begin
theorem
for
x being ( ( ) (
V22()
V23()
ext-real )
Real)
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
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 ,)
is_differentiable_in x : ( ( ) (
V22()
V23()
ext-real )
Real) &
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 ) )
> - 1 : ( ( ) (
V11()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
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 ) ) &
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 ) )
< 1 : ( ( ) (
V11()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
arcsin : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element 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_in x : ( ( ) (
V22()
V23()
ext-real )
Real) &
diff (
(arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) ,
x : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= (diff (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 ) )
/ (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) 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 ,) . 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 ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) ) ;
theorem
for
x being ( ( ) (
V22()
V23()
ext-real )
Real)
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
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 ,)
is_differentiable_in x : ( ( ) (
V22()
V23()
ext-real )
Real) &
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 ) )
> - 1 : ( ( ) (
V11()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
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 ) ) &
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 ) )
< 1 : ( ( ) (
V11()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
arccos : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element 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_in x : ( ( ) (
V22()
V23()
ext-real )
Real) &
diff (
(arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) ,
x : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= - ((diff (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 ) ) / (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) 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 ,) . 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 ) ) ) : ( ( ) (
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 ) ) * arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) &
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ,1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open 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
arcsin : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element 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 ) )
> 0 : ( ( ) (
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
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 ) )
* arcsin : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element 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 ) ) * arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
/ ((sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) : ( ( ) ( 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 ) ) * (arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) * arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) &
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ,1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open 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
arccos : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element 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 ) )
> 0 : ( ( ) (
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
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 ) )
* arccos : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element 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 ) ) * arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / ((sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) : ( ( ) ( 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 ) ) * (arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
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() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( 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() V33( 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 ) ) * arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) &
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ,1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
(#Z n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V6()
V33(
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()
V33(
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 ) )
* arcsin : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element 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() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( 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() V33( 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 ) ) * arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * ((arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) )
/ (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) : ( ( ) ( 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()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
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() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( 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() V33( 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 ) ) * arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) &
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ,1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
(#Z n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V6()
V33(
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()
V33(
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 ) )
* arccos : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element 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() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( 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() V33( 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 ) ) * arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * ((arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) / (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) : ( ( ) ( 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 ((1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) (#) ((#Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( 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() V33( 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 ) ) * arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) &
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ,1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
(1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) )
(#) ((#Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( 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() V33( 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 ) ) * arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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
(((1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) (#) ((#Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( 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() V33( 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 ) ) * arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) )
= (arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) )
/ (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) : ( ( ) ( 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 ((1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) (#) ((#Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( 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() V33( 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 ) ) * arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) &
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ,1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
(1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) )
(#) ((#Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( 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() V33( 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 ) ) * arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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
(((1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) (#) ((#Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( 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() V33( 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 ) ) * arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) )
= - ((arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) / (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) : ( ( ) ( 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
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 (arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) &
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 ) )
> - 1 : ( ( ) (
V11()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
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 ) ) &
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 ) )
< 1 : ( ( ) (
V11()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
arcsin : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element 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
((arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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)
/ (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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) * 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 ) ) ^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
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 (arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) &
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 ) )
> - 1 : ( ( ) (
V11()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
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 ) ) &
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 ) )
< 1 : ( ( ) (
V11()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
arccos : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element 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
((arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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) / (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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) * 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 ) ) ^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 ) ) : ( ( ) (
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 ((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 ) ) (#) arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) &
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ,1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open 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 ) )
(#) arcsin : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element 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 ) ) (#) arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) )
= (arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) )
+ (x : ( ( ) ( V22() V23() ext-real ) Real) / (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) : ( ( ) ( 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 ((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 ) ) (#) arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) &
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ,1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open 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 ) )
(#) arccos : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element 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 ) ) (#) arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) )
= (arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) )
- (x : ( ( ) ( V22() V23() ext-real ) Real) / (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) : ( ( ) ( 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
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 ,) (#) arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) &
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ,1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open 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 ,)
(#) arcsin : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element 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 ,) (#) arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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) * (arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) / (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) : ( ( ) ( 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
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 ,) (#) arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) &
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ,1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open 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 ,)
(#) arccos : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element 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 ,) (#) arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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) * (arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) / (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) : ( ( ) ( 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 ) )
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() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) (#) (arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) : ( ( ) (
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 ) )
= 2 : ( ( ) (
V11()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
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 ) ) &
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 ) )
> - 1 : ( ( ) (
V11()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
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 ) ) &
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 ) )
< 1 : ( ( ) (
V11()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
(1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) )
(#) (arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) )
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() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) (#) (arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) `| 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()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
/ (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - ((2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ^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 ) )
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() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) (#) (arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) : ( ( ) (
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 ) )
= 2 : ( ( ) (
V11()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
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 ) ) &
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 ) )
> - 1 : ( ( ) (
V11()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
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 ) ) &
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 ) )
< 1 : ( ( ) (
V11()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
(1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) )
(#) (arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) )
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() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) (#) (arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) `| 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() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - ((2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ^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 ) ) : ( ( ) (
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 ) )
for
f,
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 ((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) &
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 ,)
= 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 ) ) &
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()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V6()
V33(
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()
V33(
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 ) )
= 1 : ( ( ) (
V11()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
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 ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
> 0 : ( ( ) (
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
(#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element 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
(((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) )
= - (x : ( ( ) ( V22() V23() ext-real ) Real) * ((1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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) #Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) #R (- (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) : ( ( ) (
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 ) )
for
f,
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 (((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 ) ) (#) arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) + ((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ,1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
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 ,)
= 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 ) ) &
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()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V6()
V33(
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()
V33(
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 ) )
= 1 : ( ( ) (
V11()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
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 ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
> 0 : ( ( ) (
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
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)
<> 0 : ( ( ) (
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
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 ) ) (#) arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) )
+ ((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) )
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 ) ) (#) arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) + ((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) `| 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 ) )
= arcsin : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element 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 ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f,
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 (((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 ) ) (#) arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) - ((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ,1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
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 ,)
= 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 ) ) &
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()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V6()
V33(
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()
V33(
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 ) )
= 1 : ( ( ) (
V11()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
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 ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
> 0 : ( ( ) (
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
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)
<> 0 : ( ( ) (
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
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 ) ) (#) arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) )
- ((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) )
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 ) ) (#) arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) - ((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) `| 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 ) )
= arccos : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element 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 ) ) ) ) ;
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 ((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 ) ) (#) (arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) : ( ( ) (
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 ) )
