:: SERIES_1 semantic presentation

begin

theorem :: SERIES_1:1
for a being ( ( real ) ( ext-real V33() real ) number )
for s being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st 0 : ( ( ) ( empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) < a : ( ( real ) ( ext-real V33() real ) number ) & a : ( ( real ) ( ext-real V33() real ) number ) < 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) & ( for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) = a : ( ( real ) ( ext-real V33() real ) number ) to_power (n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( real ) ( ext-real V33() real ) set ) ) holds
( s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & lim s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) = 0 : ( ( ) ( empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: SERIES_1:2
for a being ( ( real ) ( ext-real V33() real ) number )
for n being ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer ) Nat) holds (abs a : ( ( real ) ( ext-real V33() real ) number ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) to_power n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer ) Nat) : ( ( real ) ( ext-real V33() real ) set ) = abs (a : ( ( real ) ( ext-real V33() real ) number ) to_power n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer ) Nat) ) : ( ( real ) ( ext-real V33() real ) set ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ;

theorem :: SERIES_1:3
for a being ( ( real ) ( ext-real V33() real ) number )
for s being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st abs a : ( ( real ) ( ext-real V33() real ) number ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) < 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) & ( for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) = a : ( ( real ) ( ext-real V33() real ) number ) to_power (n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( real ) ( ext-real V33() real ) set ) ) holds
( s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & lim s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) = 0 : ( ( ) ( empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

definition
let X be ( ( non empty complex-membered add-closed ) ( non empty complex-membered add-closed ) set ) ;
let s1, s2 be ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,X : ( ( non empty complex-membered add-closed ) ( non empty complex-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined X : ( ( non empty complex-membered add-closed ) ( non empty complex-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,X : ( ( non empty complex-membered add-closed ) ( non empty complex-membered add-closed ) set ) ) complex-valued ) sequence of X : ( ( non empty complex-membered add-closed ) ( non empty complex-membered add-closed ) set ) ) ;
:: original: +
redefine func s1 + s2 -> ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,X : ( ( ext-real ) ( ext-real ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined X : ( ( ext-real ) ( ext-real ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,X : ( ( ext-real ) ( ext-real ) set ) ) complex-valued ) sequence of X : ( ( ext-real ) ( ext-real ) set ) ) ;
end;

definition
let s be ( ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) complex-valued ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) complex-valued ) ManySortedSet of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ;
func Partial_Sums s -> ( ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) complex-valued ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) complex-valued ) ManySortedSet of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) means :: SERIES_1:def 1
( it : ( ( ext-real ) ( ext-real ) set ) . 0 : ( ( ) ( empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = s : ( ( ext-real ) ( ext-real ) set ) . 0 : ( ( ) ( empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) & ( for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds it : ( ( ext-real ) ( ext-real ) set ) . (n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = (it : ( ( ext-real ) ( ext-real ) set ) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) set ) + (s : ( ( ext-real ) ( ext-real ) set ) . (n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) );
end;

registration
let s be ( ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) real-valued ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) complex-valued ext-real-valued real-valued ) ManySortedSet of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ;
cluster Partial_Sums s : ( ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) real-valued ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) complex-valued ext-real-valued real-valued ) set ) : ( ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) complex-valued ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) complex-valued ) ManySortedSet of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) -> Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) complex-valued real-valued ;
end;

definition
let s be ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ;
:: original: Partial_Sums
redefine func Partial_Sums s -> ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ;
end;

definition
let s be ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ;
attr s is summable means :: SERIES_1:def 2
 Partial_Sums s : ( ( ) ( ) set ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent ;
func Sum s -> ( ( ) ( ext-real V33() real ) Real) equals :: SERIES_1:def 3
 lim (Partial_Sums s : ( ( ) ( ) set ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ;
end;

