begin
theorem
for
n being ( (
V21() ) (
V15()
V16()
V17()
V21()
V23()
ext-real V68() )
Nat)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) / f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) / f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
#Z n : ( (
V21() ) (
V15()
V16()
V17()
V21()
V23()
ext-real V68() )
Nat) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= ((f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) #Z n : ( ( V21() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
/ ((f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) #Z n : ( ( V21() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ;
theorem
for
a,
b being ( ( ) (
V22()
V23()
ext-real )
Real)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) / f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
PartFunc of ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= x : ( ( ) (
V22()
V23()
ext-real )
Real)
+ a : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
PartFunc of ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= x : ( ( ) (
V22()
V23()
ext-real )
Real)
- b : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
PartFunc of ,)
/ f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) / f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= ((- a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) - b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
/ ((x : ( ( ) ( V22() V23() ext-real ) Real) - b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st not
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
ln : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= - (1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
a,
b being ( ( ) (
V22()
V23()
ext-real )
Real)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
PartFunc of ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= (a : ( ( ) ( V22() V23() ext-real ) Real) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
+ b : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) holds
(
tan : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= a : ( ( ) (
V22()
V23()
ext-real )
Real)
/ ((cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . ((a : ( ( ) ( V22() V23() ext-real ) Real) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) + b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
a,
b being ( ( ) (
V22()
V23()
ext-real )
Real)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
PartFunc of ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= (a : ( ( ) ( V22() V23() ext-real ) Real) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
+ b : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) holds
(
cot : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= - (a : ( ( ) ( V22() V23() ext-real ) Real) / ((sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . ((a : ( ( ) ( V22() V23() ext-real ) Real) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) + b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st not
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
tan : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= - (1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / ((x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) * ((cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st not
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cot : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= 1 : ( ( ) (
V11()
V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
/ ((x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) * ((sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
c,
a,
b being ( ( ) (
V22()
V23()
ext-real )
Real)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) + (c : ( ( ) ( V22() V23() ext-real ) Real) (#) f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
PartFunc of ,)
= #Z 2 : ( ( ) (
V11()
V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V6()
V30(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V30(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
PartFunc of ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= a : ( ( ) (
V22()
V23()
ext-real )
Real)
+ (b : ( ( ) ( V22() V23() ext-real ) Real) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) holds
(
tan : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) + (c : ( ( ) ( V22() V23() ext-real ) Real) (#) f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) + (c : ( ( ) ( V22() V23() ext-real ) Real) (#) f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= (b : ( ( ) ( V22() V23() ext-real ) Real) + ((2 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * c : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
/ ((cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . ((a : ( ( ) ( V22() V23() ext-real ) Real) + (b : ( ( ) ( V22() V23() ext-real ) Real) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) + (c : ( ( ) ( V22() V23() ext-real ) Real) * (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
c,
a,
b being ( ( ) (
V22()
V23()
ext-real )
Real)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) + (c : ( ( ) ( V22() V23() ext-real ) Real) (#) f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
PartFunc of ,)
= #Z 2 : ( ( ) (
V11()
V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V6()
V30(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V30(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
PartFunc of ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= a : ( ( ) (
V22()
V23()
ext-real )
Real)
+ (b : ( ( ) ( V22() V23() ext-real ) Real) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) holds
(
cot : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) + (c : ( ( ) ( V22() V23() ext-real ) Real) (#) f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) + (c : ( ( ) ( V22() V23() ext-real ) Real) (#) f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= - ((b : ( ( ) ( V22() V23() ext-real ) Real) + ((2 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * c : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / ((sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . ((a : ( ( ) ( V22() V23() ext-real ) Real) + (b : ( ( ) ( V22() V23() ext-real ) Real) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) + (c : ( ( ) ( V22() V23() ext-real ) Real) * (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
tan : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* exp_R : ( (
V6()
V30(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V30(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= (exp_R : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
/ ((cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (exp_R : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cot : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* exp_R : ( (
V6()
V30(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V30(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= - ((exp_R : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / ((sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (exp_R : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
tan : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* ln : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= 1 : ( ( ) (
V11()
V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
/ (x : ( ( ) ( V22() V23() ext-real ) Real) * ((cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cot : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* ln : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= - (1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (x : ( ( ) ( V22() V23() ext-real ) Real) * ((sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (exp_R : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
exp_R : ( (
V6()
V30(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V30(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* tan : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((exp_R : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= (exp_R : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
/ ((cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (exp_R : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
exp_R : ( (
V6()
V30(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V30(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* cot : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((exp_R : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= - ((exp_R : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / ((sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
ln : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* tan : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= 1 : ( ( ) (
V11()
V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
/ ((cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) * (sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
ln : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* cot : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= - (1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / ((sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) * (cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
n being ( (
V21() ) (
V15()
V16()
V17()
V21()
V23()
ext-real V68() )
Nat)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((#Z n : ( ( V21() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) ) : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) & 1 : ( ( ) (
V11()
V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
<= n : ( (
V21() ) (
V15()
V16()
V17()
V21()
V23()
ext-real V68() )
Nat) holds
(
(#Z n : ( ( V21() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) ) : ( (
V6()
V30(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V30(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* tan : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
(((#Z n : ( ( V21() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) ) : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= (n : ( ( V21() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) * ((sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) #Z (n : ( ( V21() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) - 1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
/ ((cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) #Z (n : ( ( V21() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) + 1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
n being ( (
V21() ) (
V15()
V16()
V17()
V21()
V23()
ext-real V68() )
