begin
theorem
for
s being ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence) st ( for
n being ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
s : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence)
. n : ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
> 0 : ( ( ) (
V1()
ordinal natural V11()
real ext-real non
positive non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) &
s : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence)
. n : ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
< 1 : ( ( ) (
V1()
ordinal natural V11()
real ext-real positive non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
for
n being ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(Partial_Product s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Element of
K6(
K7(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) )
. n : ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
>= ((Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
- n : ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
set ) ;
theorem
for
s,
s1 being ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence) st ( for
n being ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
s : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence)
. n : ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
> 0 : ( ( ) (
V1()
ordinal natural V11()
real ext-real non
positive non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) &
s1 : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence)
. n : ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
= 1 : ( ( ) (
V1()
ordinal natural V11()
real ext-real positive non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
/ (s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
COMPLEX : ( ( ) (
V1()
V46()
V57()
V63() )
set ) ) ) ) holds
for
n being ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(Partial_Sums s1 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Element of
K6(
K7(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) )
. n : ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
> 0 : ( ( ) (
V1()
ordinal natural V11()
real ext-real non
positive non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
s,
s1 being ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence) st ( for
n being ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
s : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence)
. n : ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
> 0 : ( ( ) (
V1()
ordinal natural V11()
real ext-real non
positive non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) &
s1 : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence)
. n : ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
= 1 : ( ( ) (
V1()
ordinal natural V11()
real ext-real positive non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
/ (s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V11()
real ext-real )
Element of
COMPLEX : ( ( ) (
V1()
V46()
V57()
V63() )
set ) ) ) ) holds
for
n being ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) holds
((Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
* ((Partial_Sums s1 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V11()
real ext-real )
set )
>= (n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V1()
ordinal natural V11()
real ext-real positive non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
^2 : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) ;
theorem
for
s being ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence) st ( for
n being ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) st
n : ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
>= 1 : ( ( ) (
V1()
ordinal natural V11()
real ext-real positive non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
s : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence)
. n : ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
= sqrt n : ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
real ) (
V11()
real ext-real )
set ) &
s : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence)
. 0 : ( ( ) (
V1()
ordinal natural V11()
real ext-real non
positive non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
= 0 : ( ( ) (
V1()
ordinal natural V11()
real ext-real non
positive non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
for
n being ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) st
n : ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
>= 1 : ( ( ) (
V1()
ordinal natural V11()
real ext-real positive non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Element of
K6(
K7(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) )
. n : ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
< ((1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 6 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V46() V57() V63() ) set ) ) * ((4 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 3 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V1()
V11()
real ext-real positive non
negative )
set )
* (sqrt n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
real ) (
V11()
real ext-real )
set ) : ( ( ) (
V11()
real ext-real )
set ) ;
theorem
for
s being ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence) st ( for
n being ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) st
n : ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
>= 1 : ( ( ) (
V1()
ordinal natural V11()
real ext-real positive non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
s : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence)
. n : ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
= sqrt n : ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
real ) (
V11()
real ext-real )
set ) &
s : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence)
. 0 : ( ( ) (
V1()
ordinal natural V11()
real ext-real non
positive non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
= 0 : ( ( ) (
V1()
ordinal natural V11()
real ext-real non
positive non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
for
n being ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) st
n : ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
>= 1 : ( ( ) (
V1()
ordinal natural V11()
real ext-real positive non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Element of
K6(
K7(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) )
. n : ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
> ((2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V46() V57() V63() ) set ) ) * n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V11()
real ext-real non
negative )
set )
* (sqrt n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( (
real ) (
V11()
real ext-real )
set ) : ( ( ) (
V11()
real ext-real )
set ) ;
theorem
for
s being ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence) st ( for
n being ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) st
n : ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
>= 1 : ( ( ) (
V1()
ordinal natural V11()
real ext-real positive non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
s : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence)
. n : ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
= 1 : ( ( ) (
V1()
ordinal natural V11()
real ext-real positive non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
+ (1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / ((2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V1()
V11()
real ext-real positive non
negative )
Element of
COMPLEX : ( ( ) (
V1()
V46()
V57()
V63() )
set ) ) : ( ( ) (
V1()
V11()
real ext-real positive non
negative )
set ) &
s : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence)
. 0 : ( ( ) (
V1()
ordinal natural V11()
real ext-real non
positive non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
= 1 : ( ( ) (
V1()
ordinal natural V11()
real ext-real positive non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
for
n being ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) st
n : ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
>= 1 : ( ( ) (
V1()
ordinal natural V11()
real ext-real positive non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(Partial_Product s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Element of
K6(
K7(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) )
. n : ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
> (1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V1()
V11()
real ext-real positive non
negative )
Element of
COMPLEX : ( ( ) (
V1()
V46()
V57()
V63() )
set ) )
* (sqrt ((2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 3 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V11()
real ext-real )
set ) ;
theorem
for
s being ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence) st ( for
n being ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) st
n : ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
>= 1 : ( ( ) (
V1()
ordinal natural V11()
real ext-real positive non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
s : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence)
. n : ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
= sqrt (n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) &
s : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Real_Sequence)
. 0 : ( ( ) (
V1()
ordinal natural V11()
real ext-real non
positive non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
= 0 : ( ( ) (
V1()
ordinal natural V11()
real ext-real non
positive non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) holds
for
n being ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) st
n : ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
>= 1 : ( ( ) (
V1()
ordinal natural V11()
real ext-real positive non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( (
V21()
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) ) (
V16()
V19(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V20(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V21()
V26(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
V30(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
V35()
V36()
V37() )
Element of
K6(
K7(
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ,
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) (
V35()
V36()
V37() )
set ) ) : ( ( ) ( )
set ) )
. n : ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real )
Element of
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) )
> (n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ordinal natural V11() real ext-real non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
ordinal natural V11()
real ext-real non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) )
/ 2 : ( ( ) (
V1()
ordinal natural V11()
real ext-real positive non
negative V33()
V34()
V57()
V58()
V59()
V60()
V61()
V62() )
Element of
NAT : ( ( ) (
V57()
V58()
V59()
V60()
V61()
V62()
V63() )
Element of
K6(
REAL : ( ( ) (
V1()
V46()
V57()
V58()
V59()
V63() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V11()
real ext-real non
negative )
Element of
COMPLEX : ( ( ) (
V1()
V46()
V57()
V63() )
set ) ) ;