
(
cos
: ( (
Function-like
quasi_total
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
quasi_total
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
x
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
^2
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ) ) ;
theorem
:: SINCOS10:7
(
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
is_differentiable_on
].
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
.[
: ( ( ) (
open
V46
()
V47
()
V48
()
V65
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) & ( for
x
being ( ( ) (
V11
()
real
ext-real
)
Real
) st
x
: ( ( ) (
V11
()
real
ext-real
)
Real
)
in
].
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
.[
: ( ( ) (
open
V46
()
V47
()
V48
()
V65
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) holds
diff
(
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ,
x
: ( ( ) (
V11
()
real
ext-real
)
Real
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
-
(
(
cos
: ( (
Function-like
quasi_total
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
quasi_total
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
x
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
(
(
sin
: ( (
Function-like
quasi_total
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
quasi_total
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
x
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
^2
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ) ) ;
theorem
:: SINCOS10:8
(
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
is_differentiable_on
].
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
open
V46
()
V47
()
V48
()
V65
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) & ( for
x
being ( ( ) (
V11
()
real
ext-real
)
Real
) st
x
: ( ( ) (
V11
()
real
ext-real
)
Real
)
in
].
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
open
V46
()
V47
()
V48
()
V65
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) holds
diff
(
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ,
x
: ( ( ) (
V11
()
real
ext-real
)
Real
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
-
(
(
cos
: ( (
Function-like
quasi_total
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
quasi_total
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
x
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
(
(
sin
: ( (
Function-like
quasi_total
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
quasi_total
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
x
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
^2
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ) ) ;
theorem
:: SINCOS10:9
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
].
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
open
V46
()
V47
()
V48
()
V65
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
continuous
;
theorem
:: SINCOS10:10
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
].
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
open
V46
()
V47
()
V48
()
V65
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
continuous
;
theorem
:: SINCOS10:11
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
].
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
.[
: ( ( ) (
open
V46
()
V47
()
V48
()
V65
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
continuous
;
theorem
:: SINCOS10:12
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
].
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
open
V46
()
V47
()
V48
()
V65
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
continuous
;
theorem
:: SINCOS10:13
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
].
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
open
V46
()
V47
()
V48
()
V65
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
increasing
;
theorem
:: SINCOS10:14
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
].
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
open
V46
()
V47
()
V48
()
V65
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
increasing
;
theorem
:: SINCOS10:15
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
].
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
.[
: ( ( ) (
open
V46
()
V47
()
V48
()
V65
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
decreasing
;
theorem
:: SINCOS10:16
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
].
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
open
V46
()
V47
()
V48
()
V65
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
decreasing
;
theorem
:: SINCOS10:17
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
[.
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
V46
()
V47
()
V48
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
increasing
;
theorem
:: SINCOS10:18
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
].
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
V46
()
V47
()
V48
()
V65
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
increasing
;
theorem
:: SINCOS10:19
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
[.
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
.[
: ( ( ) (
V46
()
V47
()
V48
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
decreasing
;
theorem
:: SINCOS10:20
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
].
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
V46
()
V47
()
V48
()
V65
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
decreasing
;
theorem
:: SINCOS10:21
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
[.
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
V46
()
V47
()
V48
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
one-to-one
;
theorem
:: SINCOS10:22
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
].
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
V46
()
V47
()
V48
()
V65
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
one-to-one
;
theorem
:: SINCOS10:23
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
[.
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
.[
: ( ( ) (
V46
()
V47
()
V48
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
one-to-one
;
theorem
:: SINCOS10:24
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
].
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
V46
()
V47
()
V48
()
V65
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
one-to-one
;
registration
cluster
K77
(
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ,
[.
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
V46
()
V47
()
V48
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( (
V19
() ) (
V19
()
Function-like
V36
()
V37
()
V38
() )
set
)
->
V19
()
one-to-one
;
cluster
K77
(
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ,
].
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
V46
()
V47
()
V48
()
V65
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( (
V19
() ) (
V19
()
Function-like
V36
()
V37
()
V38
() )
set
)
->
V19
()
one-to-one
;
cluster
K77
(
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ,
[.
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
.[
: ( ( ) (
V46
()
V47
()
V48
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( (
V19
() ) (
V19
()
Function-like
V36
()
V37
()
V38
() )
set
)
->
V19
()
one-to-one
;
cluster
K77
(
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ,
].
