:: SERIES_3 semantic presentation

begin

registration
let x be ( ( real ) ( V11() real ext-real ) number ) ;
cluster |.x : ( ( real ) ( V11() real ext-real ) set ) .| : ( ( real ) ( V11() real ext-real non negative ) set ) -> real non negative ;
end;

theorem :: SERIES_3:1
for y, x, m being ( ( real ) ( V11() real ext-real ) number ) st y : ( ( real ) ( V11() real ext-real ) number ) > x : ( ( real ) ( V11() real ext-real ) number ) & x : ( ( real ) ( V11() real ext-real ) number ) >= 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) & m : ( ( real ) ( V11() real ext-real ) number ) >= 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
x : ( ( real ) ( V11() real ext-real ) number ) / y : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) <= (x : ( ( real ) ( V11() real ext-real ) number ) + m : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / (y : ( ( real ) ( V11() real ext-real ) number ) + m : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) : ( ( ) ( V11() real ext-real ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) ;

theorem :: SERIES_3:2
for a, b being ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) holds (a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) + b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) set ) / 2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) >= sqrt (a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) * b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) set ) : ( ( real ) ( V11() real ext-real ) set ) ;

theorem :: SERIES_3:3
for b, a being ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) holds (b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) / a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) + (a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) / b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) >= 2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: SERIES_3:4
for x, y being ( ( real ) ( V11() real ext-real ) number ) holds ((x : ( ( real ) ( V11() real ext-real ) number ) + y : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / 2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) ^2 : ( ( ) ( V11() real ext-real ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) >= x : ( ( real ) ( V11() real ext-real ) number ) * y : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real ) set ) ;

theorem :: SERIES_3:5
for x, y being ( ( real ) ( V11() real ext-real ) number ) holds ((x : ( ( real ) ( V11() real ext-real ) number ) ^2) : ( ( ) ( V11() real ext-real ) set ) + (y : ( ( real ) ( V11() real ext-real ) number ) ^2) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) / 2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) >= ((x : ( ( real ) ( V11() real ext-real ) number ) + y : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) / 2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) ^2 : ( ( ) ( V11() real ext-real ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) ;

theorem :: SERIES_3:6
for x, y being ( ( real ) ( V11() real ext-real ) number ) holds (x : ( ( real ) ( V11() real ext-real ) number ) ^2) : ( ( ) ( V11() real ext-real ) set ) + (y : ( ( real ) ( V11() real ext-real ) number ) ^2) : ( ( ) ( V11() real ext-real ) set ) : ( ( ) ( V11() real ext-real ) set ) >= (2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) * y : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ;

theorem :: SERIES_3:7
for x, y being ( ( real ) ( V11() real ext-real ) number ) holds ((x : ( ( real ) ( V11() real ext-real ) number ) ^2) : ( ( ) ( V11() real ext-real ) set ) + (y : ( ( real ) ( V11() real ext-real ) number ) ^2) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) / 2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) >= x : ( ( real ) ( V11() real ext-real ) number ) * y : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real ) set ) ;

theorem :: SERIES_3:8
for x, y being ( ( real ) ( V11() real ext-real ) number ) holds (x : ( ( real ) ( V11() real ext-real ) number ) ^2) : ( ( ) ( V11() real ext-real ) set ) + (y : ( ( real ) ( V11() real ext-real ) number ) ^2) : ( ( ) ( V11() real ext-real ) set ) : ( ( ) ( V11() real ext-real ) set ) >= (2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (abs x : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) * (abs y : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ;

theorem :: SERIES_3:9
for x, y being ( ( real ) ( V11() real ext-real ) number ) holds (x : ( ( real ) ( V11() real ext-real ) number ) + y : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ^2 : ( ( ) ( V11() real ext-real ) set ) >= (4 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) * y : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ;

theorem :: SERIES_3:10
for x, y, z being ( ( real ) ( V11() real ext-real ) number ) holds ((x : ( ( real ) ( V11() real ext-real ) number ) ^2) : ( ( ) ( V11() real ext-real ) set ) + (y : ( ( real ) ( V11() real ext-real ) number ) ^2) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) + (z : ( ( real ) ( V11() real ext-real ) number ) ^2) : ( ( ) ( V11() real ext-real ) set ) : ( ( ) ( V11() real ext-real ) set ) >= ((x : ( ( real ) ( V11() real ext-real ) number ) * y : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) + (y : ( ( real ) ( V11() real ext-real ) number ) * z : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) + (x : ( ( real ) ( V11() real ext-real ) number ) * z : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) : ( ( ) ( V11() real ext-real ) set ) ;

theorem :: SERIES_3:11
for x, y, z being ( ( real ) ( V11() real ext-real ) number ) holds ((x : ( ( real ) ( V11() real ext-real ) number ) + y : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) + z : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ^2 : ( ( ) ( V11() real ext-real ) set ) >= 3 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (((x : ( ( real ) ( V11() real ext-real ) number ) * y : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) + (y : ( ( real ) ( V11() real ext-real ) number ) * z : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) + (x : ( ( real ) ( V11() real ext-real ) number ) * z : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ;

theorem :: SERIES_3:12
for a, b, c being ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) holds ((a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) |^ 3 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) set ) + (b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) |^ 3 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) + (c : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) |^ 3 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) set ) : ( ( ) ( V11() real ext-real ) set ) >= ((3 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) * b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) * c : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ;

theorem :: SERIES_3:13
for a, b, c being ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) holds (((a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) |^ 3 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) set ) + (b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) |^ 3 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) + (c : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) |^ 3 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) / 3 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) >= (a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) * b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) set ) * c : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) : ( ( ) ( V1() V11() real ext-real positive non negative ) set ) ;

theorem :: SERIES_3:14
for a, b, c being ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) holds (((a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) / b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) |^ 3 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) set ) + ((b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) / c : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) |^ 3 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) + ((c : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) / a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) |^ 3 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) set ) : ( ( ) ( V11() real ext-real ) set ) >= ((b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) / a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) + (c : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) / b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) + (a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) / c : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) ;

theorem :: SERIES_3:15
for a, b, c being ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) holds (a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) + b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) set ) + c : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) : ( ( ) ( V1() V11() real ext-real positive non negative ) set ) >= 3 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (3 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) -root ((a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) * b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) set ) * c : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) set ) ) : ( ( real ) ( V11() real ext-real ) set ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ;

theorem :: SERIES_3:16
for a, b, c being ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) holds ((a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) + b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) set ) + c : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) set ) / 3 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) >= 3 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) -root ((a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) * b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) set ) * c : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) set ) : ( ( real ) ( V11() real ext-real ) set ) ;

theorem :: SERIES_3:17
for x, y, z being ( ( real ) ( V11() real ext-real ) number ) st (x : ( ( real ) ( V11() real ext-real ) number ) + y : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) + z : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real ) set ) = 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
((x : ( ( real ) ( V11() real ext-real ) number ) * y : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) + (y : ( ( real ) ( V11() real ext-real ) number ) * z : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) + (x : ( ( real ) ( V11() real ext-real ) number ) * z : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) : ( ( ) ( V11() real ext-real ) set ) <= 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) ;

theorem :: SERIES_3:18
for x, y being ( ( real ) ( V11() real ext-real ) number ) st x : ( ( real ) ( V11() real ext-real ) number ) + y : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real ) set ) = 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
x : ( ( real ) ( V11() real ext-real ) number ) * y : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real ) set ) <= 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 4 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) ;

theorem :: SERIES_3:19
for x, y being ( ( real ) ( V11() real ext-real ) number ) st x : ( ( real ) ( V11() real ext-real ) number ) + y : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real ) set ) = 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(x : ( ( real ) ( V11() real ext-real ) number ) ^2) : ( ( ) ( V11() real ext-real ) set ) + (y : ( ( real ) ( V11() real ext-real ) number ) ^2) : ( ( ) ( V11() real ext-real ) set ) : ( ( ) ( V11() real ext-real ) set ) >= 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) ;

theorem :: SERIES_3:20
for a, b being ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) st a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) + b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) : ( ( ) ( V1() V11() real ext-real positive non negative ) set ) = 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + (1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) * (1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + (1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) >= 9 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: SERIES_3:21
for x, y being ( ( real ) ( V11() real ext-real ) number ) st x : ( ( real ) ( V11() real ext-real ) number ) + y : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real ) set ) = 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(x : ( ( real ) ( V11() real ext-real ) number ) |^ 3 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) set ) + (y : ( ( real ) ( V11() real ext-real ) number ) |^ 3 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) set ) : ( ( ) ( V11() real ext-real ) set ) >= 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 4 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) ;

theorem :: SERIES_3:22
for a, b being ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) st a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) + b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) : ( ( ) ( V1() V11() real ext-real positive non negative ) set ) = 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) |^ 3 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) set ) + (b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) |^ 3 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) set ) : ( ( ) ( V11() real ext-real ) set ) < 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: SERIES_3:23
for a, b being ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) st a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) + b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) : ( ( ) ( V1() V11() real ext-real positive non negative ) set ) = 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) + (1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) set ) * (b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) + (1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) set ) : ( ( ) ( V1() V11() real ext-real positive non negative ) set ) >= 25 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / 4 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) ;

theorem :: SERIES_3:24
for x, a being ( ( real ) ( V11() real ext-real ) number ) st abs x : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) <= a : ( ( real ) ( V11() real ext-real ) number ) holds
x : ( ( real ) ( V11() real ext-real ) number ) ^2 : ( ( ) ( V11() real ext-real ) set ) <= a : ( ( real ) ( V11() real ext-real ) number ) ^2 : ( ( ) ( V11() real ext-real ) set ) ;

