begin
theorem
for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
cos : ( (
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
<> 0 : ( ( ) (
V1()
V4()
V5()
V6()
V8()
V9()
V10()
V11()
V12()
ext-real non
positive non
negative V47()
V62()
V63()
V64()
V65()
V66()
V67()
V68()
V69() )
Element of
NAT : ( ( ) (
V1()
V4()
V5()
V6()
V62()
V63()
V64()
V65()
V66()
V67()
V68() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
sec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
is_differentiable_in x : ( ( ) (
V11()
V12()
ext-real )
Real) &
diff (
sec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) ,
x : ( ( ) (
V11()
V12()
ext-real )
Real) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
/ ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ;
theorem
for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
sin : ( (
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
<> 0 : ( ( ) (
V1()
V4()
V5()
V6()
V8()
V9()
V10()
V11()
V12()
ext-real non
positive non
negative V47()
V62()
V63()
V64()
V65()
V66()
V67()
V68()
V69() )
Element of
NAT : ( ( ) (
V1()
V4()
V5()
V6()
V62()
V63()
V64()
V65()
V66()
V67()
V68() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
cosec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
is_differentiable_in x : ( ( ) (
V11()
V12()
ext-real )
Real) &
diff (
cosec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) ,
x : ( ( ) (
V11()
V12()
ext-real )
Real) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= - ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom sec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
sec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
(sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
/ ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom cosec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cosec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
(cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= - ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
a,
b being ( ( ) (
V11()
V12()
ext-real )
Real)
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
f : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= (a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
+ b : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) holds
(
sec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
* f : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= (a : ( ( ) ( V11() V12() ext-real ) Real) * (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . ((a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) + b : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
/ ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . ((a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) + b : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
a,
b being ( ( ) (
V11()
V12()
ext-real )
Real)
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
f : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= (a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
+ b : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) holds
(
cosec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
* f : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= - ((a : ( ( ) ( V11() V12() ext-real ) Real) * (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . ((a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) + b : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . ((a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) + b : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st not
0 : ( ( ) (
V1()
V4()
V5()
V6()
V8()
V9()
V10()
V11()
V12()
ext-real non
positive non
negative V47()
V62()
V63()
V64()
V65()
V66()
V67()
V68()
V69() )
Element of
NAT : ( ( ) (
V1()
V4()
V5()
V6()
V62()
V63()
V64()
V65()
V66()
V67()
V68() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * ((id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
sec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
* ((id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * ((id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= - ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st not
0 : ( ( ) (
V1()
V4()
V5()
V6()
V8()
V9()
V10()
V11()
V12()
ext-real non
positive non
negative V47()
V62()
V63()
V64()
V65()
V66()
V67()
V68()
V69() )
Element of
NAT : ( ( ) (
V1()
V4()
V5()
V6()
V62()
V63()
V64()
V65()
V66()
V67()
V68() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * ((id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cosec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
* ((id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * ((id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
/ ((x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
c,
a,
b being ( ( ) (
V11()
V12()
ext-real )
Real)
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
for
f1,
f2 being ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * (f1 : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) + (c : ( ( ) ( V11() V12() ext-real ) Real) (#) f2 : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) &
f2 : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
= #Z 2 : ( ( ) (
V1()
V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V47()
V62()
V63()
V64()
V65()
V66()
V67()
V69() )
Element of
NAT : ( ( ) (
V1()
V4()
V5()
V6()
V62()
V63()
V64()
V65()
V66()
V67()
V68() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
f1 : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= a : ( ( ) (
V11()
V12()
ext-real )
Real)
+ (b : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) holds
(
sec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
* (f1 : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) + (c : ( ( ) ( V11() V12() ext-real ) Real) (#) f2 : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * (f1 : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) + (c : ( ( ) ( V11() V12() ext-real ) Real) (#) f2 : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= ((b : ( ( ) ( V11() V12() ext-real ) Real) + ((2 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * c : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . ((a : ( ( ) ( V11() V12() ext-real ) Real) + (b : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) + (c : ( ( ) ( V11() V12() ext-real ) Real) * (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
/ ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . ((a : ( ( ) ( V11() V12() ext-real ) Real) + (b : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) + (c : ( ( ) ( V11() V12() ext-real ) Real) * (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
c,
a,
b being ( ( ) (
V11()
V12()
ext-real )
Real)
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
for
f1,
f2 being ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * (f1 : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) + (c : ( ( ) ( V11() V12() ext-real ) Real) (#) f2 : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) &
f2 : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
= #Z 2 : ( ( ) (
V1()
V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V47()
V62()
V63()
V64()
V65()
V66()
V67()
V69() )
Element of
NAT : ( ( ) (
V1()
V4()
V5()
V6()
V62()
V63()
V64()
V65()
V66()
V67()
V68() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
f1 : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= a : ( ( ) (
V11()
V12()
ext-real )
Real)
+ (b : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) holds
(
cosec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