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
V46
()
V47
()
V48
()
V65
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( (
V19
() ) (
V19
()
Function-like
V36
()
V37
()
V38
() )
set
)
->
V19
()
one-to-one
;
end;
definition
func
arcsec1
->
( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
PartFunc
of ,)
equals
:: SINCOS10:def 1
(
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
[.
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
V46
()
V47
()
V48
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
"
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ;
func
arcsec2
->
( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
PartFunc
of ,)
equals
:: SINCOS10:def 2
(
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
].
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
V46
()
V47
()
V48
()
V65
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
"
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ;
func
arccosec1
->
( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
PartFunc
of ,)
equals
:: SINCOS10:def 3
(
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
[.
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
.[
: ( ( ) (
V46
()
V47
()
V48
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
"
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ;
func
arccosec2
->
( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
PartFunc
of ,)
equals
:: SINCOS10:def 4
(
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
].
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
V46
()
V47
()
V48
()
V65
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
"
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ;
end;
definition
let
r
be ( ( ) (
V11
()
real
ext-real
)
Real
) ;
func
arcsec1
r
->
( ( ) ( )
set
)
equals
:: SINCOS10:def 5
arcsec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
PartFunc
of ,)
.
r
: ( ( ) ( )
set
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ;
func
arcsec2
r
->
( ( ) ( )
set
)
equals
:: SINCOS10:def 6
arcsec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
PartFunc
of ,)
.
r
: ( ( ) ( )
set
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ;
func
arccosec1
r
->
( ( ) ( )
set
)
equals
:: SINCOS10:def 7
arccosec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
PartFunc
of ,)
.
r
: ( ( ) ( )
set
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ;
func
arccosec2
r
->
( ( ) ( )
set
)
equals
:: SINCOS10:def 8
arccosec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
PartFunc
of ,)
.
r
: ( ( ) ( )
set
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ;
end;
definition
let
r
be ( ( ) (
V11
()
real
ext-real
)
Real
) ;
:: original:
arcsec1
redefine
func
arcsec1
r
->
( ( ) (
V11
()
real
ext-real
)
Real
) ;
:: original:
arcsec2
redefine
func
arcsec2
r
->
( ( ) (
V11
()
real
ext-real
)
Real
) ;
:: original:
arccosec1
redefine
func
arccosec1
r
->
( ( ) (
V11
()
real
ext-real
)
Real
) ;
:: original:
arccosec2
redefine
func
arccosec2
r
->
( ( ) (
V11
()
real
ext-real
)
Real
) ;
end;
theorem
:: SINCOS10:25
rng
arcsec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
PartFunc
of ,) : ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
=
[.
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
V46
()
V47
()
V48
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:26
rng
arcsec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
PartFunc
of ,) : ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
=
].
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
V46
()
V47
()
V48
()
V65
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:27
rng
arccosec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
PartFunc
of ,) : ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
=
[.
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
.[
: ( ( ) (
V46
()
V47
()
V48
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:28
rng
arccosec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
PartFunc
of ,) : ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
=
].
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
V46
()
V47
()
V48
()
V65
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
registration
cluster
arcsec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
PartFunc
of ,)
->
Function-like
one-to-one
;
cluster
arcsec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
PartFunc
of ,)
->
Function-like
one-to-one
;
cluster
arccosec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
PartFunc
of ,)
->
Function-like
one-to-one
;
cluster
arccosec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
PartFunc
of ,)
->
Function-like
one-to-one
;
end;
theorem
:: SINCOS10:29
(
sin
: ( (
Function-like
quasi_total
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
quasi_total
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
/
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) &
cos
: ( (
Function-like
quasi_total
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
quasi_total
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
/
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ) ;
theorem
:: SINCOS10:30
(
sin
: ( (
Function-like
quasi_total
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
quasi_total
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
-
(
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
/
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) &
cos
: ( (
Function-like
quasi_total
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
quasi_total
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
/
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) &
sin
: ( (
Function-like
quasi_total
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
quasi_total
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
(
(
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
*
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
/
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) &
cos
: ( (
Function-like
quasi_total
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
quasi_total
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
(
(
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
*
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
-
(
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
/
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ) ;
theorem
:: SINCOS10:31
(
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) &
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) &
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
(
(
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
*
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) &
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ) ;
theorem
:: SINCOS10:32
(
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) &
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) &
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) &
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ) ;
theorem
:: SINCOS10:33
for
x
being ( ( ) ( )
set
) st
x
: ( ( ) ( )
set
)
in
[.
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) holds
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
x
: ( ( ) ( )
set
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
in
[.