theorem :: SERIES_3:25
for a being ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number )
for x being ( ( real ) ( V11() real ext-real ) number ) st abs x : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) >= a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) holds
x : ( ( real ) ( V11() real ext-real ) number ) ^2 : ( ( ) ( V11() real ext-real ) set ) >= a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ^2 : ( ( ) ( V11() real ext-real ) set ) ;

theorem :: SERIES_3:26
for x, y being ( ( real ) ( V11() real ext-real ) number ) holds abs ((abs x : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) - (abs y : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) <= (abs x : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) + (abs y : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ;

theorem :: SERIES_3:27
for a, b, c being ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) st (a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) * b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) set ) * c : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) : ( ( ) ( V1() V11() real ext-real positive non negative ) set ) = 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
((1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) + (1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) + (1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / c : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) >= ((sqrt a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( real ) ( V11() real ext-real ) set ) + (sqrt b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( real ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) + (sqrt c : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( real ) ( V11() real ext-real ) set ) : ( ( ) ( V11() real ext-real ) set ) ;

theorem :: SERIES_3:28
for x, y, z being ( ( real ) ( V11() real ext-real ) number ) st x : ( ( real ) ( V11() real ext-real ) number ) > 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) & y : ( ( real ) ( V11() real ext-real ) number ) > 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) & z : ( ( real ) ( V11() real ext-real ) number ) < 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) & (x : ( ( real ) ( V11() real ext-real ) number ) + y : ( ( real ) ( V11() real ext-real ) number ) ) : ( ( ) ( V11() real ext-real ) set ) + z : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real ) set ) = 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(((x : ( ( real ) ( V11() real ext-real ) number ) |^ 2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) set ) + (y : ( ( real ) ( V11() real ext-real ) number ) |^ 2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) + (z : ( ( real ) ( V11() real ext-real ) number ) |^ 2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) |^ 3 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) set ) >= 6 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * ((((x : ( ( real ) ( V11() real ext-real ) number ) |^ 3 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) set ) + (y : ( ( real ) ( V11() real ext-real ) number ) |^ 3 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) + (z : ( ( real ) ( V11() real ext-real ) number ) |^ 3 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) ^2) : ( ( ) ( V11() real ext-real ) set ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ;

theorem :: SERIES_3:29
for a, b, c being ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) st a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) >= 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) to_power b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( real ) ( V11() real ext-real ) set ) + (a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) to_power c : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( real ) ( V11() real ext-real ) set ) : ( ( ) ( V11() real ext-real ) set ) >= 2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) to_power (sqrt (b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) * c : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) set ) ) : ( ( real ) ( V11() real ext-real ) set ) ) : ( ( real ) ( V11() real ext-real ) set ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ;

theorem :: SERIES_3:30
for a, b, c being ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) st a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) >= b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) & b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) >= c : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) holds
((a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) to_power a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( real ) ( V11() real ext-real ) set ) * (b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) to_power b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( real ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) * (c : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) to_power c : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( real ) ( V11() real ext-real ) set ) : ( ( ) ( V11() real ext-real ) set ) >= ((a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) * b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) set ) * c : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) set ) to_power (((a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) + b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) set ) + c : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) set ) / 3 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) : ( ( real ) ( V11() real ext-real ) set ) ;

theorem :: SERIES_3:31
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) )
for a, b being ( ( real non negative ) ( V11() real ext-real non negative ) number ) holds (a : ( ( real non negative ) ( V11() real ext-real non negative ) number ) + b : ( ( real non negative ) ( V11() real ext-real non negative ) number ) ) : ( ( ) ( V11() real ext-real non negative ) set ) |^ (n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) set ) >= (a : ( ( real non negative ) ( V11() real ext-real non negative ) number ) |^ (n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) set ) + (((n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( real non negative ) ( V11() real ext-real non negative ) number ) |^ (n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) * b : ( ( real non negative ) ( V11() real ext-real non negative ) number ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ;

theorem :: SERIES_3:32
for a, b being ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number )
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds ((a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) |^ n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) set ) + (b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) |^ n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) set ) ) : ( ( ) ( V11() real ext-real ) set ) / 2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) >= ((a : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) + b : ( ( real positive ) ( V1() V11() real ext-real positive non negative ) number ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) set ) / 2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) |^ n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) set ) ;

theorem :: SERIES_3:33
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) st ( for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds (Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: SERIES_3:34
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) st ( for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) >= 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds (Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) >= 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: SERIES_3:35
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) )
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) st ( for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) < 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
(Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) < 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: SERIES_3:36
for s, s1 being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) st s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) = s1 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) (#) s1 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) : ( ( V21() ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) holds
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds (Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) >= 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: SERIES_3:37
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) )
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) st ( for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) & s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) > s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . (n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ) holds
(n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . (n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) > (Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ;

