:: SEQ_2 semantic presentation

begin

theorem :: SEQ_2:1
for g, r being ( ( real ) ( ext-real complex real ) number ) holds
( ( - g : ( ( real ) ( ext-real complex real ) number ) : ( ( complex ) ( ext-real complex real ) set ) < r : ( ( real ) ( ext-real complex real ) number ) & r : ( ( real ) ( ext-real complex real ) number ) < g : ( ( real ) ( ext-real complex real ) number ) ) iff abs r : ( ( real ) ( ext-real complex real ) number ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) < g : ( ( real ) ( ext-real complex real ) number ) ) ;

theorem :: SEQ_2:2
for g, r being ( ( real ) ( ext-real complex real ) number ) st g : ( ( real ) ( ext-real complex real ) number ) <> 0 : ( ( ) ( ext-real non positive non negative zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real V33() V34() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) & r : ( ( real ) ( ext-real complex real ) number ) <> 0 : ( ( ) ( ext-real non positive non negative zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real V33() V34() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) holds
abs ((g : ( ( real ) ( ext-real complex real ) number ) ") : ( ( complex ) ( ext-real complex real ) set ) - (r : ( ( real ) ( ext-real complex real ) number ) ") : ( ( complex ) ( ext-real complex real ) set ) ) : ( ( ) ( ext-real complex real ) set ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) = (abs (g : ( ( real ) ( ext-real complex real ) number ) - r : ( ( real ) ( ext-real complex real ) number ) ) : ( ( ) ( ext-real complex real ) set ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) / ((abs g : ( ( real ) ( ext-real complex real ) number ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) * (abs r : ( ( real ) ( ext-real complex real ) number ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ;

definition
let f be ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) Function) ;
attr f is bounded_above means :: SEQ_2:def 1
 ex r being ( ( real ) ( ext-real complex real ) number ) st
for y being ( ( ) ( ) set ) st y : ( ( ) ( ) set ) in dom f : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V45() V46() V47() V48() V49() V50() ) set ) holds
f : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) . y : ( ( ) ( ) set ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) < r : ( ( real ) ( ext-real complex real ) number ) ;
attr f is bounded_below means :: SEQ_2:def 2
 ex r being ( ( real ) ( ext-real complex real ) number ) st
for y being ( ( ) ( ) set ) st y : ( ( ) ( ) set ) in dom f : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V45() V46() V47() V48() V49() V50() ) set ) holds
r : ( ( real ) ( ext-real complex real ) number ) < f : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) . y : ( ( ) ( ) set ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ;
end;

definition
let seq be ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ;
redefine attr seq is bounded_above means :: SEQ_2:def 3
 ex r being ( ( real ) ( ext-real complex real ) number ) st
for n being ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) holds seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) < r : ( ( real ) ( ext-real complex real ) number ) ;
redefine attr seq is bounded_below means :: SEQ_2:def 4
 ex r being ( ( real ) ( ext-real complex real ) number ) st
for n being ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) holds r : ( ( real ) ( ext-real complex real ) number ) < seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ;
end;

registration
cluster Relation-like Function-like real-valued bounded -> Relation-like Function-like real-valued bounded_above bounded_below for ( ( ) ( ) set ) ;
cluster Relation-like Function-like real-valued bounded_above bounded_below -> Relation-like Function-like real-valued bounded for ( ( ) ( ) set ) ;
end;

theorem :: SEQ_2:3
for seq being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) holds
( seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is bounded iff ex r being ( ( real ) ( ext-real complex real ) number ) st
( 0 : ( ( ) ( ext-real non positive non negative zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real V33() V34() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( real ) ( ext-real complex real ) number ) & ( for n being ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) holds abs (seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) < r : ( ( real ) ( ext-real complex real ) number ) ) ) ) ;

theorem :: SEQ_2:4
for seq being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence)
for n being ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) ex r being ( ( real ) ( ext-real complex real ) number ) st
( 0 : ( ( ) ( ext-real non positive non negative zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real V33() V34() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( real ) ( ext-real complex real ) number ) & ( for m being ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) st m : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) holds
abs (seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . m : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) < r : ( ( real ) ( ext-real complex real ) number ) ) ) ;