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:34
for
x
being ( ( ) ( )
set
) st
x
: ( ( ) ( )
set
)
in
[.
(
(
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
*
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) holds
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
x
: ( ( ) ( )
set
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
in
[.
(
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:35
for
x
being ( ( ) ( )
set
) st
x
: ( ( ) ( )
set
)
in
[.
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) holds
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
x
: ( ( ) ( )
set
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
in
[.
(
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:36
for
x
being ( ( ) ( )
set
) st
x
: ( ( ) ( )
set
)
in
[.
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) holds
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
x
: ( ( ) ( )
set
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
in
[.
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:37
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
[.
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
V46
()
V47
()
V48
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
continuous
;
theorem
:: SINCOS10:38
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
].
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
V46
()
V47
()
V48
()
V65
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
continuous
;
theorem
:: SINCOS10:39
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
[.
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
.[
: ( ( ) (
V46
()
V47
()
V48
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
continuous
;
theorem
:: SINCOS10:40
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
].
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
V46
()
V47
()
V48
()
V65
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
continuous
;
theorem
:: SINCOS10:41
rng
(
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
[.
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
=
[.
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:42
rng
(
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
[.
(
(
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
*
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
=
[.
(
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:43
rng
(
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
[.
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
=
[.
(
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:44
rng
(
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
[.
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
=
[.
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:45
[.
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
c=
dom
arcsec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,) : ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:46
[.
(
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
c=
dom
arcsec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,) : ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:47
[.
(
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
c=
dom
arccosec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,) : ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:48
[.
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
c=
dom
arccosec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,) : ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
registration
cluster
K77
(
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ,
[.
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( (
V19
() ) (
V19
()
Function-like
V36
()
V37
()
V38
() )
set
)
->
V19
()
one-to-one
;
cluster
K77
(
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ,
[.
(
(
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
*
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( (
V19
() ) (
V19
()
Function-like
V36
()
V37
()
V38
() )
set
)
->
V19
()
one-to-one
;
cluster
K77
(
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ,
[.
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( (
V19
() ) (
V19
()
Function-like
V36
()
V37
()
V38
() )
set
)
->
V19
()
one-to-one
;
cluster
K77
(
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ,
[.
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( (
V19
() ) (
V19
()
Function-like
V36
()
V37
()
V38
() )
set
)
->
V19
()
one-to-one
;
end;
theorem
:: SINCOS10:49
arcsec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
[.
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
=
(
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
[.
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
"
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:50
arcsec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
[.
(
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
=
(
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
[.
(
(
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
*
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
"
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:51
arccosec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
[.
(
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
=
(
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
[.
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
"
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:52
arccosec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
[.
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
=
(
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
[.
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
"
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:53
(
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
[.
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
*
(
arcsec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
[.
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
V19
() ) (
V19
()
Function-like
one-to-one
V36
()
V37
()
V38
() )
set
)
=
id
[.
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:54
(
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
[.
(
(
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
*
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
*
(
arcsec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
[.
(
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
V19
() ) (
V19
()
Function-like
one-to-one
V36
()
V37
()
V38
() )
set
)
=
id
[.
(
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:55
(
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
[.
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
*
(
arccosec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
[.
(
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
V19
() ) (
V19
()
Function-like
one-to-one
V36
()
V37
()
V38
() )
set
)
=
id
[.
(
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:56
(
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
[.
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
*
(
arccosec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
[.
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
V19
() ) (
V19
()
Function-like
one-to-one
V36
()
V37
()
V38
() )
set
)
=
id
[.
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:57
(
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
[.
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
*
(
arcsec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
[.
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
=
id
[.
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:58
(
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
[.
(
(
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
*
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
*
(
arcsec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
[.
(
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
=
id
[.
(
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:59
(
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
[.
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
*
(
arccosec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
[.
(
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
=
id
[.
(
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:60
(
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
[.
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
*
(
arccosec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
[.
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
=
id
[.
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:61
arcsec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
*
(
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
[.
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
V46
()
V47
()
V48
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
V19
() ) (
V19
()
Function-like
one-to-one
V36
()
V37
()
V38
() )
set
)
=
id
[.
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
V46
()
V47
()
V48
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:62
arcsec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
*
(
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
].
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
V46
()
V47
()
V48
()
V65
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
V19
() ) (
V19
()
Function-like
one-to-one
V36
()
V37
()
V38
() )
set
)
=
id
].