Nat)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((#Z n : ( ( V21() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) ) : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) & 1 : ( ( ) (
V11()
V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
<= n : ( (
V21() ) (
V15()
V16()
V17()
V21()
V23()
ext-real V68() )
Nat) holds
(
(#Z n : ( ( V21() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) ) : ( (
V6()
V30(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V30(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
* cot : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
(((#Z n : ( ( V21() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) ) : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= - ((n : ( ( V21() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) * ((cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) #Z (n : ( ( V21() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) - 1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / ((sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) #Z (n : ( ( V21() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) + 1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + (cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
( 1 : ( ( ) (
V11()
V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
+ (sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
<> 0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) & 1 : ( ( ) (
V11()
V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
- (sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
<> 0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
tan : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
+ (cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + (cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= 1 : ( ( ) (
V11()
V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
/ (1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - (cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
( 1 : ( ( ) (
V11()
V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
- (sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
<> 0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) & 1 : ( ( ) (
V11()
V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
+ (sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
<> 0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
tan : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
- (cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - (cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= 1 : ( ( ) (
V11()
V15()
V16()
V17()
V21()
V22()
V23()
ext-real positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
/ (1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + (sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
tan : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
- (id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= ((sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
/ ((cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((- cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
(- cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
- (id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
(((- cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= ((cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
/ ((sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
a being ( ( ) (
V22()
V23()
ext-real )
Real)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (((1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) (#) (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
PartFunc of ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= a : ( ( ) (
V22()
V23()
ext-real )
Real)
* x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) &
a : ( ( ) (
V22()
V23()
ext-real )
Real)
<> 0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
((1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) (#) (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
- (id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((((1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) (#) (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= ((sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (a : ( ( ) ( V22() V23() ext-real ) Real) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
/ ((cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (a : ( ( ) ( V22() V23() ext-real ) Real) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
a being ( ( ) (
V22()
V23()
ext-real )
Real)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (((- (1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) (#) (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
PartFunc of ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= a : ( ( ) (
V22()
V23()
ext-real )
Real)
* x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) &
a : ( ( ) (
V22()
V23()
ext-real )
Real)
<> 0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
((- (1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) (#) (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
- (id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((((- (1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) (#) (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= ((cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (a : ( ( ) ( V22() V23() ext-real ) Real) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
/ ((sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (a : ( ( ) ( V22() V23() ext-real ) Real) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
a,
b being ( ( ) (
V22()
V23()
ext-real )
Real)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
PartFunc of ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= (a : ( ( ) ( V22() V23() ext-real ) Real) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
+ b : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
PartFunc of ,)
(#) tan : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= ((a : ( ( ) ( V22() V23() ext-real ) Real) * (sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / (cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
+ (((a : ( ( ) ( V22() V23() ext-real ) Real) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) + b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / ((cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
a,
b being ( ( ) (
V22()
V23()
ext-real )
Real)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
PartFunc of ,) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
PartFunc of ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= (a : ( ( ) ( V22() V23() ext-real ) Real) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
+ b : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
PartFunc of ,)
(#) cot : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= ((a : ( ( ) ( V22() V23() ext-real ) Real) * (cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / (sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
- (((a : ( ( ) ( V22() V23() ext-real ) Real) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) + b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / ((sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (exp_R : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
exp_R : ( (
V6()
V30(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V30(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
(#) tan : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((exp_R : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= (((exp_R : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) * (sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / (cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
+ ((exp_R : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / ((cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (exp_R : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
exp_R : ( (
V6()
V30(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V30(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
(#) cot : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((exp_R : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= (((exp_R : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) * (cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / (sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
- ((exp_R : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / ((sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
ln : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
(#) tan : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= (((sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / (cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
+ ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / ((cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
ln : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
(#) cot : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= (((cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / (sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
- ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / ((sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st not
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
(#) tan : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= (- (((sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / (cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
+ ((1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / ((cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st not
0 : ( ( ) (
V11()
V15()
V16()
V17()
V19()
V20()
V21()
V22()
V23()
ext-real non
positive non
negative V49()
V50()
V51()
V52()
V53()
V54()
V55()
V68()
V69() )
Element of
NAT : ( ( ) (
V11()
V15()
V16()
V17()
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
(#) cot : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
x : ( ( ) (
V22()
V23()
ext-real )
Real)
in Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) holds
((((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element of
K19(
K20(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ,
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V39()
V40()
V41() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= (- (((cos : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / (sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
- ((1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / ((sin : ( ( V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V30( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ) ;