theorem :: SERIES_3:38
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) )
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) st ( for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) & s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) >= s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . (n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ) holds
(n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * (s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . (n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) >= (Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ;

theorem :: SERIES_3:39
for s, s1, s2 being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) st s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) = s1 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) (#) s2 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) : ( ( V21() ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) & ( for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( s1 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) >= 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) & s2 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) >= 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds (Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) <= ((Partial_Sums s1 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) * ((Partial_Sums s2 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ;

theorem :: SERIES_3:40
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) )
for s, s1, s2 being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) st s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) = s1 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) (#) s2 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) : ( ( V21() ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) & ( for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( s1 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) < 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) & s2 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) < 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
(Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) <= ((Partial_Sums s1 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) * ((Partial_Sums s2 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ;

theorem :: SERIES_3:41
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence)
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds abs ((Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) <= (Partial_Sums (abs s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ;

theorem :: SERIES_3:42
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) )
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) holds (Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) <= (Partial_Sums (abs s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ;

definition
let s be ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ;
func Partial_Product s -> ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) means :: SERIES_3:def 1
( it : ( ( ) ( ) set ) . 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) = s : ( ( ) ( ) set ) . 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) & ( for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds it : ( ( ) ( ) set ) . (n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) = (it : ( ( ) ( ) set ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) * (s : ( ( ) ( ) set ) . (n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) );
end;

theorem :: SERIES_3:43
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) )
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) st ( for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
(Partial_Product s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: SERIES_3:44
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) )
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) st ( for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) >= 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
(Partial_Product s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) >= 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: SERIES_3:45
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) st ( for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) & s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) < 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( (Partial_Product s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) & (Partial_Product s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) < 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: SERIES_3:46
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) st ( for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) >= 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds (Partial_Product s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) >= 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: SERIES_3:47
for s1, s2 being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) st ( for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( s1 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) >= 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) & s2 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) >= 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds ((Partial_Product s1 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) + ((Partial_Product s2 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) <= (Partial_Product (s1 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) + s2 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ;

theorem :: SERIES_3:48
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) )
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) st ( for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) = ((2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / ((2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V11() real ext-real positive non negative ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) ) holds
(Partial_Product s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) <= 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) / (sqrt ((3 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 4 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of COMPLEX : ( ( ) ( V1() V45() V57() V63() ) set ) ) ;

theorem :: SERIES_3:49
for s1, s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) st ( for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( s1 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) = 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + (s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) & s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) > - 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) & s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) < 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + ((Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) <= (Partial_Product s1 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ;

theorem :: SERIES_3:50
for s1, s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) st ( for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( s1 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) = 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + (s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) & s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) >= 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + ((Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) <= (Partial_Product s1 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ;

theorem :: SERIES_3:51
for s3, s1, s2, s4, s5 being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) st s3 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) = s1 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) (#) s2 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) : ( ( V21() ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) & s4 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) = s1 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) (#) s1 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) : ( ( V21() ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) & s5 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) = s2 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) (#) s2 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) : ( ( V21() ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) holds
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds ((Partial_Sums s3 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ^2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) <= ((Partial_Sums s4 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) * ((Partial_Sums s5 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ;

theorem :: SERIES_3:52
for s4, s1, s5, s2, s3 being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) st s4 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) = s1 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) (#) s1 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) : ( ( V21() ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) & s5 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) = s2 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) (#) s2 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) : ( ( V21() ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) & ( for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( s1 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) >= 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) & s2 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) >= 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) & s3 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) = ((s1 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) + (s2 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ^2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ) holds
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds sqrt ((Partial_Sums s3 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) <= (sqrt ((Partial_Sums s4 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) + (sqrt ((Partial_Sums s5 : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ;

theorem :: SERIES_3:53
for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) )
for s being ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) st ( for n being ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) > 0 : ( ( ) ( V1() ordinal natural V11() real ext-real non positive non negative V43() V56() V57() V58() V59() V60() V61() V62() V63() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) & s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) >= s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . (n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ) holds
(Partial_Sums s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Element of K6(K7(NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V33() V34() V35() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) >= (n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) * ((n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() ordinal natural V11() real ext-real positive non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) -root ((Partial_Product s : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) ) : ( ( V21() V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) ( V16() V19( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V21() V26( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) V33() V34() V35() ) Real_Sequence) . n : ( ( ) ( ordinal natural V11() real ext-real non negative V43() V56() V57() V58() V59() V60() V61() V62() ) Element of NAT : ( ( ) ( V57() V58() V59() V60() V61() V62() V63() ) Element of K6(REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( V1() V45() V57() V58() V59() V63() ) set ) ) ;