begin
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
(
sin : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
> - 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V1()
V11()
V12()
ext-real non
positive negative V60() )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) &
sin : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
< 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
arctan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
* sin : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= (cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
/ (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
(
sin : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
> - 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V1()
V11()
V12()
ext-real non
positive negative V60() )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) &
sin : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
< 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
arccot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
* sin : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= - ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
(
cos : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
> - 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V1()
V11()
V12()
ext-real non
positive negative V60() )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) &
cos : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
< 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
arctan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
* cos : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= - ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
(
cos : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
> - 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V1()
V11()
V12()
ext-real non
positive negative V60() )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) &
cos : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
< 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
arccot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
* cos : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= (sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
/ (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
(
tan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
> - 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V1()
V11()
V12()
ext-real non
positive negative V60() )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) &
tan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
< 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
arctan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
* tan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
(
tan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
> - 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V1()
V11()
V12()
ext-real non
positive negative V60() )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) &
tan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
< 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
arccot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
* tan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= - 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V1()
V11()
V12()
ext-real non
positive negative V60() )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
(
cot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
> - 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V1()
V11()
V12()
ext-real non
positive negative V60() )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) &
cot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
< 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
arctan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
* cot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= - 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V1()
V11()
V12()
ext-real non
positive negative V60() )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
(
cot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
> - 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V1()
V11()
V12()
ext-real non
positive negative V60() )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) &
cot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
< 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
arccot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
* cot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
(
arctan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
> - 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V1()
V11()
V12()
ext-real non
positive negative V60() )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) &
arctan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
< 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
arctan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
* arctan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
/ ((1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
(
arctan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
> - 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V1()
V11()
V12()
ext-real non
positive negative V60() )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) &
arctan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
< 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
arccot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
* arctan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= - (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
(
arccot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
> - 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V1()
V11()
V12()
ext-real non
positive negative V60() )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) &
arccot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
< 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
arctan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
* arccot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= - (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + ((arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
(
arccot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
> - 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V1()
V11()
V12()
ext-real non
positive negative V60() )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) &
arccot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
< 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
arccot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
* arccot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
/ ((1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + ((arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
sin : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
* arctan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= (cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
/ (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
sin : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
* arccot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= - ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cos : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
* arctan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= - ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cos : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
* arccot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= (sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
/ (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
tan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
* arctan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
/ (((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
tan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
* arccot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= - (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
* arctan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= - (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
* arccot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
/ (((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (sec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
sec : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
* arctan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((sec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= (sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
/ (((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (sec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
sec : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
* arccot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((sec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= - ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (cosec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cosec : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
* arctan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((cosec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= - ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (cosec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cosec : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
* arccot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((cosec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= (cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
/ (((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
sin : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) arctan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
+ ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
sin : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) arccot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
- ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cos : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) arctan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= (- ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
+ ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cos : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) arccot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= (- ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
- ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
tan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) arctan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
+ ((tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
tan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) arccot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= ((arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
- ((tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) arctan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= (- ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
+ ((cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) arccot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= (- ((arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
- ((cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (sec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
sec : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) arctan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((sec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= (((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
+ (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (sec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
sec : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) arccot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((sec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= (((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
- (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (cosec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cosec : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) arctan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((cosec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= (- (((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
+ (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (cosec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cosec : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) arccot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((cosec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= (- (((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
- (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
sin : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= (cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
* ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
sin : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
+ ((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * (sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cos : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= - ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cos : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= (- ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
+ ((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * (cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom tan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
tan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
/ ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom tan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
tan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= (((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
+ ((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * (tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom cot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= - (((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom cot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= (- (((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
