:: SERIES_4 semantic presentation

begin

theorem :: SERIES_4:1
for a, b, c being ( ( real ) ( ext-real V14() real ) number ) holds ((a : ( ( real ) ( ext-real V14() real ) number ) + b : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) + c : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) = (((((a : ( ( real ) ( ext-real V14() real ) number ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) + (b : ( ( real ) ( ext-real V14() real ) number ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + (c : ( ( real ) ( ext-real V14() real ) number ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + ((2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) * b : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + ((2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) * c : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + ((2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * b : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) * c : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) : ( ( ) ( ext-real V14() real ) set ) ;

theorem :: SERIES_4:2
for a, b being ( ( real ) ( ext-real V14() real ) number ) holds (a : ( ( real ) ( ext-real V14() real ) number ) + b : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) |^ 3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) = (((a : ( ( real ) ( ext-real V14() real ) number ) |^ 3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) + ((3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( real ) ( ext-real V14() real ) number ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) * b : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + ((3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (b : ( ( real ) ( ext-real V14() real ) number ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) * a : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + (b : ( ( real ) ( ext-real V14() real ) number ) |^ 3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) : ( ( ) ( ext-real V14() real ) set ) ;

theorem :: SERIES_4:3
for a, b, c being ( ( real ) ( ext-real V14() real ) number ) holds ((a : ( ( real ) ( ext-real V14() real ) number ) - b : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) + c : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) = (((((a : ( ( real ) ( ext-real V14() real ) number ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) + (b : ( ( real ) ( ext-real V14() real ) number ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + (c : ( ( real ) ( ext-real V14() real ) number ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) - ((2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) * b : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + ((2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) * c : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) - ((2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * b : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) * c : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) : ( ( ) ( ext-real V14() real ) set ) ;

theorem :: SERIES_4:4
for a, b, c being ( ( real ) ( ext-real V14() real ) number ) holds ((a : ( ( real ) ( ext-real V14() real ) number ) - b : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) - c : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) = (((((a : ( ( real ) ( ext-real V14() real ) number ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) + (b : ( ( real ) ( ext-real V14() real ) number ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + (c : ( ( real ) ( ext-real V14() real ) number ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) - ((2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) * b : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) - ((2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) * c : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + ((2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * b : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) * c : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) : ( ( ) ( ext-real V14() real ) set ) ;

theorem :: SERIES_4:5
for a, b being ( ( real ) ( ext-real V14() real ) number ) holds (a : ( ( real ) ( ext-real V14() real ) number ) - b : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) |^ 3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) = (((a : ( ( real ) ( ext-real V14() real ) number ) |^ 3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) - ((3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( real ) ( ext-real V14() real ) number ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) * b : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + ((3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (b : ( ( real ) ( ext-real V14() real ) number ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) * a : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) - (b : ( ( real ) ( ext-real V14() real ) number ) |^ 3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) : ( ( ) ( ext-real V14() real ) set ) ;

theorem :: SERIES_4:6
for a, b being ( ( real ) ( ext-real V14() real ) number ) holds (a : ( ( real ) ( ext-real V14() real ) number ) + b : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) |^ 4 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) = ((((a : ( ( real ) ( ext-real V14() real ) number ) |^ 4 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) + ((4 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( real ) ( ext-real V14() real ) number ) |^ 3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) * b : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + ((6 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( real ) ( ext-real V14() real ) number ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) * (b : ( ( real ) ( ext-real V14() real ) number ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + ((4 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (b : ( ( real ) ( ext-real V14() real ) number ) |^ 3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) * a : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + (b : ( ( real ) ( ext-real V14() real ) number ) |^ 4 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) : ( ( ) ( ext-real V14() real ) set ) ;

theorem :: SERIES_4:7
for a, b, c, d being ( ( real ) ( ext-real V14() real ) number ) holds (((a : ( ( real ) ( ext-real V14() real ) number ) + b : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) + c : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) + d : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) = ((((((a : ( ( real ) ( ext-real V14() real ) number ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) + (b : ( ( real ) ( ext-real V14() real ) number ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + (c : ( ( real ) ( ext-real V14() real ) number ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + (d : ( ( real ) ( ext-real V14() real ) number ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + ((((2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) * b : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) + ((2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) * c : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + ((2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) * d : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + (((2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * b : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) * c : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) + ((2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * b : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) * d : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + ((2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * c : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) * d : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) : ( ( ) ( ext-real V14() real ) set ) ;

theorem :: SERIES_4:8
for a, b, c being ( ( real ) ( ext-real V14() real ) number ) holds ((a : ( ( real ) ( ext-real V14() real ) number ) + b : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) + c : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) |^ 3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) = ((((((a : ( ( real ) ( ext-real V14() real ) number ) |^ 3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) + (b : ( ( real ) ( ext-real V14() real ) number ) |^ 3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + (c : ( ( real ) ( ext-real V14() real ) number ) |^ 3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + (((3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( real ) ( ext-real V14() real ) number ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) * b : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) + ((3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( real ) ( ext-real V14() real ) number ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) * c : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + (((3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (b : ( ( real ) ( ext-real V14() real ) number ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) * a : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) + ((3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (b : ( ( real ) ( ext-real V14() real ) number ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) * c : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + (((3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (c : ( ( real ) ( ext-real V14() real ) number ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) * a : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) + ((3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (c : ( ( real ) ( ext-real V14() real ) number ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) * b : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + (((6 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) * b : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) * c : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) : ( ( ) ( ext-real V14() real ) set ) ;

theorem :: SERIES_4:9
for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) )
for a being ( ( real ) ( ext-real V14() real ) number ) st a : ( ( real ) ( ext-real V14() real ) number ) <> 0 : ( ( ) ( ext-real non positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(((1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) |^ (n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) + (a : ( ( real ) ( ext-real V14() real ) number ) |^ (n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) = (((1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) |^ ((2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) + (a : ( ( real ) ( ext-real V14() real ) number ) |^ ((2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) ;

theorem :: SERIES_4:10
for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) )
for a being ( ( real ) ( ext-real V14() real ) number )
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st a : ( ( real ) ( ext-real V14() real ) number ) <> 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = a : ( ( real ) ( ext-real V14() real ) number ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) holds
(Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Element of K11(K12(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = (1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) - (a : ( ( real ) ( ext-real V14() real ) number ) |^ (n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) / (1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) - a : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) : ( ( ) ( ext-real V14() real ) set ) ;

theorem :: SERIES_4:11
for a being ( ( real ) ( ext-real V14() real ) number )
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st a : ( ( real ) ( ext-real V14() real ) number ) <> 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) & a : ( ( real ) ( ext-real V14() real ) number ) <> 0 : ( ( ) ( ext-real non positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = (1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) holds
for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds (Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Element of K11(K12(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = (((1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) - a : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) / (1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) - a : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) : ( ( ) ( ext-real V14() real ) set ) ;

theorem :: SERIES_4:12
for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) )
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = ((10 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + (2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
(Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Element of K11(K12(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = (((10 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 9 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative V14() real ) set ) - (1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 9 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + ((n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) ;

theorem :: SERIES_4:13
for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) )
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = ((2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) + ((1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() V14() real ) set ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) : ( ( ) ( ext-real V14() real ) set ) ) holds
(Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Element of K11(K12(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = ((n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) - ((1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() V14() real ) set ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) : ( ( ) ( ext-real V14() real ) set ) ;

theorem :: SERIES_4:14
for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) )
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * ((1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() V14() real ) set ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) : ( ( ) ( ext-real V14() real ) set ) ) holds
(Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Element of K11(K12(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) - ((2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * ((1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() V14() real ) set ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) : ( ( ) ( ext-real V14() real ) set ) ;

theorem :: SERIES_4:15
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = (((1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() V14() real ) set ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) + (2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) holds
for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds (Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Element of K11(K12(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = (((- (((1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 4 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() V14() real ) set ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) / 3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( V14() ) ( ext-real V14() real ) set ) + ((4 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + (2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) + 3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) ;

theorem :: SERIES_4:16
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = (((1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() V14() real ) set ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) + (3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) holds
for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds (Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Element of K11(K12(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = (((- (((1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 9 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() V14() real ) set ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) / 8 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( V14() ) ( ext-real V14() real ) set ) + ((9 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 8 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + (2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) + 3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) ;

theorem :: SERIES_4:17
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds (Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Element of K11(K12(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = ((n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) - (2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) + 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) ;

theorem :: SERIES_4:18
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = ((2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds (Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Element of K11(K12(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = (n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: SERIES_4:19
for a being ( ( real ) ( ext-real V14() real ) number )
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st a : ( ( real ) ( ext-real V14() real ) number ) <> 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( real ) ( ext-real V14() real ) number ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) : ( ( ) ( ext-real V14() real ) set ) ) holds
for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds (Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Element of K11(K12(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = ((a : ( ( real ) ( ext-real V14() real ) number ) * (1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) - (a : ( ( real ) ( ext-real V14() real ) number ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) / ((1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) - a : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) - ((n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( real ) ( ext-real V14() real ) number ) |^ (n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) / (1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) - a : ( ( real ) ( ext-real V14() real ) number ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) : ( ( ) ( ext-real V14() real ) set ) ;

theorem :: SERIES_4:20
for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) )
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / ((2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) -Root (n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) + (2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) -Root n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( real ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) : ( ( ) ( ext-real V14() real ) set ) ) holds
(Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Element of K11(K12(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) -Root (n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ;

theorem :: SERIES_4:21
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = (2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + ((1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() V14() real ) set ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) : ( ( ) ( ext-real V14() real ) set ) ) holds
for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds (Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Element of K11(K12(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = ((2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ (n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) - ((1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() V14() real ) set ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) ;

theorem :: SERIES_4:22
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = ((n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + (n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / ((n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative V14() real ) set ) : ( ( ) ( ext-real non negative V14() real ) set ) ) holds
for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Element of K11(K12(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = ((n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) - (1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / ((n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() V14() real ) set ) : ( ( ) ( ext-real V14() real ) set ) ;

theorem :: SERIES_4:23
for a being ( ( real ) ( ext-real V14() real ) number )
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st a : ( ( real ) ( ext-real V14() real ) number ) <> 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = (a : ( ( real ) ( ext-real V14() real ) number ) / (a : ( ( real ) ( ext-real V14() real ) number ) - 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) & s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . 0 : ( ( ) ( ext-real non positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = 0 : ( ( ) ( ext-real non positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Element of K11(K12(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = a : ( ( real ) ( ext-real V14() real ) number ) * (((a : ( ( real ) ( ext-real V14() real ) number ) / (a : ( ( real ) ( ext-real V14() real ) number ) - 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) - 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) : ( ( ) ( ext-real V14() real ) set ) ;

theorem :: SERIES_4:24
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = (2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (((3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) / 4 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) : ( ( ) ( ext-real V14() real ) set ) & s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . 0 : ( ( ) ( ext-real non positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = 0 : ( ( ) ( ext-real non positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Element of K11(K12(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = ((2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (((3 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) - 4 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) / 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) ) : ( ( ) ( ext-real V14() real ) set ) + 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) ;

theorem :: SERIES_4:25
for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) )
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = (n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / (n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real positive non negative V4() V14() real ) set ) ) holds
(Partial_Product s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Element of K11(K12(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / (n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real positive non negative V4() V14() real ) set ) ;

theorem :: SERIES_4:26
for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) )
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / (n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real positive non negative V4() V14() real ) set ) ) holds
(Partial_Product s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Element of K11(K12(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / ((n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real positive non negative V4() V14() real ) set ) ;

theorem :: SERIES_4:27
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) & s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . 0 : ( ( ) ( ext-real non positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(Partial_Product s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Element of K11(K12(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ! : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: SERIES_4:28
for a being ( ( real ) ( ext-real V14() real ) number )
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = a : ( ( real ) ( ext-real V14() real ) number ) / n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) & s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . 0 : ( ( ) ( ext-real non positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(Partial_Product s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Element of K11(K12(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = (a : ( ( real ) ( ext-real V14() real ) number ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V14() real ) set ) / (n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) ;

theorem :: SERIES_4:29
for a being ( ( real ) ( ext-real V14() real ) number )
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = a : ( ( real ) ( ext-real V14() real ) number ) & s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . 0 : ( ( ) ( ext-real non positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(Partial_Product s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Element of K11(K12(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = a : ( ( real ) ( ext-real V14() real ) number ) |^ n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) set ) ;

theorem :: SERIES_4:30
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) >= 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) - (1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / (n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) |^ 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative V14() real ) set ) : ( ( ) ( ext-real V14() real ) set ) & s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . 0 : ( ( ) ( ext-real non positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) & s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) . 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
for n being ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) >= 2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(Partial_Product s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) V35() V36() V37() ) Element of K11(K12(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V14() real ) Element of REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) = (n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / (2 : ( ( ) ( ext-real positive non negative V4() ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative ordinal natural V14() real V33() V34() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K11(REAL : ( ( ) ( V4() V46() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real non negative V14() real ) set ) ;