begin
theorem
for
f1,
f2 being ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,) holds
(- f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
(#) (- f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= f1 : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,)
(#) f2 : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
r,
p being ( ( ) (
V11()
V12()
ext-real )
Real)
for
n being ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(r : ( ( ) ( V11() V12() ext-real ) Real) * p : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
(#) ((#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= r : ( ( ) (
V11()
V12()
ext-real )
Real)
(#) (p : ( ( ) ( V11() V12() ext-real ) Real) (#) ((#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
x being ( ( ) (
V11()
V12()
ext-real )
Real)
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
((diff (sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
= - (sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ;
theorem
for
x being ( ( ) (
V11()
V12()
ext-real )
Real)
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
((diff (sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 3 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
= (- cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ;
theorem
for
x being ( ( ) (
V11()
V12()
ext-real )
Real)
for
n being ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
((diff (sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
= sin : ( (
V21()
V30(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ) ( non
empty V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
total V30(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
. (x : ( ( ) ( V11() V12() ext-real ) Real) + ((n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) / 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ;
theorem
for
x being ( ( ) (
V11()
V12()
ext-real )
Real)
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
((diff (cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
= - (cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ;
theorem
for
x being ( ( ) (
V11()
V12()
ext-real )
Real)
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
((diff (cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 3 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
= sin : ( (
V21()
V30(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ) ( non
empty V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
total V30(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ;
theorem
for
x being ( ( ) (
V11()
V12()
ext-real )
Real)
for
n being ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
((diff (cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
= cos : ( (
V21()
V30(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ) ( non
empty V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
total V30(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
. (x : ( ( ) ( V11() V12() ext-real ) Real) + ((n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) / 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ;
theorem
for
n being ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) )
for
f1,
f2 being ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,) st
f1 : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,)
is_differentiable_on n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) ,
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) &
f2 : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,)
is_differentiable_on n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) ,
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
(diff ((f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) + f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) (
V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) )
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= ((diff (f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
+ ((diff (f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
n being ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) )
for
f1,
f2 being ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,) st
f1 : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,)
is_differentiable_on n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) ,
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) &
f2 : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,)
is_differentiable_on n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) ,
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
(diff ((f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) - f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) (
V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) )
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= ((diff (f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
- ((diff (f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
n,
i being ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) )
for
f1,
f2 being ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,) st
f1 : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,)
is_differentiable_on n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) ,
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) &
f2 : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,)
is_differentiable_on n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) ,
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) &
i : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
<= n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(diff ((f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) + f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) (
V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) )
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. i : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= ((diff (f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . i : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
+ ((diff (f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . i : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
n,
i being ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) )
for
f1,
f2 being ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,) st
f1 : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,)
is_differentiable_on n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) ,
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) &
f2 : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,)
is_differentiable_on n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) ,
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) &
i : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
<= n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(diff ((f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) - f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) (
V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) )
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. i : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= ((diff (f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . i : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
- ((diff (f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . i : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
n being ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) )
for
f1,
f2 being ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,) st
f1 : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,)
is_differentiable_on n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) ,
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) &
f2 : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,)
is_differentiable_on n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) ,
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
f1 : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,)
+ f2 : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) ,
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) ;
theorem
for
n being ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) )
for
f1,
f2 being ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,) st
f1 : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,)
is_differentiable_on n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) ,
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) &
f2 : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,)
is_differentiable_on n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) ,
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
f1 : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,)
- f2 : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) ,
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) ;
theorem
for
r being ( ( ) (
V11()
V12()
ext-real )
Real)
for
n being ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,) st
f : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,)
is_differentiable_on n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) ,
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
(diff ((r : ( ( ) ( V11() V12() ext-real ) Real) (#) f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) (
V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) )
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= r : ( ( ) (
V11()
V12()
ext-real )
Real)
(#) ((diff (f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,) st
f : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,)
is_differentiable_on Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
(diff (f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) (
V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) )
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. 1 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= f : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,)
`| Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
n being ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) )
for
f1 being ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,) st
n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
>= 1 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) &
f1 : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,)
