:: COMSEQ_1 semantic presentation

begin

definition
mode Complex_Sequence is ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) sequence of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ;
end;

theorem :: COMSEQ_1:1
for f being ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) holds
( f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) is ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) iff ( dom f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) : ( ( ) ( ) set ) = NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . x : ( ( ) ( ) set ) : ( ( ) ( ) set ) is ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ) ) ;

theorem :: COMSEQ_1:2
for f being ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) holds
( f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) is ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) iff ( dom f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) : ( ( ) ( ) set ) = NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) & ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex V31() V32() V45() V46() V47() V48() V49() V50() V51() V52() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) holds f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex V31() V32() V45() V46() V47() V48() V49() V50() V51() V52() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) is ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ) ) ;

scheme :: COMSEQ_1:sch 1
ExComplexSeq{ F1( ( ( ) ( ) set ) ) -> ( ( complex ) ( complex ) number ) } :
ex seq being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) st
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex V31() V32() V45() V46() V47() V48() V49() V50() V51() V52() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) holds seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex V31() V32() V45() V46() V47() V48() V49() V50() V51() V52() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) = F1( ( ( complex ) ( complex ) number ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex V31() V32() V45() V46() V47() V48() V49() V50() V51() V52() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( complex ) ( complex ) number )
proof end;

notation
let f be ( ( Relation-like complex-valued ) ( Relation-like complex-valued ) Relation) ;
synonym non-zero f for non-empty ;
end;

registration
cluster Relation-like non-zero NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued for ( ( ) ( ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;
end;

theorem :: COMSEQ_1:3
for seq being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds
( seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) is non-zero iff for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) holds
seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) . x : ( ( ) ( ) set ) : ( ( ) ( complex ) set ) <> 0c : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V47() V48() V49() V50() V51() V52() V53() ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ;

theorem :: COMSEQ_1:4
for seq being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds
( seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) is non-zero iff for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex V31() V32() V45() V46() V47() V48() V49() V50() V51() V52() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) holds seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex V31() V32() V45() V46() V47() V48() V49() V50() V51() V52() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) <> 0c : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V47() V48() V49() V50() V51() V52() V53() ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ;

theorem :: COMSEQ_1:5
for IT being ( ( non-zero Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like non-zero NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds rng IT : ( ( non-zero Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like non-zero NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) : ( ( ) ( V47() ) Element of K19(COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) c= COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) \ {0c : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V47() V48() V49() V50() V51() V52() V53() ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) } : ( ( ) ( non zero V47() V48() V49() V50() V51() V52() ) set ) : ( ( ) ( V47() ) Element of K19(COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_1:6
for r being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ex seq being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) st rng seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) : ( ( ) ( V47() ) Element of K19(COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) = {r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) } : ( ( ) ( non zero V47() ) set ) ;

theorem :: COMSEQ_1:7
for seq1, seq2, seq3 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) + seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) + seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) + (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) + seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_1:8
for seq1, seq2, seq3 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) (#) seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_1:9
for seq1, seq2, seq3 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) + seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) (#) seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) + (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_1:10
for seq3, seq1, seq2 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) + seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = (seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) + (seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_1:11
for seq being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds - seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = (- 1r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_1:12
for r being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) )
for seq1, seq2 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = (r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_1:13
for r being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) )
for seq1, seq2 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) (r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_1:14
for seq1, seq2, seq3 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) - seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) (#) seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) - (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_1:15
for seq3, seq1, seq2 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds (seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) - (seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) - seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_1:16
for r being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) )
for seq1, seq2 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) + seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = (r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) + (r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_1:17
for r, p being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) )
for seq being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds (r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) * p : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) (p : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_1:18
for r being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) )
for seq1, seq2 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) - seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = (r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) - (r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_1:19
for r being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) )
for seq1, seq being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = (r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_1:20
for seq1, seq2, seq3 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) - (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) + seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) - seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) - seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_1:21
for seq being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds 1r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ;

theorem :: COMSEQ_1:22
for seq being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds - (- seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ;