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
V46
()
V47
()
V48
()
V65
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:63
arccosec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
*
(
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
[.
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
.[
: ( ( ) (
V46
()
V47
()
V48
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
V19
() ) (
V19
()
Function-like
one-to-one
V36
()
V37
()
V38
() )
set
)
=
id
[.
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
.[
: ( ( ) (
V46
()
V47
()
V48
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:64
arccosec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
*
(
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
].
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
V46
()
V47
()
V48
()
V65
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
V19
() ) (
V19
()
Function-like
one-to-one
V36
()
V37
()
V38
() )
set
)
=
id
].
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
V46
()
V47
()
V48
()
V65
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:65
arcsec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
*
(
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
[.
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
V46
()
V47
()
V48
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
=
id
[.
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
V46
()
V47
()
V48
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:66
arcsec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
*
(
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
].
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
V46
()
V47
()
V48
()
V65
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
=
id
].
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
V46
()
V47
()
V48
()
V65
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:67
arccosec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
*
(
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
[.
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
.[
: ( ( ) (
V46
()
V47
()
V48
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
=
id
[.
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
.[
: ( ( ) (
V46
()
V47
()
V48
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:68
arccosec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
*
(
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
|
].
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
V46
()
V47
()
V48
()
V65
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
=
id
].
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
V46
()
V47
()
V48
()
V65
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:69
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
<=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) holds
arcsec1
(
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Real
)
=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) ;
theorem
:: SINCOS10:70
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) st
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
<
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<=
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) holds
arcsec2
(
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Real
)
=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) ;
theorem
:: SINCOS10:71
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) st
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
<=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) holds
arccosec1
(
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Real
)
=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) ;
theorem
:: SINCOS10:72
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) st
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
<
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<=
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) holds
arccosec2
(
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Real
)
=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) ;
theorem
:: SINCOS10:73
(
arcsec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
.
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) &
arcsec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
.
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ) ;
theorem
:: SINCOS10:74
(
arcsec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
.
(
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
(
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
*
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) &
arcsec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
.
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ) ;
theorem
:: SINCOS10:75
(
arccosec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
.
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) &
arccosec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
.
(
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ) ;
theorem
:: SINCOS10:76
(
arccosec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
.
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) &
arccosec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
.
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ) ;
theorem
:: SINCOS10:77
arcsec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
(
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.:
[.
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
V46
()
V47
()
V48
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
increasing
;
theorem
:: SINCOS10:78
arcsec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
(
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.:
].
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
V46
()
V47
()
V48
()
V65
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
increasing
;
theorem
:: SINCOS10:79
arccosec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
(
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.:
[.
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
.[
: ( ( ) (
V46
()
V47
()
V48
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
decreasing
;
theorem
:: SINCOS10:80
arccosec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
(
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.:
].
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
V46
()
V47
()
V48
()
V65
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
decreasing
;
theorem
:: SINCOS10:81
arcsec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
[.
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
increasing
;
theorem
:: SINCOS10:82
arcsec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
[.
(
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
increasing
;
theorem
:: SINCOS10:83
arccosec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
[.
(
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
decreasing
;
theorem
:: SINCOS10:84
arccosec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
[.
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
decreasing
;
theorem
:: SINCOS10:85
for
x
being ( ( ) ( )
set
) st
x
: ( ( ) ( )
set
)
in
[.
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) holds
arcsec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
.
x
: ( ( ) ( )
set
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
in
[.
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:86
for
x
being ( ( ) ( )
set
) st
x
: ( ( ) ( )
set
)
in
[.
(
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) holds
arcsec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
.
x
: ( ( ) ( )
set
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
in
[.
(
(
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
*
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:87
for
x
being ( ( ) ( )
set
) st
x
: ( ( ) ( )
set
)
in
[.
(
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) holds
arccosec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
.
x
: ( ( ) ( )
set
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
in
[.
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:88
for
x
being ( ( ) ( )
set
) st
x
: ( ( ) ( )
set
)
in
[.
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) holds
arccosec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
.
x
: ( ( ) ( )
set
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
in
[.
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:89
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) st 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
<=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<=
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) holds
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
(
arcsec1
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) ;
theorem
:: SINCOS10:90
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) st
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
<=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<=
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) holds
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
(
arcsec2
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) ;
theorem
:: SINCOS10:91
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) st
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
<=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<=
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) holds
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
(
arccosec1
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) ;
theorem
:: SINCOS10:92
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) st 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
<=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<=
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) holds
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
(
arccosec2
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) ;
theorem
:: SINCOS10:93
arcsec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
[.