is_differentiable_on n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) ,
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
(diff (f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) (
V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) )
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. 1 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= f1 : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,)
`| Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
x,
r being ( ( ) (
V11()
V12()
ext-real )
Real)
for
n being ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
((diff ((r : ( ( ) ( V11() V12() ext-real ) Real) (#) sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
= r : ( ( ) (
V11()
V12()
ext-real )
Real)
* (sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V11() V12() ext-real ) Real) + ((n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) / 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ;
theorem
for
x,
r being ( ( ) (
V11()
V12()
ext-real )
Real)
for
n being ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
((diff ((r : ( ( ) ( V11() V12() ext-real ) Real) (#) cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
= r : ( ( ) (
V11()
V12()
ext-real )
Real)
* (cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V11() V12() ext-real ) Real) + ((n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) / 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ;
theorem
for
x,
r being ( ( ) (
V11()
V12()
ext-real )
Real)
for
n being ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
((diff ((r : ( ( ) ( V11() V12() ext-real ) Real) (#) exp_R : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
= r : ( ( ) (
V11()
V12()
ext-real )
Real)
* (exp_R : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ;
theorem
for
n being ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) st
n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
>= 1 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(diff ((#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) (
V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) )
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. 2 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= ((n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) (#) (#Z (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) - 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
| Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) : ( ( ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
n being ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) st
n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
>= 2 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(diff ((#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) (
V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) )
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. 3 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= (((n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) * (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) - 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) (#) (#Z (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) - 3 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
| Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) : ( ( ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
n,
m being ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) st
n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
> m : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(diff ((#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) (
V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) )
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. m : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= (((n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) choose m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * (m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) (#) (#Z (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) - m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
| Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) : ( ( ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
n being ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,) st
f : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,)
is_differentiable_on n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) ,
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
(
(diff ((- f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) (
V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) )
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= - ((diff (f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) &
- f : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) ,
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) ) ;
theorem
for
x0,
x being ( ( ) (
V11()
V12()
ext-real )
Real)
for
n being ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) st
x0 : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
(
(Taylor (sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V11() V12() ext-real ) Real) ,x : ( ( ) ( V11() V12() ext-real ) Real) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ) ( non
empty V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
total V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V74()
V75()
V76() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
. n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
= ((sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . (x0 : ( ( ) ( V11() V12() ext-real ) Real) + ((n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) / 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) * ((x : ( ( ) ( V11() V12() ext-real ) Real) - x0 : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) |^ n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
/ (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) &
(Taylor (cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V11() V12() ext-real ) Real) ,x : ( ( ) ( V11() V12() ext-real ) Real) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ) ( non
empty V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
total V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V74()
V75()
V76() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
. n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
= ((cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . (x0 : ( ( ) ( V11() V12() ext-real ) Real) + ((n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) / 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) * ((x : ( ( ) ( V11() V12() ext-real ) Real) - x0 : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) |^ n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
/ (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ) ;
theorem
for
r,
x being ( ( ) (
V11()
V12()
ext-real )
Real)
for
n being ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) st
r : ( ( ) (
V11()
V12()
ext-real )
Real)
> 0 : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
(Maclaurin (sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ,r : ( ( ) ( V11() V12() ext-real ) Real) .[ : ( ( ) ( open V84() V85() V86() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() V12() ext-real ) Real) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ) ( non
empty V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
total V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V74()
V75()
V76() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
. n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
= ((sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . ((n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) / 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) * (x : ( ( ) ( V11() V12() ext-real ) Real) |^ n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
/ (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) &
(Maclaurin (cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ,r : ( ( ) ( V11() V12() ext-real ) Real) .[ : ( ( ) ( open V84() V85() V86() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() V12() ext-real ) Real) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ) ( non
empty V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
total V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V74()
V75()
V76() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
. n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
= ((cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . ((n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) / 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) * (x : ( ( ) ( V11() V12() ext-real ) Real) |^ n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
/ (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ) ;
theorem
for
x being ( ( ) (
V11()
V12()
ext-real )
Real)
for
n,
m being ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) st
n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
> m : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) &
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
((diff ((#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
= ((n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) choose m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * (m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
* (x : ( ( ) ( V11() V12() ext-real ) Real) #Z (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) - m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ;
theorem
for
x being ( ( ) (
V11()
V12()
ext-real )
Real)
for
m being ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
((diff ((#Z m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
= m : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
! : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ;
theorem
for
x,
a being ( ( ) (
V11()
V12()
ext-real )
Real)
for
n,
m being ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) &
n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
> m : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) holds
((diff ((a : ( ( ) ( V11() V12() ext-real ) Real) (#) (#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