= 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 ) ) &
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 ) )
> - 1 : ( ( ) (
V11()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
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 ) ) &
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 ) )
< 1 : ( ( ) (
V11()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
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 ) )
(#) (arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) )
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 ) ) (#) (arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) `| 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 ) )
= (arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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) / 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 ) )
+ (x : ( ( ) ( V22() V23() ext-real ) Real) / (a : ( ( ) ( V22() V23() ext-real ) Real) * (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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) / a : ( ( ) ( 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 ) ) ) : ( ( ) ( 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 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 ((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 ) ) (#) (arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) : ( ( ) (
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 ) )
= 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 ) ) &
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 ) )
> - 1 : ( ( ) (
V11()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
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 ) ) &
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 ) )
< 1 : ( ( ) (
V11()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
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 ) )
(#) (arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) )
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 ) ) (#) (arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) `| 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 ) )
= (arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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) / 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 ) )
- (x : ( ( ) ( V22() V23() ext-real ) Real) / (a : ( ( ) ( V22() V23() ext-real ) Real) * (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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) / a : ( ( ) ( 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 ) ) ) : ( ( ) ( 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 being ( ( ) (
V22()
V23()
ext-real )
Real)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f,
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 ((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) &
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 ,)
= 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 ) ) &
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()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V6()
V33(
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()
V33(
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)
^2 : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
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 ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
> 0 : ( ( ) (
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
(#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element 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
(((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) )
= - (x : ( ( ) ( V22() V23() ext-real ) Real) * (((a : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) - (x : ( ( ) ( V22() V23() ext-real ) Real) #Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) #R (- (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) : ( ( ) (
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
f3,
f,
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 (((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 ) ) (#) (arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f3 : ( ( 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 ) ) + ((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
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 ,)
= 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 ) ) &
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()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V6()
V33(
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()
V33(
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)
^2 : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
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 ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
> 0 : ( ( ) (
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) &
f3 : ( (
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 ) ) &
f3 : ( (
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 ) )
> - 1 : ( ( ) (
V11()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
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 ) ) &
f3 : ( (
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 ) )
< 1 : ( ( ) (
V11()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
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)
<> 0 : ( ( ) (
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
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)
> 0 : ( ( ) (
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
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 ) ) (#) (arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f3 : ( ( 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 ) )
+ ((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) )
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 ) ) (#) (arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f3 : ( ( 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 ) ) + ((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) `| 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 ) )
= arcsin : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element 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) / 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 ) ) ) ) ;
theorem
for
a being ( ( ) (
V22()
V23()
ext-real )
Real)
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
for
f3,
f,
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 (((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 ) ) (#) (arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f3 : ( ( 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 ) ) - ((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) : ( ( ) (
V49()
V50()
V51() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
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 ,)
= 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 ) ) &
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()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V6()
V33(
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()
V33(
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)
^2 : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
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 ,)
. x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
> 0 : ( ( ) (
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) &
f3 : ( (
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 ) ) &
f3 : ( (
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 ) )
> - 1 : ( ( ) (
V11()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
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 ) ) &
f3 : ( (
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 ) )
< 1 : ( ( ) (
V11()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
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)
<> 0 : ( ( ) (
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
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)
> 0 : ( ( ) (
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
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 ) ) (#) (arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f3 : ( ( 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 ) )
- ((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) )
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 ) ) (#) (arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f3 : ( ( 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 ) ) - ((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) `| 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 ) )
= arccos : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element 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) / 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 ) ) ) ) ;
theorem
for
n being ( ( ) (
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((- (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) (#) ((#Z n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( 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() V33( 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 ) ) * (sin : ( ( V6() V33( 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() V33( 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 ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element 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 ) ) &
n : ( ( ) (
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
> 0 : ( ( ) (
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
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
sin : ( (
V6()
V33(
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()
V33(
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 ) )
<> 0 : ( ( ) (
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
(- (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) )
(#) ((#Z n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( 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() V33( 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 ) ) * (sin : ( ( V6() V33( 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() V33( 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 ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element 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() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) (#) ((#Z n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( 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() V33( 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 ) ) * (sin : ( ( V6() V33( 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() V33( 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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V33( 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() V33( 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() V33( 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() V33( 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() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) ) ;