theorem :: COMSEQ_1:23
for seq1, seq2 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) - (- seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) + seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_1:24
for seq1, seq2, seq3 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) - (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) - seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) - seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) + seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_1:25
for seq1, seq2, seq3 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) + (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) - seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) + seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) - seq3 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_1:26
for seq1, seq2 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds
( (- seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = - (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) & seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) (- seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = - (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: COMSEQ_1:27
for seq being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) st seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) is non-zero holds
seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) " : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) is non-zero ;

theorem :: COMSEQ_1:28
canceled;

theorem :: COMSEQ_1:29
for seq, seq1 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds
( ( seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) is non-zero & seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) is non-zero ) iff seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) is non-zero ) ;

theorem :: COMSEQ_1:30
for seq, seq1 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds (seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ") : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) (#) (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ") : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = (seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) " : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_1:31
for seq, seq1 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) st seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) is non-zero holds
(seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) (#) seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ;

theorem :: COMSEQ_1:32
for seq9, seq, seq19, seq1 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds (seq9 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) (#) (seq19 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = (seq9 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq19 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) /" (seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_1:33
for seq, seq1 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) st seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) is non-zero & seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) is non-zero holds
seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) is non-zero ;

theorem :: COMSEQ_1:34
for seq, seq1 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds (seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) " : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_1:35
for seq2, seq1, seq being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_1:36
for seq2, seq, seq1 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" (seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_1:37
for seq1, seq2, seq being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) st seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) is non-zero holds
seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = (seq2 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) /" (seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_1:38
for r being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) )
for seq being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) st r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) <> 0c : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex V47() V48() V49() V50() V51() V52() V53() ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) & seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) is non-zero holds
r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) is non-zero ;

theorem :: COMSEQ_1:39
for seq being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) st seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) is non-zero holds
- seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) is non-zero ;

theorem :: COMSEQ_1:40
for r being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) )
for seq being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds (r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) " : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = (r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ") : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) (seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ") : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_1:41
for seq being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) st seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) is non-zero holds
(- seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) " : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = (- 1r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) (seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ") : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_1:42
for seq, seq1 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) st seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) is non-zero holds
( - (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = (- seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) & seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" (- seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = - (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: COMSEQ_1:43
for seq1, seq, seq19 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds
( (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) + (seq19 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) + seq19 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) & (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) - (seq19 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) - seq19 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: COMSEQ_1:44
for seq, seq9, seq1, seq19 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) st seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) is non-zero & seq9 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) is non-zero holds
( (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) + (seq19 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq9 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = ((seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq9 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) + (seq19 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) /" (seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq9 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) & (seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) - (seq19 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq9 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = ((seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq9 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) - (seq19 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) /" (seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq9 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: COMSEQ_1:45
for seq19, seq, seq9, seq1 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds (seq19 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) /" (seq9 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) = (seq19 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq1 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) /" (seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq9 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_1:46
for seq, seq9 being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds |.(seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) (#) seq9 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) .| : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) = |.seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) .| : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) |.seq9 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) .| : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_1:47
for seq being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) st seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) is non-zero holds
|.seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) .| : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is non-zero ;

theorem :: COMSEQ_1:48
for seq being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds |.seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) .| : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) " : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) = |.(seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ") : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) .| : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_1:49
for seq9, seq being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds |.(seq9 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) /" seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) .| : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) = |.seq9 : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) .| : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) /" |.seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) .| : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMSEQ_1:50
for r being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) )
for seq being ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) holds |.(r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) (#) seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) : ( ( ) ( complex-valued ) set ) ) : ( ( ) ( ) set ) ) .| : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) = |.r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) .| : ( ( ) ( complex V31() V32() ) Element of REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) (#) |.seq : ( ( Function-like V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , COMPLEX : ( ( ) ( non zero V47() V53() V54() ) set ) ) complex-valued ) Complex_Sequence) .| : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) -valued Function-like non zero V14( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) V18( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K19(K20(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V47() V48() V49() V50() V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V47() V48() V49() V53() V54() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ;