SERIES

theorem :: SERIES_2:2

for n being ( ( real ) ( ext-real V14() real ) number ) holds
( (n : ( ( real ) ( ext-real V14() real ) number ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) |^ 3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) = (((n : ( ( real ) ( ext-real V14() real ) number ) |^ 3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) + (3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( real ) ( ext-real V14() real ) number ) |^ 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + (3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) & (n : ( ( real ) ( ext-real V14() real ) number ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) |^ 4 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) = ((((n : ( ( real ) ( ext-real V14() real ) number ) |^ 4 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) + (4 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( real ) ( ext-real V14() real ) number ) |^ 3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + (6 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( real ) ( ext-real V14() real ) number ) |^ 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + (4 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) & (n : ( ( real ) ( ext-real V14() real ) number ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) |^ 5 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) = (((((n : ( ( real ) ( ext-real V14() real ) number ) |^ 5 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) + (5 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( real ) ( ext-real V14() real ) number ) |^ 4 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + (10 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( real ) ( ext-real V14() real ) number ) |^ 3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + (10 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( real ) ( ext-real V14() real ) number ) |^ 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + (5 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) ;

theorem :: SERIES_2:3

theorem :: SERIES_2:4

theorem :: SERIES_2:5

theorem :: SERIES_2:6

theorem :: SERIES_2:7

theorem :: SERIES_2:8

for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = ((n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
for n being ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = ((((n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 4 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 5 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) ;

theorem :: SERIES_2:9

theorem :: SERIES_2:10

theorem :: SERIES_2:11

for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / (((n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) ) holds
for n being ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = (1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 18 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) - (1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / (((3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) ;

theorem :: SERIES_2:12

theorem :: SERIES_2:13

theorem :: SERIES_2:14

theorem :: SERIES_2:15

theorem :: SERIES_2:16

theorem :: SERIES_2:17

theorem :: SERIES_2:18

theorem :: SERIES_2:19

theorem :: SERIES_2:20

for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ 6 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
for n being ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = (((n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * ((2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * ((((3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ 4 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + (6 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ 3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) - (3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) / 42 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) ;

theorem :: SERIES_2:21

for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ 7 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
for n being ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = (((n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * ((n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (((((3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ 4 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + (6 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ 3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) - (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) - (4 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) + 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) / 24 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) ;

theorem :: SERIES_2:22

theorem :: SERIES_2:23

for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * ((n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
for n being ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = ((((n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * ((2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 10 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) ;

theorem :: SERIES_2:24

theorem :: SERIES_2:25

for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / ((n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) & s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . 0 : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = 0 : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
for n being ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) >= 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = ((3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 4 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) - (1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / (2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) - (1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / (2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) ;

theorem :: SERIES_2:26

theorem :: SERIES_2:27

theorem :: SERIES_2:28

for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / ((((2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) * ((2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) * ((2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) & s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . 0 : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = 0 : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
for n being ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = (1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 12 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) - (1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / ((4 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * ((2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * ((2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) ;

theorem :: SERIES_2:29

for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / ((((3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) - 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) * ((3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) * ((3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 4 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) & s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . 0 : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = 0 : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
for n being ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = (1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 24 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) - (1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / ((6 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * ((3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * ((3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 4 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) ;

theorem :: SERIES_2:30

for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = ((2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) / ((n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) ) holds
for n being ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = ((3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 4 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) - (2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) + (1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / ((2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) ;

theorem :: SERIES_2:31

for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / ((n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) ) holds
for n being ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = (((29 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 36 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) - (1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) - (3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / ((2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) - (4 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / (((3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) ;

theorem :: SERIES_2:32

theorem :: SERIES_2:33

for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = ((n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (4 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / ((n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) ) holds
for n being ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = (2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) + (((n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) * (4 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) / (3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) ;

theorem :: SERIES_2:34

theorem :: SERIES_2:35

theorem :: SERIES_2:36

for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = (((- 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V14() ) ( ext-real V14() real ) set ) |^ n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) * (2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) / (((2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + ((- 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V14() ) ( ext-real V14() real ) set ) |^ (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) * ((2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + ((- 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V14() ) ( ext-real V14() real ) set ) |^ (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) ) holds
for n being ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = (1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) + (((- 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V14() ) ( ext-real V14() real ) set ) |^ (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) / (3 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * ((2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + ((- 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V14() ) ( ext-real V14() real ) set ) |^ (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) ;

theorem :: SERIES_2:37

theorem :: SERIES_2:38

theorem :: SERIES_2:39

for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = (((n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) / ((n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( ext-real positive natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) & s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . 0 : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = 0 : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
for n being ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = (1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) - ((n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / ((n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( ext-real positive natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) ;

theorem :: SERIES_2:40

theorem :: SERIES_2:41

for a, b 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = (a : ( ( real ) ( ext-real V14() real ) number ) * n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) + b : ( ( real ) ( ext-real V14() real ) number ) : ( ( ) ( ext-real V14() real ) set ) ) holds
for n being ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = ((((a : ( ( real ) ( ext-real V14() real ) number ) * (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) * n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) / 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) + (n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * b : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + b : ( ( real ) ( ext-real V14() real ) number ) : ( ( ) ( ext-real V14() real ) set ) ;

theorem :: SERIES_2:42

for a, b 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = (a : ( ( real ) ( ext-real V14() real ) number ) * n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) + b : ( ( real ) ( ext-real V14() real ) number ) : ( ( ) ( ext-real V14() real ) set ) ) holds
for n being ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() 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() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) = ((n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * ((s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . 0 : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) + (s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) / 2 : ( ( ) ( ext-real positive V4() natural V14() real V33() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of COMPLEX : ( ( ) ( V4() V45() V57() V63() ) set ) ) ;