theorem :: SERIES_1:4
for s being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable holds
( s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & lim s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) = 0 : ( ( ) ( empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: SERIES_1:5
for X being ( ( non empty complex-membered add-closed ) ( non empty complex-membered add-closed ) set )
for s1, s2 being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty complex-membered add-closed ) ( non empty complex-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined b1 : ( ( non empty complex-membered add-closed ) ( non empty complex-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty complex-membered add-closed ) ( non empty complex-membered add-closed ) set ) ) complex-valued ) sequence of X : ( ( non empty complex-membered add-closed ) ( non empty complex-membered add-closed ) set ) ) holds (Partial_Sums s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty complex-membered add-closed ) ( non empty complex-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined b1 : ( ( non empty complex-membered add-closed ) ( non empty complex-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty complex-membered add-closed ) ( non empty complex-membered add-closed ) set ) ) complex-valued ) sequence of b1 : ( ( non empty complex-membered add-closed ) ( non empty complex-membered add-closed ) set ) ) ) : ( ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) complex-valued ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) complex-valued ) ManySortedSet of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) + (Partial_Sums s2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty complex-membered add-closed ) ( non empty complex-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined b1 : ( ( non empty complex-membered add-closed ) ( non empty complex-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty complex-membered add-closed ) ( non empty complex-membered add-closed ) set ) ) complex-valued ) sequence of b1 : ( ( non empty complex-membered add-closed ) ( non empty complex-membered add-closed ) set ) ) ) : ( ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) complex-valued ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) complex-valued ) ManySortedSet of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) complex-valued ) set ) = Partial_Sums (s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty complex-membered add-closed ) ( non empty complex-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined b1 : ( ( non empty complex-membered add-closed ) ( non empty complex-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty complex-membered add-closed ) ( non empty complex-membered add-closed ) set ) ) complex-valued ) sequence of b1 : ( ( non empty complex-membered add-closed ) ( non empty complex-membered add-closed ) set ) ) + s2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty complex-membered add-closed ) ( non empty complex-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined b1 : ( ( non empty complex-membered add-closed ) ( non empty complex-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty complex-membered add-closed ) ( non empty complex-membered add-closed ) set ) ) complex-valued ) sequence of b1 : ( ( non empty complex-membered add-closed ) ( non empty complex-membered add-closed ) set ) ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty complex-membered add-closed ) ( non empty complex-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined b1 : ( ( non empty complex-membered add-closed ) ( non empty complex-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty complex-membered add-closed ) ( non empty complex-membered add-closed ) set ) ) complex-valued ) sequence of b1 : ( ( non empty complex-membered add-closed ) ( non empty complex-membered add-closed ) set ) ) : ( ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) complex-valued ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) complex-valued ) ManySortedSet of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: SERIES_1:6
for s1, s2 being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) holds (Partial_Sums s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) - (Partial_Sums s2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) = Partial_Sums (s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) - s2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ;

theorem :: SERIES_1:7
for s1, s2 being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable & s2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable holds
( s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) + s2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) sequence of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) is summable & Sum (s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) + s2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) sequence of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ext-real V33() real ) Real) = (Sum s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( ext-real V33() real ) Real) + (Sum s2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( ext-real V33() real ) Real) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ;

theorem :: SERIES_1:8
for s1, s2 being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable & s2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable holds
( s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) - s2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is summable & Sum (s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) - s2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ext-real V33() real ) Real) = (Sum s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( ext-real V33() real ) Real) - (Sum s2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( ext-real V33() real ) Real) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ;

theorem :: SERIES_1:9
for r being ( ( real ) ( ext-real V33() real ) number )
for s being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) holds Partial_Sums (r : ( ( real ) ( ext-real V33() real ) number ) (#) s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) = r : ( ( real ) ( ext-real V33() real ) number ) (#) (Partial_Sums s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SERIES_1:10
for r being ( ( real ) ( ext-real V33() real ) number )
for s being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable holds
( r : ( ( real ) ( ext-real V33() real ) number ) (#) s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is summable & Sum (r : ( ( real ) ( ext-real V33() real ) number ) (#) s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ext-real V33() real ) Real) = r : ( ( real ) ( ext-real V33() real ) number ) * (Sum s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( ext-real V33() real ) Real) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ;

theorem :: SERIES_1:11
for s, s1 being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) = s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . 0 : ( ( ) ( empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) holds
Partial_Sums (s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ^\ 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) subsequence of b1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) = ((Partial_Sums s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ^\ 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) subsequence of Partial_Sums b1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) - s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SERIES_1:12
for s being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable holds
for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ^\ n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) subsequence of b1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) is summable ;

theorem :: SERIES_1:13
for s being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st ex n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) st s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ^\ n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) subsequence of b1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) is summable holds
s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable ;

theorem :: SERIES_1:14
for s1, s2 being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) <= s2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) holds
for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds (Partial_Sums s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) <= (Partial_Sums s2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ;

theorem :: SERIES_1:15
for s being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable holds
for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds Sum s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real V33() real ) Real) = ((Partial_Sums s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) + (Sum (s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ^\ (n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) subsequence of b1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) ) : ( ( ) ( ext-real V33() real ) Real) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ;

theorem :: SERIES_1:16
for s being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds 0 : ( ( ) ( empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) <= s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) holds
Partial_Sums s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is V41() ;