theorem
for
n being ( ( ) (
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) (#) ((#Z n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( 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() V33( 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 ) ) * (cos : ( ( V6() V33( 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() V33( 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 ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element 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 ) ) &
n : ( ( ) (
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
> 0 : ( ( ) (
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
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
cos : ( (
V6()
V33(
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()
V33(
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 ) )
<> 0 : ( ( ) (
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
(1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) )
(#) ((#Z n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( 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() V33( 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 ) ) * (cos : ( ( V6() V33( 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() V33( 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 ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element 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() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) (#) ((#Z n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( 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() V33( 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 ) ) * (cos : ( ( V6() V33( 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() V33( 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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V33( 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() V33( 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() V33( 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() V33( 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() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (sin : ( ( V6() V33( 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() V33( 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 ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element 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
x : ( ( ) (
V22()
V23()
ext-real )
Real)
> 0 : ( ( ) (
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
sin : ( (
V6()
V33(
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()
V33(
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 ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element 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
((sin : ( ( V6() V33( 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() V33( 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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V33( 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() V33( 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 ) )
/ x : ( ( ) (
V22()
V23()
ext-real )
Real) : ( ( ) (
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 (cos : ( ( V6() V33( 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() V33( 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 ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element 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
x : ( ( ) (
V22()
V23()
ext-real )
Real)
> 0 : ( ( ) (
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
cos : ( (
V6()
V33(
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()
V33(
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 ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V5(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
V6()
V39()
V40()
V41() )
Element 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
((cos : ( ( V6() V33( 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() V33( 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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V33( 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() V33( 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 ) ) / 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
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (sin : ( ( V6() V33( 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() V33( 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() V33( 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() V33( 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
(
sin : ( (
V6()
V33(
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()
V33(
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()
V33(
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()
V33(
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
((sin : ( ( V6() V33( 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() V33( 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() V33( 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() V33( 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() V33( 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() V33( 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() V33( 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() V33( 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() V33( 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() V33( 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 (cos : ( ( V6() V33( 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() V33( 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() V33( 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() V33( 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
(
cos : ( (
V6()
V33(
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()
V33(
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()
V33(
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()
V33(
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
((cos : ( ( V6() V33( 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() V33( 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() V33( 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() V33( 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() V33( 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() V33( 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() V33( 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() V33( 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() V33( 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() V33( 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
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() V33( 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() V33( 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 ) ) * cos : ( ( V6() V33( 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() V33( 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
(
exp_R : ( (
V6()
V33(
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()
V33(
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 ) )
* cos : ( (
V6()
V33(
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()
V33(
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
((exp_R : ( ( V6() V33( 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() V33( 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 ) ) * cos : ( ( V6() V33( 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() V33( 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() V33( 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() V33( 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 ) ) . (cos : ( ( V6() V33( 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() V33( 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() V33( 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() V33( 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 (exp_R : ( ( V6() V33( 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() V33( 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 ) ) * sin : ( ( V6() V33( 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() V33( 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
(
exp_R : ( (
V6()
V33(
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()
V33(
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 ) )
* sin : ( (
V6()
V33(
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()
V33(
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
((exp_R : ( ( V6() V33( 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() V33( 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 ) ) * sin : ( ( V6() V33( 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() V33( 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() V33( 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() V33( 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 ) ) . (sin : ( ( V6() V33( 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() V33( 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() V33( 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() V33( 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 ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (sin : ( ( V6() V33( 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() V33( 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 ) ) + cos : ( ( V6() V33( 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() V33( 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
(
sin : ( (
V6()
V33(
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()
V33(
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 ) )
+ cos : ( (
V6()
V33(
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()
V33(
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
((sin : ( ( V6() V33( 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() V33( 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 ) ) + cos : ( ( V6() V33( 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() V33( 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 ) )