= ((a : ( ( ) ( V11() V12() ext-real ) Real) * (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) choose m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) * (m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
* (x : ( ( ) ( V11() V12() ext-real ) Real) #Z (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) - m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ;
theorem
for
x,
a being ( ( ) (
V11()
V12()
ext-real )
Real)
for
n being ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
((diff ((a : ( ( ) ( V11() V12() ext-real ) Real) (#) (#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
= a : ( ( ) (
V11()
V12()
ext-real )
Real)
* (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ;
theorem
for
x0,
x being ( ( ) (
V11()
V12()
ext-real )
Real)
for
n,
m being ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) st
x0 : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) &
n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
> m : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
(Taylor ((#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V11() V12() ext-real ) Real) ,x : ( ( ) ( V11() V12() ext-real ) Real) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ) ( non
empty V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
total V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V74()
V75()
V76() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
. m : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
= ((n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) choose m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * (x0 : ( ( ) ( V11() V12() ext-real ) Real) #Z (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) - m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
* ((x : ( ( ) ( V11() V12() ext-real ) Real) - x0 : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) |^ m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) &
(Taylor ((#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V11() V12() ext-real ) Real) ,x : ( ( ) ( V11() V12() ext-real ) Real) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ) ( non
empty V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
total V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V74()
V75()
V76() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
. n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
= (x : ( ( ) ( V11() V12() ext-real ) Real) - x0 : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
|^ n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ) ;
theorem
for
n,
m being ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
for
r,
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
> m : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) &
r : ( ( ) (
V11()
V12()
ext-real )
Real)
> 0 : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
(Maclaurin ((#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ,r : ( ( ) ( V11() V12() ext-real ) Real) .[ : ( ( ) ( open V84() V85() V86() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() V12() ext-real ) Real) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ) ( non
empty V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
total V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V74()
V75()
V76() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
. m : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
= 0 : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) &
(Maclaurin ((#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ,r : ( ( ) ( V11() V12() ext-real ) Real) .[ : ( ( ) ( open V84() V85() V86() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() V12() ext-real ) Real) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ) ( non
empty V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
total V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V74()
V75()
V76() )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
. n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
= x : ( ( ) (
V11()
V12()
ext-real )
Real)
|^ n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ) ;
theorem
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) st not
0 : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
in Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
((id REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( total ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V22() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() V78() V80() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
`| Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= ((- 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) (#) ((#Z 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
| Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) : ( ( ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) st not
0 : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
in Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
((#Z 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
`| Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= ((- 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) (#) ((#Z 3 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
| Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) : ( ( ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
n being ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) st not
0 : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
in Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) &
n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
>= 1 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) holds
((#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
`| Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= ((- n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) (#) ((#Z (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
| Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) : ( ( ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) )
for
f1,
f2 being ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,) st
f1 : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,)
is_differentiable_on 2 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) ,
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) &
f2 : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,)
is_differentiable_on 2 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) ,
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
(diff ((f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) (#) f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) (
V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) )
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. 2 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= ((((diff (f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) + (2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) `| Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) (f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) `| Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
+ (f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) (#) ((diff (f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) )
c= dom ln : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V84()
V85()
V86() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) )
c= dom ((id REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( total ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V22() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() V78() V80() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V84()
V85()
V86() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) holds
ln : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
`| Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= ((id REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( total ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V22() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() V78() V80() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
| Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) : ( ( ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
x0 being ( ( ) (
V11()
V12()
ext-real )
Real)
for
n being ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) st
n : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
>= 1 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) &
x0 : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) & not
0 : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
in Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
((diff (((#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
. x0 : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
= (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
* (((#Z (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . x0 : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ;
theorem
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
(diff ((sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) (
V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) )
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. 2 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= (2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
+ ((- 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) (#) ((sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
(diff ((cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) (
V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) )
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. 2 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= (2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
+ ((- 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) (#) ((cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
(diff ((sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) (
V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) )
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. 2 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= 4 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
(#) (((- sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) )
c= dom tan : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V84()