+ ((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * (cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom sec : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
sec : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((sec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= (((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
/ ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom sec : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
sec : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((sec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= ((((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
+ ((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * (sec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom cosec : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cosec : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((cosec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= - ((((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom cosec : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cosec : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((cosec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= (- ((((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
+ ((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * (cosec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
exp_R : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= (exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
* ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
exp_R : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= ((exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
+ ((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * (exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
(arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
/ exp_R : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
(((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= - (((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
(arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
/ exp_R : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
(((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= (((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
/ (exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
exp_R : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
* (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= 0 : ( ( ) (
V10()
V11()
V12()
ext-real V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
exp_R : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
* (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= (2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * (exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
/ (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
sin : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
* (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= 0 : ( ( ) (
V10()
V11()
V12()
ext-real V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
sin : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
* (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= (2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * (cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
/ (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cos : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
* (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= 0 : ( ( ) (
V10()
V11()
V12()
ext-real V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
cos : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
* (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= - ((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * (sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
arctan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) arccot : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= ((arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
/ (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st not
0 : ( ( ) (
V10()
V11()
V12()
ext-real V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
(
((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
> - 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V1()
V11()
V12()
ext-real non
positive negative V60() )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) &
((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
< 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
(arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
(((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
/ (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st not
0 : ( ( ) (
V10()
V11()
V12()
ext-real V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
(
((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
> - 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V1()
V11()
V12()
ext-real non
positive negative V60() )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) &
((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
< 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
(id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
(((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
- (x : ( ( ) ( V11() V12() ext-real ) Real) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st not
0 : ( ( ) (
V10()
V11()
V12()
ext-real V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
(
((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
> - 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V1()
V11()
V12()
ext-real non
positive negative V60() )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) &
((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
< 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
(id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
(((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
+ (x : ( ( ) ( V11() V12() ext-real ) Real) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
for
g being ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
PartFunc of ,) st not
0 : ( ( ) (
V10()
V11()
V12()
ext-real V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (g : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) PartFunc of ,) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
g : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
PartFunc of ,)
= #Z 2 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
(
((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
> - 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V1()
V11()
V12()
ext-real non
positive negative V60() )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) &
((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
< 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
g : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
PartFunc of ,)
(#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((g : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) PartFunc of ,) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= ((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
- ((x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
for
g being ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
PartFunc of ,) st not
0 : ( ( ) (
V10()
V11()
V12()
ext-real V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (g : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) PartFunc of ,) (#) (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
g : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
PartFunc of ,)
= #Z 2 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V33(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
(
((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
> - 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V1()
V11()
V12()
ext-real non
positive negative V60() )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) &
((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
< 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
(
g : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
PartFunc of ,)
(#) (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((g : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) PartFunc of ,) (#) (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= ((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
+ ((x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
n being ( ( ) (
V10()
V11()
V12()
ext-real V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom ((1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) (#) ((#Z n : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
n : ( ( ) (
V10()
V11()
V12()
ext-real V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
> 0 : ( ( ) (
V10()
V11()
V12()
ext-real V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
arctan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
<> 0 : ( ( ) (
V10()
V11()
V12()
ext-real V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
(1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
(#) ((#Z n : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
(((1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) (#) ((#Z n : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= - (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) #Z (n : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
n being ( ( ) (
V10()
V11()
V12()
ext-real V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom ((1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) (#) ((#Z n : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
n : ( ( ) (
V10()
V11()
V12()
ext-real V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
> 0 : ( ( ) (
V10()
V11()
V12()
ext-real V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
(1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
(#) ((#Z n : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
(((1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) (#) ((#Z n : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= 1 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
/ (((arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) #Z (n : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((#R (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
arctan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
> 0 : ( ( ) (
V10()
V11()
V12()
ext-real V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
( 2 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
(#) ((#R (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((#R (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) #R (- (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
/ (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom (2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((#R (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
( 2 : ( ( ) (
V1()
V10()
V11()
V12()
ext-real positive non
negative V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) )
(#) ((#R (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((#R (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= - (((arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) #R (- (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom ((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) (#) ((#R (3 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
arctan : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
> 0 : ( ( ) (
V10()
V11()
V12()
ext-real V46()
V47()
V48()
V49()
V50()
V51()
V60()
V61() )
Element of
NAT : ( ( ) (
V46()
V47()
V48()
V49()
V50()
V51()
V52() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
(
(2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V1()
V11()
V12()
ext-real positive non
negative )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
(#) ((#R (3 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
(((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) (#) ((#R (3 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) #R (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
/ (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= dom ((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) (#) ((#R (3 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) &
Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) )
c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) (
open V46()
V47()
V48() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) ( )
set ) ) holds
(
(2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V1()
V11()
V12()
ext-real positive non
negative )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
(#) ((#R (3 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V11()
V12()
ext-real )
Real) st
x : ( ( ) (
V11()
V12()
ext-real )
Real)
in Z : ( (
open ) (
open V46()
V47()
V48() )
Subset of ( ( ) ( )
set ) ) holds
(((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) (#) ((#R (3 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V24() ) (
V19()
V22(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V23(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
V24()
V36()
V37()
V38() )
Element of
K6(
K7(
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ,
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V36()
V37()
V38() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V11()
V12()
ext-real )
Real) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) )
= - (((arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) #R (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) : ( ( ) (
V11()
V12()
ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V47()
V48()
V52()
V62() )
set ) ) ) ) ;