= (cos : ( ( V6() V33( 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() V33( 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() V33( 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() V33( 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 ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom (sin : ( ( V6() V33( 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() V33( 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 ) ) - cos : ( ( V6() V33( 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() V33( 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
(
sin : ( (
V6()
V33(
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()
V33(
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 ) )
- cos : ( (
V6()
V33(
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()
V33(
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
((sin : ( ( V6() V33( 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() V33( 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 ) ) - cos : ( ( V6() V33( 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() V33( 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 ) )
= (cos : ( ( V6() V33( 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() V33( 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() V33( 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() V33( 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 ) ) ) ) ;
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() V33( 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() V33( 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 ) ) (#) (sin : ( ( V6() V33( 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() V33( 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 ) ) - cos : ( ( V6() V33( 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() V33( 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 ) ) holds
(
exp_R : ( (
V6()
V33(
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()
V33(
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 ) )
(#) (sin : ( ( V6() V33( 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() V33( 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 ) ) - cos : ( ( V6() V33( 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() V33( 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
((exp_R : ( ( V6() V33( 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() V33( 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 ) ) (#) (sin : ( ( V6() V33( 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() V33( 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 ) ) - cos : ( ( V6() V33( 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() V33( 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 ) )
= (2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (exp_R : ( ( V6() V33( 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() V33( 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() V33( 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() V33( 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 ) ) ) ) ;
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() V33( 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() V33( 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 ) ) (#) (sin : ( ( V6() V33( 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() V33( 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 ) ) + cos : ( ( V6() V33( 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() V33( 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 ) ) holds
(
exp_R : ( (
V6()
V33(
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()
V33(
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 ) )
(#) (sin : ( ( V6() V33( 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() V33( 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 ) ) + cos : ( ( V6() V33( 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() V33( 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
((exp_R : ( ( V6() V33( 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() V33( 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 ) ) (#) (sin : ( ( V6() V33( 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() V33( 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 ) ) + cos : ( ( V6() V33( 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() V33( 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 ) )
= (2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (exp_R : ( ( V6() V33( 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() V33( 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() V33( 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() V33( 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 ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((sin : ( ( V6() V33( 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() V33( 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 ) ) + cos : ( ( V6() V33( 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() V33( 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 ) ) / exp_R : ( ( V6() V33( 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() V33( 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
(
(sin : ( ( V6() V33( 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() V33( 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 ) ) + cos : ( ( V6() V33( 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() V33( 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 ) )
/ exp_R : ( (
V6()
V33(
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()
V33(
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
(((sin : ( ( V6() V33( 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() V33( 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 ) ) + cos : ( ( V6() V33( 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() V33( 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 ) ) / exp_R : ( ( V6() V33( 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() V33( 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 ) )
= - ((2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (sin : ( ( V6() V33( 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() V33( 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() V33( 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() V33( 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 ((sin : ( ( V6() V33( 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() V33( 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 ) ) - cos : ( ( V6() V33( 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() V33( 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 ) ) / exp_R : ( ( V6() V33( 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() V33( 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
(
(sin : ( ( V6() V33( 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() V33( 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 ) ) - cos : ( ( V6() V33( 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() V33( 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 ) )
/ exp_R : ( (
V6()
V33(
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()
V33(
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
(((sin : ( ( V6() V33( 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() V33( 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 ) ) - cos : ( ( V6() V33( 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() V33( 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 ) ) / exp_R : ( ( V6() V33( 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() V33( 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 ) )
= (2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (cos : ( ( V6() V33( 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() V33( 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() V33( 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() V33( 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 ) ) ) ) ;
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() V33( 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() V33( 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 ) ) (#) sin : ( ( V6() V33( 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() V33( 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
(
exp_R : ( (
V6()
V33(
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()
V33(
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 ) )
(#) sin : ( (
V6()
V33(
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()
V33(
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
((exp_R : ( ( V6() V33( 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() V33( 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 ) ) (#) sin : ( ( V6() V33( 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() V33( 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() V33( 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() V33( 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() V33( 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() V33( 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() V33( 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() V33( 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 (exp_R : ( ( V6() V33( 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() V33( 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 ) ) (#) cos : ( ( V6() V33( 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() V33( 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
(
exp_R : ( (
V6()
V33(
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()
V33(
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 ) )
(#) cos : ( (
V6()
V33(
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()
V33(
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
((exp_R : ( ( V6() V33( 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() V33( 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 ) ) (#) cos : ( ( V6() V33( 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() V33( 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() V33( 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() V33( 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() V33( 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() V33( 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() V33( 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() V33( 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