* (f1 : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) + (c : ( ( ) ( V11() V12() ext-real ) Real) (#) f2 : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * (f1 : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) + (c : ( ( ) ( V11() V12() ext-real ) Real) (#) f2 : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= - (((b : ( ( ) ( V11() V12() ext-real ) Real) + ((2 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * c : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . ((a : ( ( ) ( V11() V12() ext-real ) Real) + (b : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) + (c : ( ( ) ( V11() V12() ext-real ) Real) * (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . ((a : ( ( ) ( V11() V12() ext-real ) Real) + (b : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) + (c : ( ( ) ( V11() V12() ext-real ) Real) * (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
sec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
* exp_R : ( (
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= ((exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
/ ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cosec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
* exp_R : ( (
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= - (((exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
sec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
* ln : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
/ (x : ( ( ) ( V11() V12() ext-real ) Real) * ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cosec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
* ln : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= - ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / (x : ( ( ) ( V11() V12() ext-real ) Real) * ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) * sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
exp_R : ( (
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
* sec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) * sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= ((exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
/ ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
exp_R : ( (
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
* cosec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= - (((exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) * sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
ln : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
* sec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) * sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
/ (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
ln : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
* cosec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= - ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
n being ( (
V10() ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real V47() )
Nat)
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom ((#Z n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real V47() ) Nat) ) : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) * sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) & 1 : ( ( ) (
V1()
V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V47()
V62()
V63()
V64()
V65()
V66()
V67()
V69() )
Element of
NAT : ( ( ) (
V1()
V4()
V5()
V6()
V62()
V63()
V64()
V65()
V66()
V67()
V68() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
<= n : ( (
V10() ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real V47() )
Nat) holds
(
(#Z n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real V47() ) Nat) ) : ( (
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
* sec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
(((#Z n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real V47() ) Nat) ) : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) * sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= (n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real V47() ) Nat) * (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
/ ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) #Z (n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real V47() ) Nat) + 1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
n being ( (
V10() ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real V47() )
Nat)
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom ((#Z n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real V47() ) Nat) ) : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) & 1 : ( ( ) (
V1()
V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V47()
V62()
V63()
V64()
V65()
V66()
V67()
V69() )
Element of
NAT : ( ( ) (
V1()
V4()
V5()
V6()
V62()
V63()
V64()
V65()
V66()
V67()
V68() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
<= n : ( (
V10() ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real V47() )
Nat) holds
(
(#Z n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real V47() ) Nat) ) : ( (
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
* cosec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
(((#Z n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real V47() ) Nat) ) : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= - ((n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real V47() ) Nat) * (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) #Z (n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real V47() ) Nat) + 1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) - (id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
sec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
- (id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) - (id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) - ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
/ ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom ((- cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
(- cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
- (id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
(((- cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) - ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
/ ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
exp_R : ( (
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
(#) sec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= ((exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
+ (((exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
exp_R : ( (
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
(#) cosec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= ((exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
- (((exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
a being ( ( ) (
V11()
V12()
ext-real )
Real)
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (((1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) (#) (sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
(
f : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= a : ( ( ) (
V11()
V12()
ext-real )
Real)
* x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) &
a : ( ( ) (
V11()
V12()
ext-real )
Real)
<> 0 : ( ( ) (
V1()
V4()
V5()
V6()
V8()
V9()
V10()
V11()
V12()
ext-real non
positive non
negative V47()
V62()
V63()
V64()
V65()
V66()
V67()
V68()
V69() )
Element of
NAT : ( ( ) (
V1()
V4()
V5()
V6()
V62()
V63()
V64()
V65()
V66()
V67()
V68() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
((1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) (#) (sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
- (id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((((1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) (#) (sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) - ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
/ ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