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
continuous
;
theorem
:: SINCOS10:94
arcsec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
[.
(
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
continuous
;
theorem
:: SINCOS10:95
arccosec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
[.
(
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
continuous
;
theorem
:: SINCOS10:96
arccosec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
[.
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
continuous
;
theorem
:: SINCOS10:97
rng
(
arcsec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
[.
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
=
[.
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:98
rng
(
arcsec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
[.
(
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
=
[.
(
(
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
*
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:99
rng
(
arccosec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
[.
(
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
=
[.
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:100
rng
(
arccosec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
[.
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
=
[.
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.]
: ( ( ) (
closed
V46
()
V47
()
V48
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:101
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) holds
( ( 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
<=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<=
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) &
arcsec1
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Real
)
=
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) implies
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ) & ( 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
<=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<=
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) &
arcsec1
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Real
)
=
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) implies
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
=
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ) ) ;
theorem
:: SINCOS10:102
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) holds
( (
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
<=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<=
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) &
arcsec2
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Real
)
=
(
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
*
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) implies
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
=
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ) & (
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
<=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<=
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) &
arcsec2
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Real
)
=
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) implies
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
=
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ) ) ;
theorem
:: SINCOS10:103
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) holds
( (
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
<=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<=
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) &
arccosec1
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Real
)
=
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) implies
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
=
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ) & (
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
<=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<=
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) &
arccosec1
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Real
)
=
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) implies
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
=
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ) ) ;
theorem
:: SINCOS10:104
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) holds
( ( 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
<=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<=
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) &
arccosec2
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Real
)
=
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) implies
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
=
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ) & ( 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
<=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<=
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) &
arccosec2
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Real
)
=
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) implies
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ) ) ;
theorem
:: SINCOS10:105
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) st 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
<=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<=
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) holds
(
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
<=
arcsec1
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Real
) &
arcsec1
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Real
)
<=
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ) ;
theorem
:: SINCOS10:106
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) st
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
<=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<=
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) holds
(
(
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
*
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
<=
arcsec2
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Real
) &
arcsec2
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Real
)
<=
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ) ;
theorem
:: SINCOS10:107
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) st
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
<=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<=
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) holds
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
<=
arccosec1
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Real
) &
arccosec1
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Real
)
<=
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ) ;
theorem
:: SINCOS10:108
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) st 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
<=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<=
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) holds
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
<=
arccosec2
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Real
) &
arccosec2
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Real
)
<=
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ) ;
theorem
:: SINCOS10:109
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) st 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
<
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) holds
(
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
<
arcsec1
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Real
) &
arcsec1
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Real
)
<
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ) ;
theorem
:: SINCOS10:110
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) st
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
<
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) holds
(
(
3 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
*
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
<
arcsec2
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Real
) &
arcsec2
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Real
)
<
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ) ;
theorem
:: SINCOS10:111
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) st
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
<
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) holds
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
<
arccosec1
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Real
) &
arccosec1
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Real
)
<
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ) ;
theorem
:: SINCOS10:112
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) st 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
<
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) holds
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
4 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
<
arccosec2
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Real
) &
arccosec2
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Real
)
<
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ) ;
theorem
:: SINCOS10:113
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) st 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
<=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<=
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) holds
(
sin
: ( (
Function-like
quasi_total
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
quasi_total
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
(
arcsec1
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
(
sqrt
(
(
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
^2
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) &
cos
: ( (
Function-like
quasi_total
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
quasi_total
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
(
arcsec1
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
/
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ) ;
theorem
:: SINCOS10:114
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) st
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
<=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<=
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) holds
(
sin
: ( (
Function-like
quasi_total
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
quasi_total
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
(
arcsec2
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
-
(
(
sqrt
(
(
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
^2
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) &
cos
: ( (
Function-like
quasi_total
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
quasi_total
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
(
arcsec2
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
/
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ) ;
theorem
:: SINCOS10:115
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) st
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
<=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<=
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) holds
(
sin
: ( (
Function-like
quasi_total
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
quasi_total
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
(
arccosec1
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
/
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) &
cos
: ( (
Function-like
quasi_total
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
quasi_total
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
(
arccosec1
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
-
(
(
sqrt
(
(
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
^2
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ) ;
theorem
:: SINCOS10:116
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) st 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
<=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<=
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) holds
(
sin
: ( (
Function-like
quasi_total
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
quasi_total
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
(
arccosec2
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
/
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) &
cos
: ( (
Function-like
quasi_total
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
quasi_total
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
(
arccosec2
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
(
sqrt
(
(
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
^2
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ) ;
theorem
:: SINCOS10:117
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) st 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
<
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) holds
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
(
arcsec1
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
/
(
sqrt
(
(
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
^2
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ;
theorem
:: SINCOS10:118
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) st
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
<
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) holds
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
(
arcsec2
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
-
(
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
/
(
sqrt
(
(
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
^2
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ;
theorem
:: SINCOS10:119
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) st
-
(
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
<
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
V71
() )
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) holds
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
(
arccosec1
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
-
(
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
/
(
sqrt
(
(
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
^2
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ;
theorem
:: SINCOS10:120
for
r
being ( ( ) (
V11
()
real
ext-real
)
Real
) st 1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
<
r
: ( ( ) (
V11
()
real
ext-real
)
Real
) &
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
<
sqrt
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) holds
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.