theorem :: SERIES_1:17
for s being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds 0 : ( ( ) ( empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) <= s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) holds
( Partial_Sums s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is bounded_above iff s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable ) ;

theorem :: SERIES_1:18
for s being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable & ( for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds 0 : ( ( ) ( empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) <= s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) holds
0 : ( ( ) ( empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) <= Sum s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real V33() real ) Real) ;

theorem :: SERIES_1:19
for s2, s1 being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds 0 : ( ( ) ( empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) <= s2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) & s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable & ex m being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) st
for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) st m : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds
s2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) <= s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) holds
s2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable ;

theorem :: SERIES_1:20
for s1, s2 being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds
( 0 : ( ( ) ( empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) <= s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) & s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) <= s2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ) & s2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable holds
( s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable & Sum s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real V33() real ) Real) <= Sum s2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real V33() real ) Real) ) ;

theorem :: SERIES_1:21
for s being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) holds
( s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable iff for r being ( ( real ) ( ext-real V33() real ) number ) st 0 : ( ( ) ( empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( real ) ( ext-real V33() real ) number ) holds
ex n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) st
for m being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds
abs (((Partial_Sums s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . m : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) - ((Partial_Sums s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) < r : ( ( real ) ( ext-real V33() real ) number ) ) ;

theorem :: SERIES_1:22
for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) )
for a being ( ( real ) ( ext-real V33() real ) number ) st a : ( ( real ) ( ext-real V33() real ) number ) <> 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds
(Partial_Sums (a : ( ( real ) ( ext-real V33() real ) number ) GeoSeq) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) = (1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) - (a : ( ( real ) ( ext-real V33() real ) number ) to_power (n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( real ) ( ext-real V33() real ) set ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) / (1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) - a : ( ( real ) ( ext-real V33() real ) number ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ;

theorem :: SERIES_1:23
for a being ( ( real ) ( ext-real V33() real ) number )
for s being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st a : ( ( real ) ( ext-real V33() real ) number ) <> 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) & ( for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . (n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) = a : ( ( real ) ( ext-real V33() real ) number ) * (s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) holds
for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds (Partial_Sums s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) = ((s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . 0 : ( ( ) ( empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) * (1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) - (a : ( ( real ) ( ext-real V33() real ) number ) to_power (n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( real ) ( ext-real V33() real ) set ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) / (1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) - a : ( ( real ) ( ext-real V33() real ) number ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ;

theorem :: SERIES_1:24
for a being ( ( real ) ( ext-real V33() real ) number ) st abs a : ( ( real ) ( ext-real V33() real ) number ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) < 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds
( a : ( ( real ) ( ext-real V33() real ) number ) GeoSeq : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is summable & Sum (a : ( ( real ) ( ext-real V33() real ) number ) GeoSeq) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ext-real V33() real ) Real) = 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) / (1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) - a : ( ( real ) ( ext-real V33() real ) number ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ;

theorem :: SERIES_1:25
for a being ( ( real ) ( ext-real V33() real ) number )
for s being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st abs a : ( ( real ) ( ext-real V33() real ) number ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) < 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) & ( for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . (n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) = a : ( ( real ) ( ext-real V33() real ) number ) * (s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) holds
( s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable & Sum s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real V33() real ) Real) = (s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . 0 : ( ( ) ( empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) / (1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) - a : ( ( real ) ( ext-real V33() real ) number ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ;

theorem :: SERIES_1:26
for s, s1 being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds
( s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) > 0 : ( ( ) ( empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) & s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) = (s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . (n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) / (s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ) & s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & lim s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) < 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds
s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable ;

theorem :: SERIES_1:27
for s being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) > 0 : ( ( ) ( empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) & ex m being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) st
for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) >= m : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds
(s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . (n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) / (s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) >= 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds
not s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable ;

theorem :: SERIES_1:28
for s, s1 being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds
( s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) >= 0 : ( ( ) ( empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) & s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) = n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) -root (s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ) & s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & lim s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) < 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds
s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable ;

theorem :: SERIES_1:29
for s, s1 being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds
( s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) >= 0 : ( ( ) ( empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) & s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) = n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) -root (s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ) & ex m being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) st
for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) st m : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds
s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) >= 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds
not s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable ;

theorem :: SERIES_1:30
for s, s1 being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds
( s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) >= 0 : ( ( ) ( empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) & s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) = n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) -root (s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ) & s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & lim s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) > 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds
not s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable ;