definition
canceled;
let seq be ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ;
redefine attr seq is convergent means :: SEQ_2:def 6
 ex g being ( ( real ) ( ext-real complex real ) number ) st
for p being ( ( real ) ( ext-real complex real ) number ) st 0 : ( ( ) ( ext-real non positive non negative zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real V33() V34() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < p : ( ( real ) ( ext-real complex real ) number ) holds
ex n being ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) st
for m being ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) holds
abs ((seq : ( ( ) ( ) set ) . m : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) - g : ( ( real ) ( ext-real complex real ) number ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) < p : ( ( real ) ( ext-real complex real ) number ) ;
end;

definition
let seq be ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ;
assume seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent ;
func lim seq -> ( ( real ) ( ext-real complex real ) number ) means :: SEQ_2:def 7
 for p being ( ( real ) ( ext-real complex real ) number ) st 0 : ( ( ) ( ext-real non positive non negative zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real V33() V34() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) < p : ( ( real ) ( ext-real complex real ) number ) holds
ex n being ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) st
for m being ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) holds
abs ((seq : ( ( ) ( ) set ) . m : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) - it : ( ( non zero ) ( non zero ) set ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) < p : ( ( real ) ( ext-real complex real ) number ) ;
end;

definition
let f be ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) Function) ;
redefine attr f is bounded means :: SEQ_2:def 8
( f : ( ( ) ( ) set ) is bounded_above & f : ( ( ) ( ) set ) is bounded_below );
end;

definition
let seq be ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ;
:: original: lim
redefine func lim seq -> ( ( ) ( ext-real complex real ) Real) ;
end;

registration
cluster Function-like constant V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) -> Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) convergent for ( ( ) ( ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ;
end;

registration
cluster non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like constant V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued for ( ( ) ( ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ;
end;

theorem :: SEQ_2:5
for seq, seq9 being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & seq9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent holds
seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) + seq9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is convergent ;

registration
let seq1, seq2 be ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ;
cluster seq1 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued convergent ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) + seq2 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued convergent ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) complex-valued ext-real-valued real-valued ) set ) -> Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) convergent for ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ;
end;

theorem :: SEQ_2:6
for seq, seq9 being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & seq9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent holds
lim (seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) + seq9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ext-real complex real ) Real) = (lim seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( ext-real complex real ) Real) + (lim seq9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( ext-real complex real ) Real) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ;

theorem :: SEQ_2:7
for r being ( ( real ) ( ext-real complex real ) number )
for seq being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent holds
r : ( ( real ) ( ext-real complex real ) number ) (#) seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is convergent ;

registration
let r be ( ( real ) ( ext-real complex real ) number ) ;
let seq be ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ;
cluster r : ( ( real ) ( ext-real complex real ) set ) (#) seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued convergent ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) complex-valued ext-real-valued real-valued ) set ) -> Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) convergent for ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ;
end;

theorem :: SEQ_2:8
for r being ( ( real ) ( ext-real complex real ) number )
for seq being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent holds
lim (r : ( ( real ) ( ext-real complex real ) number ) (#) seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ext-real complex real ) Real) = r : ( ( real ) ( ext-real complex real ) number ) * (lim seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( ext-real complex real ) Real) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ;

theorem :: SEQ_2:9
for seq being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent holds
- seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is convergent ;

registration
let seq be ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ;
cluster - seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued convergent ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) complex-valued ext-real-valued real-valued ) set ) -> Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) convergent for ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ;
end;

theorem :: SEQ_2:10
for seq being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent holds
lim (- seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ext-real complex real ) Real) = - (lim seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( ext-real complex real ) Real) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ;

theorem :: SEQ_2:11
for seq, seq9 being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & seq9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent holds
seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) - seq9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is convergent ;

registration
let seq1, seq2 be ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued convergent ) Real_Sequence) ;
cluster seq1 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued convergent ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - seq2 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued convergent ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) complex-valued ext-real-valued real-valued ) set ) -> Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) convergent for ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ;
end;

theorem :: SEQ_2:12
for seq, seq9 being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & seq9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent holds
lim (seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) - seq9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ext-real complex real ) Real) = (lim seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( ext-real complex real ) Real) - (lim seq9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( ext-real complex real ) Real) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ;

