begin
theorem
for
a being ( (
real ) (
ext-real V14()
real )
number )
for
s being ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence) st
a : ( (
real ) (
ext-real V14()
real )
number )
<> 1 : ( ( ) (
ext-real positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) & ( for
n being ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) st
n : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
>= 1 : ( ( ) (
ext-real positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
s : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence)
. n : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
ext-real V14()
real )
Element of
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
= (a : ( ( real ) ( ext-real V14() real ) number ) / (a : ( ( real ) ( ext-real V14() real ) number ) - 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) (
ext-real V14()
real )
set )
|^ n : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
ext-real V14()
real )
set ) &
s : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence)
. 0 : ( ( ) (
ext-real non
positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
ext-real V14()
real )
Element of
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
= 0 : ( ( ) (
ext-real non
positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
for
n being ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) st
n : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
>= 1 : ( ( ) (
ext-real positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Element of
K11(
K12(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) )
. n : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
ext-real V14()
real )
Element of
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
= a : ( (
real ) (
ext-real V14()
real )
number )
* (((a : ( ( real ) ( ext-real V14() real ) number ) / (a : ( ( real ) ( ext-real V14() real ) number ) - 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) - 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
ext-real V14()
real )
set ) : ( ( ) (
ext-real V14()
real )
set ) ;
theorem
for
s being ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence) st ( for
n being ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) st
n : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
>= 1 : ( ( ) (
ext-real positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
s : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence)
. n : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
ext-real V14()
real )
Element of
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
= (2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
* (((3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) / 4 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
ext-real V14()
real )
set ) : ( ( ) (
ext-real V14()
real )
set ) &
s : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence)
. 0 : ( ( ) (
ext-real non
positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
ext-real V14()
real )
Element of
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
= 0 : ( ( ) (
ext-real non
positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
for
n being ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) st
n : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
>= 1 : ( ( ) (
ext-real positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Element of
K11(
K12(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) )
. n : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
ext-real V14()
real )
Element of
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
= ((2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (((3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) - 4 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) / 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) (
ext-real V14()
real )
set )
+ 2 : ( ( ) (
ext-real positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
ext-real V14()
real )
set ) ;
theorem
for
s being ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence) st ( for
n being ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) st
n : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
>= 1 : ( ( ) (
ext-real positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
s : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence)
. n : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
ext-real V14()
real )
Element of
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
= n : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) &
s : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence)
. 0 : ( ( ) (
ext-real non
positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
ext-real V14()
real )
Element of
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
= 1 : ( ( ) (
ext-real positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
for
n being ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) st
n : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
>= 1 : ( ( ) (
ext-real positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(Partial_Product s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Element of
K11(
K12(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) )
. n : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
ext-real V14()
real )
Element of
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
= n : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
! : ( ( ) (
ext-real positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
a being ( (
real ) (
ext-real V14()
real )
number )
for
s being ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence) st ( for
n being ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) st
n : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
>= 1 : ( ( ) (
ext-real positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
s : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence)
. n : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
ext-real V14()
real )
Element of
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
= a : ( (
real ) (
ext-real V14()
real )
number )
/ n : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
ext-real V14()
real )
set ) &
s : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence)
. 0 : ( ( ) (
ext-real non
positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
ext-real V14()
real )
Element of
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
= 1 : ( ( ) (
ext-real positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
for
n being ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) st
n : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
>= 1 : ( ( ) (
ext-real positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(Partial_Product s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Element of
K11(
K12(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) )
. n : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
ext-real V14()
real )
Element of
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
= (a : ( ( real ) ( ext-real V14() real ) number ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
ext-real V14()
real )
set )
/ (n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) (
ext-real positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
ext-real V14()
real )
set ) ;
theorem
for
a being ( (
real ) (
ext-real V14()
real )
number )
for
s being ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence) st ( for
n being ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) st
n : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
>= 1 : ( ( ) (
ext-real positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
s : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence)
. n : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
ext-real V14()
real )
Element of
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
= a : ( (
real ) (
ext-real V14()
real )
number ) &
s : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence)
. 0 : ( ( ) (
ext-real non
positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
ext-real V14()
real )
Element of
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
= 1 : ( ( ) (
ext-real positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
for
n being ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) st
n : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
>= 1 : ( ( ) (
ext-real positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(Partial_Product s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Element of
K11(
K12(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) )
. n : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
ext-real V14()
real )
Element of
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
= a : ( (
real ) (
ext-real V14()
real )
number )
|^ n : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
ext-real V14()
real )
set ) ;
theorem
for
s being ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence) st ( for
n being ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) st
n : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
>= 2 : ( ( ) (
ext-real positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
s : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence)
. n : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
ext-real V14()
real )
Element of
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
= 1 : ( ( ) (
ext-real positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
- (1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / (n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
ext-real non
negative V14()
real )
set ) : ( ( ) (
ext-real V14()
real )
set ) &
s : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence)
. 0 : ( ( ) (
ext-real non
positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
ext-real V14()
real )
Element of
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
= 1 : ( ( ) (
ext-real positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) &
s : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence)
. 1 : ( ( ) (
ext-real positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
ext-real V14()
real )
Element of
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
= 1 : ( ( ) (
ext-real positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
for
n being ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) st
n : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
>= 2 : ( ( ) (
ext-real positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(Partial_Product s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Element of
K11(
K12(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) )
. n : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
ext-real V14()
real )
Element of
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) )
= (n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
ext-real positive non
negative V4()
ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
/ (2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
ext-real non
negative ordinal natural V14()
real V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K11(
REAL : ( ( ) (
V4()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
ext-real non
negative V14()
real )
set ) ;