a being ( ( ) (
V11()
V12()
ext-real )
Real)
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (((- (1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) (#) (cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
(
f : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= a : ( ( ) (
V11()
V12()
ext-real )
Real)
* x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) &
a : ( ( ) (
V11()
V12()
ext-real )
Real)
<> 0 : ( ( ) (
V1()
V4()
V5()
V6()
V8()
V9()
V10()
V11()
V12()
ext-real non
positive non
negative V47()
V62()
V63()
V64()
V65()
V66()
V67()
V68()
V69() )
Element of
NAT : ( ( ) (
V1()
V4()
V5()
V6()
V62()
V63()
V64()
V65()
V66()
V67()
V68() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
((- (1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) (#) (cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
- (id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((((- (1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) (#) (cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) - ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
/ ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
a,
b being ( ( ) (
V11()
V12()
ext-real )
Real)
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) (#) sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
f : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= (a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
+ b : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) holds
(
f : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
(#) sec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) (#) sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= (a : ( ( ) ( V11() V12() ext-real ) Real) / (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
+ ((((a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) + b : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
a,
b being ( ( ) (
V11()
V12()
ext-real )
Real)
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) (#) cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
f : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= (a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
+ b : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) holds
(
f : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
(#) cosec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) (#) cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= (a : ( ( ) ( V11() V12() ext-real ) Real) / (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
- ((((a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) + b : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
ln : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
(#) sec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= ((1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
+ (((ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
ln : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
(#) cosec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= ((1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
- (((ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st not
0 : ( ( ) (
V1()
V4()
V5()
V6()
V8()
V9()
V10()
V11()
V12()
ext-real non
positive non
negative V47()
V62()
V63()
V64()
V65()
V66()
V67()
V68()
V69() )
Element of
NAT : ( ( ) (
V1()
V4()
V5()
V6()
V62()
V63()
V64()
V65()
V66()
V67()
V68() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (((id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
((id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
(#) sec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((((id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= (- ((1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
+ (((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st not
0 : ( ( ) (
V1()
V4()
V5()
V6()
V8()
V9()
V10()
V11()
V12()
ext-real non
positive non
negative V47()
V62()
V63()
V64()
V65()
V66()
V67()
V68()
V69() )
Element of
NAT : ( ( ) (
V1()
V4()
V5()
V6()
V62()
V63()
V64()
V65()
V66()
V67()
V68() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) )
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (((id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
((id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
(#) cosec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((((id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= (- ((1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
- (((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
sec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
* sin : ( (
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
/ ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
sec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
* cos : ( (
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= - (((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cosec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
* sin : ( (
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= - (((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cosec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
* cos : ( (
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V27(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
/ ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
sec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
* tan : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
/ ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
sec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
* cot : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= - (((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cosec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
* tan : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= - (((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cosec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,)
* cot : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
/ ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
tan : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
(#) sec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= ((1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
+ (((tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cot : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
(#) sec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= (- ((1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
+ (((cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
tan : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
(#) cosec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= ((1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
- (((tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) )
c= dom (cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V62()
V63()
V64() )
Element of
K6(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cot : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
(#) cosec : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
PartFunc of ,) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V62()
V63()
V64() )
Subset of ( ( ) ( )
set ) ) holds
((cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V18() ) (
V13()
V16(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V17(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
V18()
V52()
V53()
V54() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ,
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V52()
V53()
V54() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
= (- ((1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) )
- (((cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V62()
V63()
V64()
V68()
V70() )
set ) ) ) ) ;