theorem :: SEQ_2:13
for seq being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent holds
seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is bounded ;

registration
cluster Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) convergent -> Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) bounded for ( ( ) ( ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ;
end;

theorem :: SEQ_2:14
for seq, seq9 being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & seq9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent holds
seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is convergent ;

registration
let seq1, seq2 be ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued bounded convergent bounded_above bounded_below ) Real_Sequence) ;
cluster seq1 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued bounded convergent bounded_above bounded_below ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) seq2 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued bounded convergent bounded_above bounded_below ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) complex-valued ext-real-valued real-valued ) set ) -> Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) convergent for ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ;
end;

theorem :: SEQ_2:15
for seq, seq9 being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & seq9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent holds
lim (seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ext-real complex real ) Real) = (lim seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( ext-real complex real ) Real) * (lim seq9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( ext-real complex real ) Real) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ;

theorem :: SEQ_2:16
for seq being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & lim seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real complex real ) Real) <> 0 : ( ( ) ( ext-real non positive non negative zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real V33() V34() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) holds
ex n being ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) st
for m being ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(abs (lim seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( ext-real complex real ) Real) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) / 2 : ( ( ) ( ext-real positive non negative non zero epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) < abs (seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . m : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ;

theorem :: SEQ_2:17
for seq being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & ( for n being ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) holds 0 : ( ( ) ( ext-real non positive non negative zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real V33() V34() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) <= seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) holds
0 : ( ( ) ( ext-real non positive non negative zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real V33() V34() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) <= lim seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real complex real ) Real) ;

theorem :: SEQ_2:18
for seq, seq9 being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & seq9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & ( for n being ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) holds seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) <= seq9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) holds
lim seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real complex real ) Real) <= lim seq9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real complex real ) Real) ;

theorem :: SEQ_2:19
for seq, seq9, seq1 being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & seq9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & ( for n being ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) <= seq1 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) & seq1 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) <= seq9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ) & lim seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real complex real ) Real) = lim seq9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real complex real ) Real) holds
seq1 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent ;

theorem :: SEQ_2:20
for seq, seq9, seq1 being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & seq9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & ( for n being ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) <= seq1 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) & seq1 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) <= seq9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural complex real V33() V34() V45() V46() V47() V48() V49() V50() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ) & lim seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real complex real ) Real) = lim seq9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real complex real ) Real) holds
lim seq1 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real complex real ) Real) = lim seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real complex real ) Real) ;

theorem :: SEQ_2:21
for seq being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & lim seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real complex real ) Real) <> 0 : ( ( ) ( ext-real non positive non negative zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real V33() V34() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) & seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is non-zero holds
seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) " : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is convergent ;

theorem :: SEQ_2:22
for seq being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & lim seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real complex real ) Real) <> 0 : ( ( ) ( ext-real non positive non negative zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real V33() V34() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) & seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is non-zero holds
lim (seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ") : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ext-real complex real ) Real) = (lim seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( ext-real complex real ) Real) " : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ;

theorem :: SEQ_2:23
for seq9, seq being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & lim seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real complex real ) Real) <> 0 : ( ( ) ( ext-real non positive non negative zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real V33() V34() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) & seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is non-zero holds
seq9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) /" seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is convergent ;

theorem :: SEQ_2:24
for seq9, seq being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & lim seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real complex real ) Real) <> 0 : ( ( ) ( ext-real non positive non negative zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real V33() V34() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) & seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is non-zero holds
lim (seq9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) /" seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ext-real complex real ) Real) = (lim seq9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( ext-real complex real ) Real) / (lim seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( ) ( ext-real complex real ) Real) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ;

theorem :: SEQ_2:25
for seq, seq1 being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & seq1 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is bounded & lim seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real complex real ) Real) = 0 : ( ( ) ( ext-real non positive non negative zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real V33() V34() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) holds
seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq1 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is convergent ;

theorem :: SEQ_2:26
for seq, seq1 being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & seq1 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is bounded & lim seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( ext-real complex real ) Real) = 0 : ( ( ) ( ext-real non positive non negative zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real V33() V34() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) holds
lim (seq : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) (#) seq1 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ) : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ext-real complex real ) Real) = 0 : ( ( ) ( ext-real non positive non negative zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real V33() V34() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) ;