V85()
V86() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) holds
(
tan : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) &
tan : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
`| Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= ((cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
| Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) : ( ( ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) )
c= dom tan : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V84()
V85()
V86() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cos : ( (
V21()
V30(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ) ( non
empty V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
total V30(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
^ : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) &
(cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
`| Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= ((cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) tan : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
| Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) : ( ( ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) )
c= dom tan : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V84()
V85()
V86() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) holds
(diff (tan : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) (
V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) )
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. 2 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= 2 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
(#) (((tan : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) (cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) (cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) )
c= dom cot : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V84()
V85()
V86() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cot : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) &
cot : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
`| Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= ((- 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) (#) ((sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
| Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) : ( ( ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) )
c= dom cot : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V84()
V85()
V86() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) holds
(
sin : ( (
V21()
V30(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) ) ( non
empty V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
total V30(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
^ : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) &
(sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
`| Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= (- ((sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) cot : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
| Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) : ( ( ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) )
c= dom cot : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V84()
V85()
V86() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) holds
(diff (cot : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) (
V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) )
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. 2 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= 2 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
(#) (((cot : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) (sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) (sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
(diff ((exp_R : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) (
V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) )
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. 2 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= 2 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
(#) ((exp_R : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
(diff ((exp_R : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) (
V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) )
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. 2 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= 2 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
(#) ((exp_R : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) (- sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) )
for
f1,
f2 being ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,) st
f1 : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,)
is_differentiable_on 3 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) ,
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) &
f2 : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,)
is_differentiable_on 3 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) ,
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
(diff ((f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) (#) f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) (
V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) )
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. 3 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= ((((diff (f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 3 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) + ((3 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) (#) (((diff (f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) (f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) `| Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) + (3 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) `| Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) ((diff (f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
+ (f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) (#) ((diff (f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 3 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
(diff ((sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) (
V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) )
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. 3 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= (- 8 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V11()
V12()
ext-real V48() )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
(#) ((cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,) st
f : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,)
is_differentiable_on 2 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) ,
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
(diff ((f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ^2) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) (
V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) )
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. 2 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= (2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) (#) (f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) (#) ((diff (f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
+ (2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) `| Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,) st
f : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,)
is_differentiable_on 2 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) ,
Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) & ( for
x0 being ( ( ) (
V11()
V12()
ext-real )
Real) st
x0 : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
f : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
PartFunc of ,)
. x0 : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
<> 0 : ( ( ) (
V4()
V5()
V6()
V10()
V11()
V12()
ext-real non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(diff ((f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) (
V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) )
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. 2 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= (((2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) (#) (f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) `| Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) (f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) `| Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) - (((diff (f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
/ (f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) (#) (f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) (#) f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
Z being ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) holds
(diff ((exp_R : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( (
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) (
V16()
V19(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) )
V21()
V30(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
PFuncs (
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) )
Element of
K6(
K7(
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ,
(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. 3 : ( ( ) ( non
empty V4()
V5()
V6()
V10()
V11()
V12()
ext-real positive non
negative V48()
V84()
V85()
V86()
V87()
V88()
V89()
V91() )
Element of
NAT : ( ( ) ( non
empty V4()
V5()
V6()
V84()
V85()
V86()
V87()
V88()
V89()
V90() )
Element of
K6(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
= (2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) (#) (exp_R : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) ((- sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V21() ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) )
| Z : ( (
open ) (
open V84()
V85()
V86() )
Subset of ( ( ) ( )
set ) ) : ( ( ) (
V16()
V19(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V20(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) )
V21()
V74()
V75()
V76() )
Element of
K6(
K7(
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ,
REAL : ( ( ) ( non
empty V57()
V84()
V85()
V86()
V90() )
set ) ) : ( ( ) (
V74()
V75()
V76() )
set ) ) : ( ( ) ( )
set ) ) ;