registration
let k, n be ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer ) Nat) ;
cluster k : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer ) set ) to_power n : ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer ) set ) : ( ( real ) ( ext-real V33() real ) set ) -> natural real ;
end;

definition
let k, n be ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer ) Nat) ;
:: original: to_power
redefine func k to_power n -> ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ;
end;

theorem :: SERIES_1:31
for s, s1 being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is V42() & ( for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds
( s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) >= 0 : ( ( ) ( empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) & s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) = (2 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) to_power n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) * (s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . (2 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) to_power n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ) holds
( s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable iff s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable ) ;

theorem :: SERIES_1:32
for p being ( ( real ) ( ext-real V33() real ) number )
for s being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st p : ( ( real ) ( ext-real V33() real ) number ) > 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) & ( for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds
s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) = 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) / (n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) to_power p : ( ( real ) ( ext-real V33() real ) number ) ) : ( ( real ) ( ext-real V33() real ) set ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) holds
s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable ;

theorem :: SERIES_1:33
for p being ( ( real ) ( ext-real V33() real ) number )
for s being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st p : ( ( real ) ( ext-real V33() real ) number ) <= 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) & ( for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds
s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) = 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) / (n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) to_power p : ( ( real ) ( ext-real V33() real ) number ) ) : ( ( real ) ( ext-real V33() real ) set ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) holds
not s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable ;

definition
let s be ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ;
attr s is absolutely_summable means :: SERIES_1:def 4
 abs s : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is summable ;
end;

theorem :: SERIES_1:34
for s being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence)
for n, m being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds
abs (((Partial_Sums s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . m : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) - ((Partial_Sums s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) <= abs (((Partial_Sums (abs s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . m : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) - ((Partial_Sums (abs s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ;

registration
cluster Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) absolutely_summable -> Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) summable for ( ( ) ( ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ;
end;

theorem :: SERIES_1:35
for s being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is absolutely_summable holds
s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable ;

theorem :: SERIES_1:36
for s being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds 0 : ( ( ) ( empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) <= s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) & s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable holds
s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is absolutely_summable ;

theorem :: SERIES_1:37
for s, s1 being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds
( s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) <> 0 : ( ( ) ( empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) & s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) = ((abs s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) / ((abs s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ) & s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & lim s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) < 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds
s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is absolutely_summable ;

theorem :: SERIES_1:38
for r being ( ( real ) ( ext-real V33() real ) number )
for s being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st r : ( ( real ) ( ext-real V33() real ) number ) > 0 : ( ( ) ( empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) & ex m being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) st
for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) >= m : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds
abs (s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) >= r : ( ( real ) ( ext-real V33() real ) number ) & s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent holds
lim s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) <> 0 : ( ( ) ( empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: SERIES_1:39
for s being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) <> 0 : ( ( ) ( empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) & ex m being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) st
for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) >= m : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds
((abs s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) / ((abs s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) >= 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds
not s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable ;

theorem :: SERIES_1:40
for s1, s being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) = n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) -root ((abs s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) & s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & lim s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) < 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds
s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is absolutely_summable ;

theorem :: SERIES_1:41
for s1, s being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) = n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) -root ((abs s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) & ex m being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) st
for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) st m : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds
s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) >= 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds
not s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable ;

theorem :: SERIES_1:42
for s1, s being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st ( for n being ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) = n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) -root ((abs s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) & s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & lim s1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) > 1 : ( ( ) ( non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer V61() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) holds
not s : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable ;

begin

definition
let s be ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ;
let n be ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer ) Nat) ;
func Sum (s,n) -> ( ( ) ( ext-real V33() real ) Real) equals :: SERIES_1:def 5
(Partial_Sums s : ( ( ) ( ) set ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ;
end;

definition
let s be ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) -valued Function-like non empty V14( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered add-closed ) Element of K19(REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ;
let n, m be ( ( natural ) ( ext-real non negative epsilon-transitive epsilon-connected ordinal natural V33() real integer ) Nat) ;
func Sum (s,n,m) -> ( ( ) ( ext-real V33() real ) Real) equals :: SERIES_1:def 6
(Sum (s : ( ( ) ( ) set ) ,n : ( ( non empty ) ( non empty ) set ) )) : ( ( ) ( ext-real V33() real ) Real) - (Sum (s : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) )) : ( ( ) ( ext-real V33() real ) Real) : ( ( ) ( ext-real V33() real ) Element of REAL : ( ( ) ( non empty V50() complex-membered ext-real-membered real-membered add-closed ) set ) ) ;
end;