registration
let s be ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) ;
cluster |.s : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued bounded convergent ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) .| : ( ( Relation-like Function-like real-valued ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) complex-valued ext-real-valued real-valued ) set ) -> Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) convergent for ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ;
end;

theorem :: SEQ_2:27
for s being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Complex_Sequence) st s : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Complex_Sequence) is convergent holds
lim |.s : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Complex_Sequence) .| : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ext-real complex real ) Real) = |.(lim s : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) .| : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ;

theorem :: SEQ_2:28
for s, s9 being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) holds lim |.(s : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) + s9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) ) : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued bounded convergent ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) .| : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued bounded convergent bounded_above bounded_below ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ext-real complex real ) Real) = |.((lim s : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) + (lim s9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) .| : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ;

theorem :: SEQ_2:29
for r being ( ( real ) ( ext-real complex real ) number )
for s being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) holds lim |.(r : ( ( real ) ( ext-real complex real ) number ) (#) s : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) ) : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued bounded convergent ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) .| : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued bounded convergent bounded_above bounded_below ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ext-real complex real ) Real) = |.r : ( ( real ) ( ext-real complex real ) number ) .| : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) * |.(lim s : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) .| : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ;

theorem :: SEQ_2:30
for s being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) holds lim |.(- s : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) ) : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued bounded convergent ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) .| : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued bounded convergent bounded_above bounded_below ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ext-real complex real ) Real) = |.(lim s : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) .| : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ;

theorem :: SEQ_2:31
for s, s9 being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) holds lim |.(s : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) - s9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) ) : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued bounded convergent ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) .| : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued bounded convergent bounded_above bounded_below ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ext-real complex real ) Real) = |.((lim s : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) - (lim s9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) .| : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ;

theorem :: SEQ_2:32
for s, s9 being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) holds lim |.(s : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) (#) s9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) ) : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued bounded convergent ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) .| : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued bounded convergent bounded_above bounded_below ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ext-real complex real ) Real) = |.(lim s : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) .| : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) * |.(lim s9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) convergent ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued bounded convergent ) Complex_Sequence) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) .| : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ;

theorem :: SEQ_2:33
for s being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Complex_Sequence) st s : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Complex_Sequence) is convergent & lim s : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Complex_Sequence) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) <> 0c : ( ( ) ( ext-real non positive non negative zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real V45() V46() V47() V48() V49() V50() V51() ) Element of COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) & s : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Complex_Sequence) is non-zero holds
lim |.(s : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Complex_Sequence) ") : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) .| : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ext-real complex real ) Real) = |.(lim s : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) .| : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) " : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ;

theorem :: SEQ_2:34
for s9, s being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Complex_Sequence) st s9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Complex_Sequence) is convergent & s : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Complex_Sequence) is convergent & lim s : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Complex_Sequence) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) <> 0c : ( ( ) ( ext-real non positive non negative zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real V45() V46() V47() V48() V49() V50() V51() ) Element of COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) & s : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Complex_Sequence) is non-zero holds
lim |.(s9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Complex_Sequence) /" s : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) .| : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ext-real complex real ) Real) = |.(lim s9 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) .| : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) / |.(lim s : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) .| : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) ;

theorem :: SEQ_2:35
for s, s1 being ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Complex_Sequence) st s : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Complex_Sequence) is convergent & s1 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Complex_Sequence) is bounded & lim s : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Complex_Sequence) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) = 0c : ( ( ) ( ext-real non positive non negative zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real V45() V46() V47() V48() V49() V50() V51() ) Element of COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) holds
lim |.(s : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Complex_Sequence) (#) s1 : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) complex-valued ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V45() V51() V52() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) .| : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) complex-valued ext-real-valued real-valued bounded convergent bounded_above bounded_below ) Element of K11(K12(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ext-real complex real ) Real) = 0 : ( ( ) ( ext-real non positive non negative zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real V33() V34() V45() V46() V47() V48() V49() V50() V51() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V45() V46() V47() V48() V49() V50() V51() ) Element of K11(REAL : ( ( ) ( non zero V45() V46() V47() V51() V52() ) set ) ) : ( ( ) ( ) set ) ) ) ;