(
arccosec2
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
)
: ( ( ) (
V11
()
real
ext-real
)
Real
) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
=
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
/
(
sqrt
(
(
r
: ( ( ) (
V11
()
real
ext-real
)
Real
)
^2
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
-
1 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V11
()
real
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ;
theorem
:: SINCOS10:121
arcsec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
is_differentiable_on
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.:
].
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
open
V46
()
V47
()
V48
()
V65
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:122
arcsec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
is_differentiable_on
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.:
].
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
open
V46
()
V47
()
V48
()
V65
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:123
arccosec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
is_differentiable_on
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.:
].
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
.[
: ( ( ) (
open
V46
()
V47
()
V48
()
V65
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:124
arccosec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
is_differentiable_on
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.:
].
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
open
V46
()
V47
()
V48
()
V65
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ;
theorem
:: SINCOS10:125
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.:
].
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
open
V46
()
V47
()
V48
()
V65
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) is
open
;
theorem
:: SINCOS10:126
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.:
].
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
open
V46
()
V47
()
V48
()
V65
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) is
open
;
theorem
:: SINCOS10:127
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.:
].
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
.[
: ( ( ) (
open
V46
()
V47
()
V48
()
V65
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) is
open
;
theorem
:: SINCOS10:128
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.:
].
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
open
V46
()
V47
()
V48
()
V65
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) is
open
;
theorem
:: SINCOS10:129
arcsec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
(
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.:
].
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
open
V46
()
V47
()
V48
()
V65
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
continuous
;
theorem
:: SINCOS10:130
arcsec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
(
sec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.:
].
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
open
V46
()
V47
()
V48
()
V65
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
continuous
;
theorem
:: SINCOS10:131
arccosec1
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
(
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.:
].
(
-
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
non
positive
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) ,
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
.[
: ( ( ) (
open
V46
()
V47
()
V48
()
V65
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
continuous
;
theorem
:: SINCOS10:132
arccosec2
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
PartFunc
of ,)
|
(
cosec
: ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) )
.:
].
0
: ( ( ) (
empty
epsilon-transitive
epsilon-connected
ordinal
T-Sequence-like
c=-linear
natural
V11
()
real
ext-real
non
positive
non
negative
Function-like
functional
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V67
()
V70
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) ) ,
(
PI
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
/
2 : ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
natural
V11
()
real
ext-real
positive
non
negative
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V65
()
V67
()
V71
() )
Element
of
NAT
: ( ( ) ( non
empty
epsilon-transitive
epsilon-connected
ordinal
V46
()
V47
()
V48
()
V49
()
V50
()
V51
()
V52
()
V65
()
V67
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) )
)
: ( ( ) ( non
empty
V11
()
real
ext-real
positive
non
negative
)
Element
of
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
.[
: ( ( ) (
open
V46
()
V47
()
V48
()
V65
()
V66
()
V70
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) )
)
: ( ( ) (
V46
()
V47
()
V48
() )
Element
of
K6
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) ( )
set
) ) : ( (
Function-like
) (
V19
()
V22
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
V23
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) )
Function-like
one-to-one
V36
()
V37
()
V38
() )
Element
of
K6
(
K7
(
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ,
REAL
: ( ( ) ( non
empty
V46
()
V47
()
V48
()
V52
()
V67
()
V68
()
V70
()
V73
() )
set
) ) : ( ( ) (
V36
()
V37
()
V38
() )
set
) ) : ( ( ) ( )
set
) ) is
continuous
;