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
cos : ( (
V6()
V33(
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()
V33(
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 ) )
<> 0 : ( ( ) (
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
sin : ( (
V6()
V33(
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()
V33(
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 ) )
/ cos : ( (
V6()
V33(
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()
V33(
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_in x : ( ( ) (
V22()
V23()
ext-real )
Real) &
diff (
(sin : ( ( V6() V33( 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() V33( 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 ) ) / cos : ( ( V6() V33( 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() V33( 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 ) ) ,
x : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= 1 : ( ( ) (
V11()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
/ ((cos : ( ( V6() V33( 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() V33( 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
x being ( ( ) (
V22()
V23()
ext-real )
Real) st
sin : ( (
V6()
V33(
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()
V33(
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 ) )
<> 0 : ( ( ) (
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
cos : ( (
V6()
V33(
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()
V33(
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 ) )
/ sin : ( (
V6()
V33(
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()
V33(
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_in x : ( ( ) (
V22()
V23()
ext-real )
Real) &
diff (
(cos : ( ( V6() V33( 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() V33( 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 ) ) / sin : ( ( V6() V33( 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() V33( 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 ) ) ,
x : ( ( ) (
V22()
V23()
ext-real )
Real) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
= - (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / ((sin : ( ( V6() V33( 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() V33( 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 ((#Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( 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() V33( 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 ) ) * (sin : ( ( V6() V33( 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() V33( 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 ) ) / cos : ( ( V6() V33( 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() V33( 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
cos : ( (
V6()
V33(
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()
V33(
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 ) )
<> 0 : ( ( ) (
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
(#Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V6()
V33(
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()
V33(
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 ) )
* (sin : ( ( V6() V33( 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() V33( 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 ) ) / cos : ( ( V6() V33( 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() V33( 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
(((#Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( 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() V33( 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 ) ) * (sin : ( ( V6() V33( 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() V33( 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 ) ) / cos : ( ( V6() V33( 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() V33( 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 ) )
= (2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (sin : ( ( V6() V33( 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() V33( 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() V33( 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() V33( 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 3 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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
Z being ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V49()
V50()
V51() )
Subset of ( ( ) ( )
set ) )
c= dom ((#Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( 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() V33( 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 ) ) * (cos : ( ( V6() V33( 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() V33( 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 ) ) / sin : ( ( V6() V33( 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() V33( 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
sin : ( (
V6()
V33(
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()
V33(
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 ) )
<> 0 : ( ( ) (
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
(#Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V6()
V33(
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()
V33(
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 ) )
* (cos : ( ( V6() V33( 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() V33( 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 ) ) / sin : ( ( V6() V33( 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() V33( 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
(((#Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( 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() V33( 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 ) ) * (cos : ( ( V6() V33( 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() V33( 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 ) ) / sin : ( ( V6() V33( 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() V33( 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 ) )
= - ((2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (cos : ( ( V6() V33( 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() V33( 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() V33( 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() V33( 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 3 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( 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 ) )
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 ((sin : ( ( V6() V33( 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() V33( 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 ) ) / cos : ( ( V6() V33( 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() V33( 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 ) ) * 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 ) )
= x : ( ( ) (
V22()
V23()
ext-real )
Real)
/ 2 : ( ( ) (
V11()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
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 ) ) &
cos : ( (
V6()
V33(
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()
V33(
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 ) )
. (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 ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
<> 0 : ( ( ) (
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
(sin : ( ( V6() V33( 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() V33( 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 ) ) / cos : ( ( V6() V33( 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() V33( 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 ) )
* 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
(((sin : ( ( V6() V33( 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() V33( 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 ) ) / cos : ( ( V6() V33( 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() V33( 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 ) ) * 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 ) )
= 1 : ( ( ) (
V11()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
/ (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + (cos : ( ( V6() V33( 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() V33( 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 ) )
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 ((cos : ( ( V6() V33( 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() V33( 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 ) ) / sin : ( ( V6() V33( 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() V33( 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 ) ) * 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 ) )
= x : ( ( ) (
V22()
V23()
ext-real )
Real)
/ 2 : ( ( ) (
V11()
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
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 ) ) &
sin : ( (
V6()
V33(
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()
V33(
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 ) )
. (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 ) ) : ( ( ) (
V22()
V23()
ext-real )
Element of
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) )
<> 0 : ( ( ) (
V21()
V22()
V23()
ext-real V49()
V50()
V51()
V52()
V53()
V54()
V68()
V69() )
Element of
NAT : ( ( ) (
V49()
V50()
V51()
V52()
V53()
V54()
V55() )
Element of
K19(
REAL : ( ( ) (
V11()
V49()
V50()
V51()
V55()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
(cos : ( ( V6() V33( 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() V33( 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 ) ) / sin : ( ( V6() V33( 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() V33( 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 ) )
* 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
(((cos : ( ( V6() V33( 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() V33( 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 ) ) / sin : ( ( V6() V33( 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() V33( 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 ) ) * 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 ) )
= - (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (cos : ( ( V6() V33( 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() V33( 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 ) ) ) ) ;